Recent zbMATH articles in MSC 17 https://zbmath.org/atom/cc/17 2021-11-25T18:46:10.358925Z Werkzeug On the isomorphisms between evolution algebras of graphs and random walks https://zbmath.org/1472.05134 2021-11-25T18:46:10.358925Z "Cadavid, Paula" https://zbmath.org/authors/?q=ai:cadavid.paula "Rodiño Montoya, Mary Luz" https://zbmath.org/authors/?q=ai:rodino-montoya.mary-luz "Rodriguez, Pablo M." https://zbmath.org/authors/?q=ai:rodriguez.pablo-m Summary: Evolution algebras are non-associative algebras inspired from biological phenomena, with applications to or connections with different mathematical fields. There are two natural ways to define an evolution algebra associated to a given graph. While one takes into account only the adjacencies of the graph, the other includes probabilities related to the symmetric random walk on the same graph. In this work we state new properties related to the relation between these algebras, which is one of the open problems in the interplay between evolution algebras and graphs. On the one hand, we show that for any graph both algebras are strongly isotopic. On the other hand, we provide conditions under which these algebras are or are not isomorphic. For the case of finite non-singular graphs we provide a complete description of the problem, while for the case of finite singular graphs we state a conjecture supported by examples and partial results. The case of graphs with an infinite number of vertices is also discussed. As a sideline of our work, we revisit a result existing in the literature about the identification of the automorphism group of an evolution algebra, and we give an improved version of it. Crystal structures for symmetric Grothendieck polynomials https://zbmath.org/1472.05152 2021-11-25T18:46:10.358925Z "Monical, Cara" https://zbmath.org/authors/?q=ai:monical.cara "Pechenik, Oliver" https://zbmath.org/authors/?q=ai:pechenik.oliver "Scrimshaw, Travis" https://zbmath.org/authors/?q=ai:scrimshaw.travis Summary: The symmetric Grothendieck polynomials representing Schubert classes in the K theory of Grassmannians are generating functions for semistandard set-valued tableaux. We construct a type $$A_n$$ crystal structure on these tableaux. This crystal yields a new combinatorial formula for decomposing symmetric Grothendieck polynomials into Schur polynomials. For single-columns and single-rows, we give a new combinatorial interpretation of Lascoux polynomials (K-analogs of Demazure characters) by constructing a K-theoretic analog of crystals with an appropriate analog of a Demazure crystal. We relate our crystal structure to combinatorial models using excited Young diagrams, Gelfand-Tsetlin patterns via the 5-vertex model, and biwords via Hecke insertion to compute symmetric Grothendieck polynomials. Alcove paths and Gelfand-Tsetlin patterns https://zbmath.org/1472.05154 2021-11-25T18:46:10.358925Z "Watanabe, Hideya" https://zbmath.org/authors/?q=ai:watanabe.hideya "Yamamura, Keita" https://zbmath.org/authors/?q=ai:yamamura.keita Summary: In their study of the equivariant K-theory of the generalized flag varieties $$G/P$$, where $$G$$ is a complex semisimple Lie group, and $$P$$ is a parabolic subgroup of $$G$$, \textit{C. Lenart} and \textit{A. Postnikov} [Trans. Am. Math. Soc. 360, No. 8, 4349--4381 (2008; Zbl 1211.17021)] introduced a combinatorial tool, called the alcove path model. It provides a model for the highest weight crystals with dominant integral highest weights, generalizing the model by semistandard Young tableaux. In this paper, we prove a simple and explicit formula describing the crystal isomorphism between the alcove path model and the Gelfand-Tsetlin pattern model for type $$A$$. Abelian doppelsemigroups https://zbmath.org/1472.08008 2021-11-25T18:46:10.358925Z "Zhuchok, Anatolii V." https://zbmath.org/authors/?q=ai:zhuchok.anatolii-v "Knauer, Kolja" https://zbmath.org/authors/?q=ai:knauer.kolja-b Summary: A doppelsemigroup is an algebraic system consisting of a set with two binary associative operations satisfying certain identities. Doppelsemigroups are a generalization of semigroups and they have relationships with such algebraic structures as doppelalgebras, duplexes, interassociative semigroups, restrictive bisemigroups, dimonoids and trioids. This paper is devoted to the study of abelian doppelsemigroups. We show that every abelian doppelsemigroup can be constructed from a left and right commutative semigroup and describe the free abelian doppelsemigroup. We also characterize the least abelian congruence on the free doppelsemigroup, give examples of abelian doppelsemigroups and find conditions under which the operations of an abelian doppelsemigroup coincide. On Weierstrass mock modular forms and a dimension formula for certain vertex operator algebras https://zbmath.org/1472.11114 2021-11-25T18:46:10.358925Z "Beneish, Lea" https://zbmath.org/authors/?q=ai:beneish.lea "Mertens, Michael H." https://zbmath.org/authors/?q=ai:mertens.michael-h A dimension formula concerning weight spaces of a holomorphic strongly rational vertex operator algebra (VOA) with central charge $$24$$ and its twisted modules was developed by \textit{J. van Ekeren} et al. [Int. Math. Res. Not. 2020, No. 7, 2145--2204 (2020; Zbl 07195654)]. The useful formula was established for twisted modules of an automorphisms with certain order, and in particular, an order $$N$$ which essentially allowed for the character of the fixed-point subVOA to be written in terms of the Hauptmodul for the group $$\Gamma_0(N)$$. The paper under review considers the cases of other finite-order automorphisms that correspond to levels $$N$$ where there is no Hauptmodul, but where the modular curve $$X_0(N)$$ has genus $$1$$. In this case, the authors essentially express the character of the fixed-point subVOA in terms of Weierstrass mock modular forms, a Fricke involution, and Hecke operators. The resulting dimension formula found in Theorem 1.1 includes a constant $$C_E$$ that contains the term $$\widehat{\zeta}(\Lambda_E ;L(E,1))$$, where $$E$$ is an elliptic curve of conductor $$N$$ isomorphic to the torus $$\mathbb{C}/\Lambda_E$$ for a full lattice $$\Lambda_E\subset \mathbb{C}$$, and $$\widehat{\zeta}(\Lambda_E ;z)$$ is the associated completed Weierstrass zeta function. While a more general dimension VOA dimension formula independent of the order of the automorphism has since been obtained by \textit{S. Möller} and \textit{N. R. Scheithauer} in [Dimension formulae and generalised deep holes of the Leech lattice vertex operator algebra'', Preprint, \url{arXiv:1910.04947}], the paper under review utilizes its own results to provide criteria for when the value $$\widehat{\zeta}(\Lambda_E ;L(E,1))$$ is rational (see Corollary 1.2 for details). In addition to the developments described above, a dimension formula for automorphisms of prime order is also obtained in Theorem 1.3 utilizing the so-called Bruinier-Funke pairing and incorporating newforms. The core results of the paper rely of the authors expressing a harmonic Maaßform of weight $$0$$ for $$\Gamma_0(N)$$ as a linear combination of images of the completed Weierstrass mock modular form associated with the $$\Gamma_0(N)$$-optimal elliptic curve $$E$$. More explanation on this is found in Theorem 1.4 and in Sections 2.3 and 2.4. The proofs of all results are found in Section 3. The abelian part of a compatible system and $$\ell$$-independence of the Tate conjecture https://zbmath.org/1472.11168 2021-11-25T18:46:10.358925Z "Hui, Chun Yin" https://zbmath.org/authors/?q=ai:hui.chun-yin Summary: Let $$K$$ be a number field and $$\{V_\ell \}_\ell$$ a rational strictly compatible system of semisimple Galois representations of $$K$$ arising from geometry. Let $$\mathbf{G}_\ell$$ and $$V_\ell^{{\text{ab}}}$$ be respectively the algebraic monodromy group and the maximal abelian subrepresentation of $$V_\ell$$ for all $$\ell$$. We prove that the system $$\{V_\ell^{{\text{ab}}}\}_\ell$$ is also a rational strictly compatible system under some group theoretic conditions, e.g., when $$\mathbf{G}_{\ell '}$$ is connected and satisfies \textit{Hypothesis A} for some prime $$\ell '$$. As an application, we prove that the Tate conjecture for abelian variety $$X/K$$ is independent of $$\ell$$ if the algebraic monodromy groups of the Galois representations of $$X$$ satisfy the required conditions. Ramification filtration via deformations https://zbmath.org/1472.11289 2021-11-25T18:46:10.358925Z "Abrashkin, Victor A." https://zbmath.org/authors/?q=ai:abrashkin.victor-a Let $$k$$ be the finite field with $$p^{N_0}$$ elements and let $$\mathcal{K}$$ be the field of formal Laurent series over $$k$$. Thus $$\mathcal{K}$$ is a local field of characteristic $$p$$ with residue field $$k$$. Let $$\mathcal{K}_{<p}$$ be the maximum Galois extension of $$\mathcal{K}$$ whose Galois group has exponent $$p$$ and is nilpotent of class $$<p$$. Set $$\mathcal{G}_{<p} =\text{Gal}(\mathcal{K}_{<p}/\mathcal{K})$$. In [Ann. Sci. Éc. Norm. Supér. (3) 71, 101--190 (1954; Zbl 0055.25103)] \textit{M. Lazard} proved that there is a Lie algebra $$\mathcal{L}$$ over $$\mathbb{F}_p$$ such that $$\mathcal{G}_{<p}\cong G(\mathcal{L})$$, where $$G(\mathcal{L})$$ is a group with same elements as $$\mathcal{L}$$, but whose operation is defined using the Baker-Campbell-Hausdorff formula. For $$v\ge1$$ let $$\mathcal{G}_{<p}^{(v)}$$ denote the $$v$$th ramification subgroup of $$\mathcal{G}_{<p}$$ with respect to the upper numbering. Then there is an ideal $$\mathcal{L}^{(v)}$$ in $$\mathcal{L}$$ which is mapped onto $$\mathcal{G}_{<p}^{(v)}$$ by the isomorphism $$G(\mathcal{L})\cong\mathcal{G}_{<p}$$. In [Transl., Ser. 2, Am. Math. Soc. 166, 35--100 (1995; Zbl 0873.11063)], the author gave a description of the ideal $$\mathcal{L}^{(v)}$$. This description was not entirely satisfactory, however, in that it was nonlinear, i.e., it depended on the universal enveloping algebra of $$\mathcal{L}$$. In the present paper, the author gives a new description of $$\mathcal{L}^{(v)}$$ strictly in terms of Lie algebras. The proof relies on an auxiliary Lie algebra $$\overline{\mathcal{L}}^{\dagger}$$ and an onto Lie algebra homomorphism $$\overline{\eta}^{\dagger}: \mathcal{L}\rightarrow\overline{\mathcal{L}}^{\dagger}$$, both of which depend on $$v$$. Using deformation theory, the author constructs an ideal $$\overline{\mathcal{L}}^{\dagger}[v]$$ in $$\overline{\mathcal{L}}^{\dagger}$$ such that $$\mathcal{L}^{(v)}=(\overline{\eta}^{\dagger})^{-1} (\overline{\mathcal{L}}^{\dagger}[v])$$. This allows one to find explicit generators for $$\mathcal{L}^{(v)}$$. Quantum Grothendieck rings as quantum cluster algebras https://zbmath.org/1472.13039 2021-11-25T18:46:10.358925Z "Bittmann, Léa" https://zbmath.org/authors/?q=ai:bittmann.lea \textit{D. Hernandez} and \textit{B. Leclerc} [Duke Math. J. 154, No. 2, 265--341 (2010; Zbl 1284.17010)] first realized that the Grothendieck ring of a certain monoidal subcategory $$\mathcal{C}_1$$ of the category $$\mathcal{C}$$ of finite-dimensional $$U_q(L\mathfrak{g})$$-modules had the structure of a cluster algebra. They thus proved that the Grothendieck ring of a certain monoidal subcategory $$\mathcal{O}^+_\mathbb{Z}$$ of the category $$\mathcal{O}$$ had a cluster algebra structure of infinite rank, for which one can take as initial seed the classes of the positive prefundamental representations. That is, the category $$\mathcal{O}^+_{\mathbb{Z}}$$ contains the finite-dimensional representations and the positive prefundamental representations whose spectral parameter satisfy an integrality condition. Moreover, certain exchange relations, such as the Baxter relation, coming from cluster mutations appear naturally. In order to construct of quantum Grothendieck ring for the category $$\mathcal{O}$$ of representations of the quantum loop algebra introduced by \textit{D. Hernandez} and \textit{M. Jimbo} [Compos. Math. 148, No. 5, 1593--1623 (2012; Zbl 1266.17010)], previous approaches were no longer applicable. The geometrical approach of \textit{H. Nakajima} [Ann. Math. (2) 160, No. 3, 1057--1097 (2005; Zbl 1140.17015)] and \textit{M. Varagnolo} and \textit{E. Vasserot} [Prog. Math. 210, 345--365 (2003; Zbl 1162.17307)] (in which the $$t$$-graduation naturally comes from the graduation of cohomological complexes) requires a geometric interpretation of the objects in the category $$\mathcal{O}$$, which is yet to be found. The more algebraic approach consisting of realizing the (quantum) Grothendieck ring as an invariant under a sort of Weyl symmetry, which allowed \textit{D. Hernandez} [Adv. Math. 187, No. 1, 1--52 (2004; Zbl 1098.17009)] to define a quantum Grothendieck ring of finite-dimensional representations in non-simply laced types, is again no longer relevant for the category $$\mathcal{O}$$. However, only the cluster algebra approach yields results in this context. The author constructs a quantum Grothendieck ring for a certain monoidal subcategory of the category $$\mathcal{O}$$ (Theorem 5.2.1, page 180). She uses the cluster algebra structure of the Grothendieck ring of this category to define the quantum Grothendieck ring as a quantum cluster algebra. When the underlying simple Lie algebra is of type $$A$$, she proves that this quantum Grothendieck ring contains the quantum Grothendieck ring of the category of finite-dimensional representations of the associated quantum affine algebra (Theorem 8.1.1, page 193). Hurwitz theory of elliptic orbifolds. I https://zbmath.org/1472.14034 2021-11-25T18:46:10.358925Z "Engel, Philip" https://zbmath.org/authors/?q=ai:engel.philip-milton Summary: An \textit{elliptic orbifold} is the quotient of an elliptic curve by a finite group. In [Invent. Math. 145, No. 1, 59--103 (2001; Zbl 1019.32014)], \textit{A. Eskin} and \textit{A. Okounkov} proved that generating functions for the number of branched covers of an elliptic curve with specified ramification are quasimodular forms for $$\operatorname{SL}2_(\mathbb{Z})$$. In [Prog. Math. 253, 1--25 (2006; Zbl 1136.14039)], they generalized this theorem to branched covers of the quotient of an elliptic curve by $$\pm 1$$, proving quasimodularity for $$\Gamma_0(2)$$. We generalize their work to the quotient of an elliptic curve by $$\langle\zeta N\rangle$$ for $$N=3, 4, 6$$, proving quasimodularity for $$\Gamma(N)$$, and extend their work in the case $$N=2$$. It follows that certain generating functions of hexagon, square and triangle tilings of compact surfaces are quasimodular forms. These tilings enumerate lattice points in moduli spaces of flat surfaces. We analyze the asymptotics as the number of tiles goes to infinity, providing an algorithm to compute the Masur-Veech volumes of strata of cubic, quartic, and sextic differentials. We conclude a generalization of the Kontsevich-Zorich conjecture: these volumes are polynomial in $$\pi$$. On Jordan-Clifford algebras, three fermion generations with Higgs fields and a $$\mathrm{SU}(3) \times \mathrm{SU}(2)_L \times \mathrm{SU}(2)_R \times \mathrm{U}(1)$$ model https://zbmath.org/1472.15033 2021-11-25T18:46:10.358925Z "Castro Perelman, Carlos" https://zbmath.org/authors/?q=ai:castro-perelman.carlos Summary: Previously we have shown that the algebra $J_3 [{\mathbb{C}}\otimes{\mathbb{O}}] \otimes C \ell (4,{\mathbb{C}}),$ given by the tensor product of the complex exceptional Jordan $$J_3 [\mathbb{C}\otimes\mathbb{O}]$$ and the complex Clifford algebra $$C \ell (4,\mathbb{C})$$, can describe all of the spinorial degrees of freedom of three generations of fermions in four-space-time dimensions. We extend our construction to show that it also includes the degrees of freedom of three sets of pairs of complex scalar Higgs-doublets $$\{\mathbf{H}^{(m)}_L, \mathbf{H}^{(m)}_R\}$$; $$m = 1,2,3$$, and their $$\mathrm{CPT}$$ conjugates. Furthermore, a close inspection of the fermion structure of each generation reveals that it fits naturally with the sixteen complex-dimensional representation of the internal left/right symmetric gauge group $$G_{LR} = \mathrm{SU}(3)_C \times \mathrm{SU}(2)_L \times \mathrm{SU}(2)_R \times \mathrm{U}(1)$$. It is reviewed how the latter group emerges from the intersection of $$\mathrm{SO}(10)$$ and $$\mathrm{SU}(3) \times \mathrm{SU}(3) \times \mathrm{SU}(3)$$ in $$E_6$$. In the concluding remarks we briefly discuss the role that the extra Higgs fields may have as dark matter candidates; the construction of Chern-Simons-like matrix cubic actions; hexaquarks; supersymmetry and Clifford bundles over the complex-octonionic projective plane $$(\mathbb{C}\otimes\mathbb{O}) \mathbb{P}^2$$ whose isometry group is $$E_6$$. Hopf algebras and tensor categories. International workshop, Nanjing University, Nanjing, China, September 9--13, 2019 https://zbmath.org/1472.16001 2021-11-25T18:46:10.358925Z "Andruskiewitsch, Nicolás" https://zbmath.org/authors/?q=ai:andruskiewitsch.nicolas "Liu, Gongxiang" https://zbmath.org/authors/?q=ai:liu.gongxiang "Montgomery, Susan" https://zbmath.org/authors/?q=ai:montgomery.susan "Zhang, Yinhuo" https://zbmath.org/authors/?q=ai:zhang.yinhuo Publisher's description: Articles in this volume are based on talks given at the International Workshop on Hopf Algebras and Tensor Categories, held from September 9--13, 2019, at Nanjing University, Nanjing, China. The articles highlight the latest advances and further research directions in a variety of subjects related to tensor categories and Hopf algebras. Primary topics discussed in the text include the classification of Hopf algebras, structures and actions of Hopf algebras, algebraic supergroups, representations of quantum groups, quasi-quantum groups, algebras in tensor categories, and the construction method of fusion categories. The articles of this volume will be reviewed individually. Ringel-Hall algebras beyond their quantum groups. I: Restriction functor and Green formula https://zbmath.org/1472.16014 2021-11-25T18:46:10.358925Z "Xiao, Jie" https://zbmath.org/authors/?q=ai:xiao.jie "Xu, Fan" https://zbmath.org/authors/?q=ai:xu.fan "Zhao, Minghui" https://zbmath.org/authors/?q=ai:zhao.minghui In the paper under review, the authors generalize the categorical construction of a quantum group and its canonical basis introduced by Lusztig to the generic form of the whole Ringel-Hall algebra. They clarify the explicit relation between the Green formula and the restriction functor. By a geometric way to prove the Green formula, they show that the compatibility of multiplication and comultiplication of a Ringel-Hall algebra can be categorified under Lusztig's framework. Bounds of Gelfand-Tsetlin multiplicities and tableaux realizations of Verma modules https://zbmath.org/1472.16016 2021-11-25T18:46:10.358925Z "Futorny, Vyacheslav" https://zbmath.org/authors/?q=ai:futorny.vyacheslav-m "Grantcharov, Dimitar" https://zbmath.org/authors/?q=ai:grantcharov.dimitar "Ramirez, Luis Enrique" https://zbmath.org/authors/?q=ai:ramirez.luis-enrique "Zadunaisky, Pablo" https://zbmath.org/authors/?q=ai:zadunaisky.pablo Summary: We introduce the notion of essential support of a simple Gelfand-Tsetlin $$\mathfrak{gl}_n$$-module as an attempt to understand the character formula of such module. This support detects the weights having maximal possible Gelfand-Tsetlin multiplicities. Using combinatorial tools we describe the essential supports of the simple socles of the universal tableaux modules. We also prove that every simple Verma module appears as the socle of a universal tableaux module. As a consequence, we prove the Strong Futorny-Ovsienko Conjecture on the sharpness of the upper bounds of the Gelfand-Tsetlin multiplicities. We also give a very explicit description of the support and essential support of the simple singular Verma module $$M(- \rho)$$. Brauer-Clifford group of $$(S, \mathcal G, H)$$-Azumaya comodule algebras https://zbmath.org/1472.16019 2021-11-25T18:46:10.358925Z "Guédénon, T." https://zbmath.org/authors/?q=ai:guedenon.thomas The Brauer-Clifford group of $$(G,Z)$$-central simple algebras for $$G$$ a finite group and $$Z$$ a commutative central simple $$G$$-algebra was first introduced by \textit{A. Turull} in [J. Algebra 321, No. 12, 3620--3642 (2009; Zbl 1186.20011)]. This notion has since been extended to various cases, including, but not limited to: \begin{itemize} \item $$G$$-algebras over commutative rings [\textit{A. Herman} and \textit{D. Mitra}, Commun. Algebra 39, No. 10, 3905--3915 (2011; Zbl 1246.16017)], \item $$(S,H)$$-Azumaya algebras for $$H$$ a cocommutative Hopf algebra and $$S$$ a commutative $$H$$-module algebra [the author and \textit{A. Herman}, Algebr. Represent. Theory 16, No. 1, 101--127 (2013; Zbl 1270.16015)], and \item $$(S,\mathcal{G},H)$$-Azumaya algebras with $$H$$ a cocommutative Hopf algebra, $$\mathcal{G}$$ a Lie algebra in the symmetric monoidal category of left $$H$$-modules, and $$S$$ a commutative algebra which is an $$H$$-module algebra [the author, Adv. Appl. Clifford Algebr. 30, No. 3, Paper No. 34, 24 p. (2020; Zbl 1454.16026)]. \end{itemize} In the paper under review, the Brauer-Clifford group of $$(S,\mathcal{G},H)$$-comodule algebras is given, where $$H$$ is a commutative Hopf algebra, $$\mathcal{G}$$ is a Lie algebra in the symmetric monoidal category of right $$H$$-comodules, and $$S$$ is a commutative algebra that is an $$H$$-comodule algebra. A construction of the group and some general properties are presented. The construction is analogous to that of the Brauer-Clifford group considered in [the author, loc. cit.]. Some finitely generated associative algebras with a Lie nilpotency identity https://zbmath.org/1472.16022 2021-11-25T18:46:10.358925Z "Glizburg, Vita" https://zbmath.org/authors/?q=ai:glizburg.vita-i "Pchelintsev, Sergey" https://zbmath.org/authors/?q=ai:pchelintsev.sergei-valentinovich A unital associative algebra is said to be Lie nilpotent if it satisfies the identity $$[..[[x_1,x_2],x_3],...,x_n]=0$$, and strongly Lie nilpotent if it satisfies $$[[..[[x_0,x_1]y_1,x_2]y_3,...,x_{n-1}]y_{n-1},x_n]=0$$ for some $$n$$. \par The paper under review deals with several results related to (strongly) Lie nilpotency of certain finitely generated unital associative algebras. In particular, it is proved that the (unital associative) multiplication algebra of the free commutative alternative algebra on $$r\geq 3$$ generators over a field of characteristic $$3$$ is strongly Lie nilpotent of class $$2(r-1)$$. (Note that if the characteristic is not $$3$$, any commutative alternative algebra is associative.) \par Also, let $$T^{(n)}$$ be the $$T$$-ideal of the free unital associative algebra $$F_r$$ on $$r$$ generators, over a unital commutative associative ring containing $$1/6$$, generated by $$[..[[x_1,x_2],x_3],...,x_n]$$. Let $$F_r^{(n)}$$ be the quotient $$F_r/T^{(n)}$$. The class of nilpotency of the Lie algebra $$[F_r^{(n)},F_r^{(n)}]$$ is carefully determined for $$r\leq 5$$ or $$n\leq 5$$. Rota-Baxter operators on a sum of fields https://zbmath.org/1472.16044 2021-11-25T18:46:10.358925Z "Gubarev, V." https://zbmath.org/authors/?q=ai:gubarev.vyacheslav-f|gubarev.vsevolod-yurevich|gubarev.v-v Classification of left octonionic modules https://zbmath.org/1472.17001 2021-11-25T18:46:10.358925Z "Huo, Qinghai" https://zbmath.org/authors/?q=ai:huo.qinghai "Li, Yong" https://zbmath.org/authors/?q=ai:li.yong.5|li.yong.6|li.yong.3|li.yong.8|li.yong.9|li.yong.4|li.yong.7 "Ren, Guangbin" https://zbmath.org/authors/?q=ai:ren.guangbin The article is devoted to left $$\mathbb{O}$$-modules, where $$\mathbb{O}$$ denotes the classical octonion (Cayley) algebra over the real field $$\mathbb{R}$$. This is a particular case of modules over alternative algebras. A review of some previous publications is given. The octonion algebra contains the classical quaternion skew field $$\mathbb{H}$$ of Hamilton. The octonion algebra considered as the vector space over $$\mathbb{R}$$ has the basis $$i_0$$, $$i_1,\ldots,i_7$$ such that $$i_0=1$$, $$i_k^2=-1$$ for each $$k=1,\ldots,7$$, $$i_ki_l=-i_li_k$$ for each $$k\ne l$$ such that $$k\ge 1$$ and $$l\ge 1$$. The subsequent doubling procedures are: $$i_1$$ is the doubling generator of the complex field $$\mathbb{C}$$ over the real field $$\mathbb{R}$$, $$i_2$$ is the doubling generator of $$\mathbb{H}$$ generated from $$\mathbb{C}$$ and $$\mathbb{C}i_2$$ such that $$i_3=i_1i_2$$, then $$i_4$$ denotes the doubling generator of $$\mathbf{O}$$ generated from $$\mathbb{H}$$ and $$\mathbb{H}i_4$$ by the smashed product, where $$i_5$$, $$i_6$$, $$i_7$$ are obtained by multiplication of $$i_1$$, $$i_2$$, $$i_3$$ respectively on $$i_4$$ with the corresponding order up to a notation choice and an automorphism of $$\mathbb{O}$$. It is nonassociative, for example, $$(i_1i_2)i_4=-i_1(i_2i_4)$$. The commutator $$(i_k,i_j)$$ and the associator $$(i_k,i_j,i_l)$$ belong to $$\mathbb{Z}_2$$ for each $$k$$, $$j$$, $$l$$, where $$ab=(ba)(a,b)$$ and $$(ab)c=(a(bc))(a,b,c)$$ for each $$a$$, $$b$$, $$c$$ in $$\mathbf{O}\setminus \{ 0 \}$$, $$\mathbb{Z}_2= \{ -1, 1 \}$$. There is an involution $$\mathbb{O}\ni z\mapsto \bar{z}\in \mathbb{O}$$ such that $$\overline{ab}=\bar{b} \bar{a}$$ for each $$a$$, $$b$$ in $$\mathbb{O}$$, $$|b|^2=b\bar{b}$$. The octonion division algebra is nonassociative alternative with center $$\mathbb{ R}$$ and the multiplicative norm. It is shown in the article that left $$\mathbb{O}$$-modules are of the type $$M=\mathbb{O}^n\bigoplus \bar{\mathbb{O}}^m$$. This induces the algebra structure on $$\mathbb{O}^n$$. This matter is also described in: $$$$ [\textit{N. Bourbaki}, Éléments de mathématique. Algèbre. Chapitres 1 à 3. Reprint of the 1970 original. Berlin: Springer (2007; Zbl 1111.00001)]. $$$$ [\textit{R. D. Schafer}, An introduction to nonassociative algebras. New York and London: Academic Press (1966; Zbl 0145.25601)]. $$$$ [\textit{R. H. Bruck}, A survey of binary systems. Berlin: Springer-Verlag (1958; Zbl 0081.01704)]. Using the opposite algebra $$\mathbb{O}_o$$, or $$\overline{\mathbb{O}}$$ obtained by the involution from $$\mathbb{O}$$, one gets the standard correspondence between left and right modules, the left module over the enveloping algebra $$\mathbb{O}_e$$ also corresponds to the two-sided $$\mathbb{O}$$-module as in [\textit{N. Bourbaki}, (loc. cit.); \textit{R. D. Schafer}, (loc. cit.)]. The algebra $$L(\mathbb{O})$$ generated by left multipliers $$L_b$$ on $$\mathbb{O}$$, $$b\in \mathbb{O}$$, with the associative composition obtained by the set-theoretic composition of maps, is isomorphic to the proper subalgebra of the matrix algebra $$Mat_{8\times 8}(\mathbb{R})$$ (see [\textit{R. D. Schafer}, (loc. cit.)]) satisfying relations $$(5.14)$$-$$(5.20)$$ in [\textit{R. H. Bruck}, (loc. cit.)] implying particularly that $$i_lL_{i_j}L_{i_k}\mathbb{Z}_2=i_lL_{i_ki_j}\mathbb{Z}_2$$ for each $$l$$, $$j$$, $$k$$ in $$\{ 0,...,7 \}$$. On the other hand, $$\mathbb{Z}_2$$ is the normal subgroup in the Moufang multiplicative loop $$G= \{ \pm i_k: k=0,...,7 \}$$ such that its quotient by $$\mathbb{Z}_2$$ is the commutative group $$G/\mathbb{Z}_2$$ by Theorem IV.1.1 in [loc. cit.]. It is proposed in the article to use the Clifford algebra $$Cl_7=Cl(0,7,\mathbb{R})$$ over $$\mathbb{R}$$ for studying the octonion modules by using $$\hat{e}_1, ...,\hat{e_7}$$ as generators of $$Cl_7$$ with Clifford multiplication $$\hat{e}_{k_1}\cdot ... \cdot \hat{e}_{k_m}$$ instead of $$L_{i_1},...,L_{i_7}$$. The Clifford algebra is associative semisimple and isomorphic to $$Mat_{8\times 8}(\mathbb{R})\bigoplus Mat_{8\times 8}(\mathbb{R})$$. The octonion algebra is simple. There is no any nontrivial homomorphism from $$Cl_7$$ to $$L(\mathbb{O})$$, or from $$Cl_7$$ to $$\mathbf{O}$$. For comparison $$L(\mathbb{H})=\mathbb{H}$$, since $$\mathbb{H}$$ is associative. Then Theorem 4.1 of Huo, Li, Ren contradicts to [\textit{N. Bourbaki}, (loc. cit.)] and the Cartan-Jacobson Theorem 3.28 and Corollary 3.29 in [\textit{R. D. Schafer}, (loc. cit.)]. In the article under review there is wrongly cited reference $$$$ in Russian. It is available also in English translation: [\textit{S. V. Ludkovsky}, J. Math. Sci., New York 144, No. 4, 4301--4366 (2007; Zbl 1178.47057); translation from Sovrem. Mat. Prilozh. 35 (2005)]. In the latter paper were considered vector spaces $$X$$ over $$\mathbb{O}$$, which also have the structure of the two-sided octonion modules, such that $$X=X_0i_0\oplus X_1i_1\oplus ... \oplus X_7i_7$$, where $$X_0$$, ...,$$X_7$$ are real vector spaces such that $$X_l$$ is isomorphic to $$X_k$$ for each $$l$$, $$k$$, $$(ab)x_k=a(bx_k)$$, $$bx_k=x_kb$$, $$x_k(ab)=(x_ka)b$$ for each $$a$$,$$b$$ in $$\mathbb{O}$$, $$x_k\in X_k$$, $$k$$, $$l$$ in $$\{ 0,...,7 \}$$. It has properties: $$(bb)x=b(bx)$$, $$x(bb)=(xb)b$$, $$[a,b,x_ki_k]=[a,b,i_k]x_k$$ for each $$x_k\in X_k$$, $$k\in \{ 0,...,7 \}$$, $$a$$, $$b$$ in $$\mathbb{O}$$, implying by the $$\mathbb{R}$$-linearity in $$X$$ and the corresponding identities in $$\mathbb{O}$$ that $$[a,b,x]=[b,x,a]=[x,a,b]$$ for each $$x\in X$$, $$a$$, $$b$$ in $$\mathbb{O}$$, where $$[a,b,x]=(ab)x-a(bx)$$. An approach between the multiplicative and additive structure of a Jordan ring https://zbmath.org/1472.17002 2021-11-25T18:46:10.358925Z "Ferreira, Bruno Leonardo Macedo" https://zbmath.org/authors/?q=ai:ferreira.bruno-leonardo-macedo "Guzzo, Henrique jun." https://zbmath.org/authors/?q=ai:guzzo.henrique-jun "Ferreira, Ruth Nascimento" https://zbmath.org/authors/?q=ai:ferreira.ruth-nascimento Summary: Let $$\mathfrak{J}$$ and $$\mathfrak{J}'$$ be Jordan rings. In this paper we study the additivity of $$n$$-multiplicative isomorphisms from $$\mathfrak{J}$$ onto $$\mathfrak{J}'$$ and of $$n$$-multiplicative derivations of $$\mathfrak{J}$$. Suppose that $$\mathfrak{J}$$ contains a nontrivial idempotent; we prove that if $$\mathfrak{J}$$ satisfying certain conditions, then $$n$$-multiplicative maps and $$n$$-multiplicative derivations from $$\mathfrak{J}$$ to $$\mathfrak{J}'$$ are additive maps. Simple finite-dimensional modular noncommutative Jordan superalgebras https://zbmath.org/1472.17003 2021-11-25T18:46:10.358925Z "Pozhidaev, A. P." https://zbmath.org/authors/?q=ai:pozhidaev.aleksandr-petrovich "Shestakov, I. P." https://zbmath.org/authors/?q=ai:shestakov.ivan-p Summary: We classify the central simple finite-dimensional noncommutative Jordan superalgebras over an algebraically closed field of characteristic $$p > 2$$. The case of characteristic 0 was considered by the authors in the previous paper [Sib. Math. J. 54, No. 2, 301--316 (2013; Zbl 1276.17017); translation from Sib. Mat. Zh. 54, No. 2, 389--406 (2013)]. In particular, we describe Leibniz brackets on all finite dimensional central simple Jordan superalgebras except mixed (nor vector neither Poisson) Kantor doubles of the supercommutative superalgebra $$B(m, n)$$. Polynomial identities of bicommutative algebras, Lie and Jordan elements https://zbmath.org/1472.17004 2021-11-25T18:46:10.358925Z "Dzhumadil'daev, Askar S." https://zbmath.org/authors/?q=ai:dzhumadildaev.a-s "Ismailov, Nurlan A." https://zbmath.org/authors/?q=ai:ismailov.nurlan-a Summary: An algebra with identities $$a(bc)=b(ac)$$, $$(ab) c=(ac)b$$ is called bicommutative. We construct list of identities satisfied by commutator and anti-commutator products in a free bicommutative algebra. We give criterions for elements of a free bicommutative algebra to be Lie or Jordan. Embedding of pre-Lie algebras into preassociative algebras https://zbmath.org/1472.17005 2021-11-25T18:46:10.358925Z "Gubarev, Vsevolod" https://zbmath.org/authors/?q=ai:gubarev.vsevolod-yurevich In the paper under review a solution of the following problem is given: prove that every pre-Lie algebra injectively embeds into its universal enveloping preassociative algebra. Extending structures and classifying complements for left-symmetric algebras https://zbmath.org/1472.17006 2021-11-25T18:46:10.358925Z "Hong, Yanyong" https://zbmath.org/authors/?q=ai:hong.yanyong Summary: Let $$A$$ be a left-symmetric (resp. Novikov) algebra, $$E$$ be a vector space containing $$A$$ as a subspace and $$V$$ be a complement of $$A$$ in $$E$$. The extending structures problem which asks for the classification of all left-symmetric (resp. Novikov) algebra structures on $$E$$ up to an isomorphism which stabilizes $$A$$ such that $$A$$ is a subalgebra of $$E$$ is studied. In this paper, the definition of the unified product for left-symmetric (resp. Novikov) algebras is introduced. It is shown that there exists a left-symmetric (resp. Novikov) algebra structure on $$E$$ such that $$A$$ is a subalgebra of $$E$$ if and only if $$E$$ is isomorphic to a unified product of $$A$$ and $$V$$. A cohomological type object $$\mathcal{H}_A^2(V,A)$$ is constructed to give a theoretical answer to the extending structures problem. Furthermore, given an extension $$A\subset E$$ of left-symmetric (resp. Novikov) algebras, another cohomological type object is constructed to classify all complements of $$A$$ in $$E$$. Several examples are provided in detail. Classification of compatible left-symmetric conformal algebraic structures on the Lie conformal algebra $$\mathcal{W} (a,b)$$ https://zbmath.org/1472.17007 2021-11-25T18:46:10.358925Z "Liu, Deng" https://zbmath.org/authors/?q=ai:liu.deng "Hong, Yanyong" https://zbmath.org/authors/?q=ai:hong.yanyong "Zhou, Hao" https://zbmath.org/authors/?q=ai:zhou.hao "Zhang, Nuan" https://zbmath.org/authors/?q=ai:zhang.nuan Summary: In this article, under some natural condition, a complete classification of compatible left-symmetric conformal algebraic structures on the Lie conformal algebra $$\mathcal{W} (a,b)$$ is presented. Moreover, applying this result, we obtain a class of compatible left-symmetric algebraic structures on the coefficient algebra of $$\mathcal{W} (a,b)$$. Correspondence between some metabelian varieties and left nilpotent varieties https://zbmath.org/1472.17008 2021-11-25T18:46:10.358925Z "Mishchenko, S. P." https://zbmath.org/authors/?q=ai:mishchenko.sergei-petrovich "Valenti, A." https://zbmath.org/authors/?q=ai:valenti.angela Let $$_2 \mathcal N$$ denote the variety of \textit{left nilpotent} algebras of index two, that is the variety of algebras satisfying the identity $$x(yz) \equiv 0$$ in the class of absolutely free algebras. The exponent of the codimension growth of subvarieties of such type can be non-integral (namely, an example with the exponent 7/2 was constructed) [\textit{M. V. Zaĭtsev} and \textit{S. P. Mishchenko}, Mosc. Univ. Math. Bull. 63, No. 1, 25--31 (2008; Zbl 1199.17001); translation from Vestn. Mosk. Univ., Ser. I 2008, No. 1, 25--31 (2008)]. It was shown that there are no varieties of left nilpotent algebras of index two with the codimension growth such that $$c_ n (V) \le Cn^\alpha$$ with $$1 < \alpha < 2$$ and $$2 < \alpha < 3$$. [\textit{S. Mishchenko} and \textit{A. Valenti}, J. Algebra 518, 321--342 (2019; Zbl 1459.17002)]. Now the authors construct a correspondence between left nilpotent algebras of index two and commutative metabelian algebras or anticommutative metabelian algebras and prove that the codimensions sequences of the corresponding algebras coincide up to a constant. This allows them to transfer the above results concerning varieties of left nilpotent algebras of index two to varieties of commutative or anticommutative metabelian algebras. Algebraic deformation quantization of Leibniz algebras https://zbmath.org/1472.17009 2021-11-25T18:46:10.358925Z "Alexandre, Charles" https://zbmath.org/authors/?q=ai:alexandre.charles "Bordemann, Martin" https://zbmath.org/authors/?q=ai:bordemann.martin "Rivière, Salim" https://zbmath.org/authors/?q=ai:riviere.salim "Wagemann, Friedrich" https://zbmath.org/authors/?q=ai:wagemann.friedrich Summary: In this paper we focus on a certain self-distributive multiplication on coalgebras, which leads to so-called rack bialgebra. We construct canonical rack bialgebras (some kind of enveloping algebras) for any Leibniz algebra. Our motivation is deformation quantization of Leibniz algebras in the sense of \textit{B. Dherin} and \textit{F. Wagemann} [ Adv. Math. 270, 21--48 (2015; Zbl 1362.17004)]. Namely, the canonical rack bialgebras we have constructed for any Leibniz algebra lead to a simple explicit formula of the rack-star-product on the dual of a Leibniz algebra recently constructed by Dherin and Wagemann [loc. cit.]. We clarify this framework setting up a general deformation theory for rack bialgebras and show that the rack-star-product turns out to be a deformation of the trivial rack bialgebra product. Odd-quadratic Leibniz superalgebras https://zbmath.org/1472.17010 2021-11-25T18:46:10.358925Z "Benayadi, Saïd" https://zbmath.org/authors/?q=ai:benayadi.said "Mhamdi, Fahmi" https://zbmath.org/authors/?q=ai:mhamdi.fahmi Summary: An odd-quadratic Leibniz superalgebra is a (left or right) Leibniz superalgebra with an odd, supersymmetric, non-degenerate and invariant bilinear form. In this paper, we prove that a left (resp. right) Leibniz superalgebra that carries this structure is symmetric (meaning that it is simultaneously a left and a right Leibniz superalgebra). Moreover, we show that any non-abelian (left or right) Leibniz superalgebra does not possess simultaneously a quadratic and an odd-quadratic structure. Further, we obtain an inductive description of odd-quadratic Leibniz superalgebras using the procedure of generalized odd double extension and we reduce the study of this class of Leibniz superalgebras to that of odd-quadratic Lie superalgebras. Finally, several non-trivial examples of odd-quadratic Leibniz superalgebras are included. \textsf{Lie}-isoclinism of pairs of Leibniz algebras https://zbmath.org/1472.17011 2021-11-25T18:46:10.358925Z "Riyahi, Zahra" https://zbmath.org/authors/?q=ai:riyahi.zahra "Casas Mirás, José Manuel" https://zbmath.org/authors/?q=ai:casas-miras.jose-manuel Summary: The aim of this paper is to consider the relation between \textsf{Lie}-isoclinism and isomorphism of two pairs of Leibniz algebras. We show that, unlike the absolute case for finite dimensional \textsf{Lie} algebras, these concepts are not identical, even if the pairs of Leibniz algebras are \textsf{Lie}-stem. Moreover, throughout the paper, we provide some conditions under which \textsf{Lie}-isoclinism and isomorphism of \textsf{Lie}-stem Leibniz algebras are equal. In order to get this equality, the concept of factor set is studied as well. Exhaustive construction of four-dimensional unital division algebras over finite fields with Kleinian automorphism group https://zbmath.org/1472.17012 2021-11-25T18:46:10.358925Z "Al-Ali Bani-Ata, Mashhour" https://zbmath.org/authors/?q=ai:bani-ata.mashhour Summary: The purpose of this paper is to give an exhaustive construction of the four-dimensional unital division algebras $$A$$ over finite fields $$\mathbb {F}_q$$, $$q$$ is an odd prime, admitting an elementary abelian four-group of automorphisms $$E\leq \Aut(A)$$, using tools from algebraic geometry. Nonassociative cyclic extensions of fields and central simple algebras https://zbmath.org/1472.17013 2021-11-25T18:46:10.358925Z "Brown, C." https://zbmath.org/authors/?q=ai:brown.christian "Pumplün, S." https://zbmath.org/authors/?q=ai:pumplun.susanne Summary: We define nonassociative cyclic extensions of degree $$m$$ of both fields and central simple algebras over fields. If a suitable field contains a primitive $$m$$th (resp., $$q$$th) root of unity, we show that suitable nonassociative generalized cyclic division algebras yield nonassociative cyclic extensions of degree $$m$$ (resp., $$qs$$). Some of Amitsur's classical results on non-commutative associative cyclic extensions of both fields and central simple algebras are obtained as special cases. Multiplicative Lie-type derivations on alternative rings https://zbmath.org/1472.17014 2021-11-25T18:46:10.358925Z "Macedo Ferreira, Bruno Leonardo" https://zbmath.org/authors/?q=ai:ferreira.bruno-leonardo-macedo "Guzzo, Henrique Jr." https://zbmath.org/authors/?q=ai:guzzo.henrique-jun "Wei, Feng" https://zbmath.org/authors/?q=ai:wei.feng Summary: Let $$\mathfrak R$$ be an alternative ring containing a nontrivial idempotent and $$\mathfrak D$$ be a multiplicative Lie-type derivation from $$\mathfrak R$$ into itself. Under certain assumptions on $$\mathfrak R,$$ we prove that $$\mathfrak D$$ is almost additive. Let $$p_n (x_1, x_2, \ldots, x_n)$$ be the $$(n - 1)$$-th commutator defined by $$n$$ indeterminates $$x_1, \ldots, x_n.$$ If $$\mathfrak R$$ is a unital alternative ring with a nontrivial idempotent and is $$\{2, 3, n - 1, n - 3\}$$-torsion free, it is shown under certain condition of $$\mathfrak R$$ and $$\mathfrak D$$ that $$\mathfrak D = \delta + \tau,$$ where $$\delta$$ is a derivation and $$\tau : \mathfrak R \to \mathcal Z (\mathfrak R)$$ such that $$\tau (p_n(a_1, \ldots, a_n)) = 0$$ for all $$a_1, \ldots, a_n \in \mathfrak R.$$ On the stability of left $$\delta$$-centralizers on Banach Lie triple systems https://zbmath.org/1472.17015 2021-11-25T18:46:10.358925Z "Ghobadipour, Norouz" https://zbmath.org/authors/?q=ai:ghobadipour.norouz "Sepasian, Ali Reza" https://zbmath.org/authors/?q=ai:sepasian.ali-reza Summary: In this paper under a condition, we prove that every Jordan left $$\delta$$-centralizer on a Lie triple system is a left $$\delta$$-centralizer. Moreover, we use a fixed point method to prove the generalized Hyers-Ulam-Rassias stability associated with the Pexiderized Cauchy-Jensen type functional equation $rf\left(\frac{x+y}{r}\right)+sg\left(\frac{x-y}{s}\right)=2h(x),$ for $$r,s \in \mathbb R \setminus \{0\}$$ in Banach Lie triple systems. The construction of 3-Bihom-Lie algebras https://zbmath.org/1472.17016 2021-11-25T18:46:10.358925Z "Li, Juan" https://zbmath.org/authors/?q=ai:li.juan.1|li.juan "Chen, Liangyun" https://zbmath.org/authors/?q=ai:chen.liangyun Summary: The purpose of this article is to study the construction of 3-Bihom-Lie algebras. We give some ways of constructing 3-Bihom-Lie algebras from 3-Bihom-Lie algebras and 3-totally Bihom-associative algebras. Furthermore, we introduce $$T_\theta$$-extensions and $$T_\theta^*$$-extensions of 3-Bihom-Lie algebras and prove the necessary and sufficient conditions for a $$2n$$-dimensional quadratic 3-Bihom-Lie algebra to be isomorphic to a $$T^*_\theta$$-extension. Gelfand-Dorfman-Novikov-Poisson superalgebras and their envelopes https://zbmath.org/1472.17017 2021-11-25T18:46:10.358925Z "Zakharov, Anton Stanislavovich" https://zbmath.org/authors/?q=ai:zakharov.anton-s Summary: We prove some result of $$\mathcal{GDNP}$$-algebras for $$\mathcal{GDNP}$$-superalgebras and embeding some $$\mathcal{GDNP}$$-superalgebras into $$\mathcal{GDNP}$$-superalgebras of vector type. Also we prove that not every $$\mathcal{GDNP}$$-superalgebra can be embeded into associative supercommutative superalgebra with even derivation $$D$$ and product given by $$a\circ b = aD(b)$$. Abelian product of free abelian and free Lie algebras https://zbmath.org/1472.17018 2021-11-25T18:46:10.358925Z "Özkurt, Zeynep" https://zbmath.org/authors/?q=ai:ozkurt.zeynep "Ekici, Naime" https://zbmath.org/authors/?q=ai:ekici.naime Summary: Let $$F_n$$ be a free Lie algebra of finite rank $$n$$ and $$A$$ be a free abelian Lie algebra of finite rank $$m\geq 0$$. We investigate the properties of the generating sets and subalgebras of the abelian product $$A*_{ab}F_n$$. Moreover these properties are used to solve the membership problem for $$A*_{ab}F_n$$. Bosonic-fermionic realizations of root spaces and bilinear forms for Lie superalgebras https://zbmath.org/1472.17019 2021-11-25T18:46:10.358925Z "Kwon, Namhee" https://zbmath.org/authors/?q=ai:kwon.namhee In the paper under review, the authors realize the root spaces and bilinear forms on a Lie superalgebras by some normally ordered quadratic elements in a vertex algebra. Then, as applications, they gave bosonic-fermionic realizations of the general linear Lie superalgebras and vertex representations of the affine ortho-symplectic Lie superalgebras. This is a generalization of the work [\textit{A. J. Feingold} and \textit{I. B. Frenkel}, Adv. Math. 56, 117--172 (1985; Zbl 0601.17012)] on the cases of classical Lie algebras. Classification of nilpotent Lie superalgebras of multiplier-rank less than or equal to 2 https://zbmath.org/1472.17020 2021-11-25T18:46:10.358925Z "Liu, Wende" https://zbmath.org/authors/?q=ai:liu.wende "Zhang, Yanling" https://zbmath.org/authors/?q=ai:zhang.yanling The Lie algebra concepts of cover and multiplier are generalized to the Lie superalgebra case. The algebras considered are over an algebraically closed field of characteristic 0. Let $$\mathrm{sdim}(V)$$ be the superdimension of the superspace $$V$$. Some classical results for the Lie algebra case are extended to the present case, including the superdimension of the multiplier of the sum of Lie superalgebras .and the superdimension of the multiplier of a Heisenberg Lie superalgebra. For a superalgebra $$L$$ of superdimension $$(s,t)$$, define the super-rank of $$L$$ to be $$\mathrm{smr}(L)= ((1/2)s(s-1) +(1/2)t(t+1),st) - \mathrm{sdim } M(L)$$ and the multiplier-rank to be $$\mathrm{mr}(L)= \vert\mathrm{smr}(L)\vert$$. All Lie superalgebras of multiplier-rank less than or equal to 2 are determined. Orthogonal abelian Cartan subalgebra decomposition of $\mathfrak{sl}_n$ over a finite commutative ring https://zbmath.org/1472.17021 2021-11-25T18:46:10.358925Z "Sriwongsa, Songpon" https://zbmath.org/authors/?q=ai:sriwongsa.songpon "Zou, Yi Ming" https://zbmath.org/authors/?q=ai:zou.yiming Summary: Orthogonal decomposition of the special linear Lie algebra over the complex numbers was studied in the early 1980s and attracted further attentions in the past decade due to its application in quantum information theory. In this paper, we study this decomposition problem of the special linear Lie algebra over a finite commutative ring with identity. Nilpotent orbits for Borel subgroups of $$\mathrm{SO}_5(k)$$ https://zbmath.org/1472.17022 2021-11-25T18:46:10.358925Z "Burkhart, Madeleine" https://zbmath.org/authors/?q=ai:burkhart.madeleine "Vella, David" https://zbmath.org/authors/?q=ai:vella.david-c Summary: Let $$G$$ be a quasisimple algebraic group defined over an algebraically closed field $$k$$ and $$B$$ a Borel subgroup of $$G$$ acting on the nilradical $$\mathfrak{n}$$ of its Lie algebra $$\mathfrak{b}$$ via the adjoint representation. It is known that $$B$$ has only finitely many orbits in only five cases: when $$G$$ is type $$A_{n}$$ for $$n \leq 4$$, and when $$G$$ is type $$B_{2}$$. We elaborate on this work in the case when $$G =\mathrm{SO}_{5}(k)$$ (type $$B_{2})$$ by finding the defining equations of each orbit. We use these equations to determine the dimension of the orbits and the closure ordering on the set of orbits. The other four cases, when $$G$$ is type $$A_{n}$$, can be approached the same way and are treated in a separate paper. The universal enveloping algebra of the Schrödinger algebra and its prime spectrum https://zbmath.org/1472.17023 2021-11-25T18:46:10.358925Z "Bavula, V. V." https://zbmath.org/authors/?q=ai:bavula.vladimir-v "Lu, T." https://zbmath.org/authors/?q=ai:lu.tao Summary: The prime, completely prime, maximal, and primitive spectra are classified for the universal enveloping algebra of the Schrödinger algebra. The explicit generators are given for all of these ideals. A counterexample is constructed to the conjecture of Cheng and Zhang about nonexistence of simple singular Whittaker modules for the Schrödinger algebra (and all such modules are classified). It is proved that the conjecture holds generically'. Representations of the Lie algebra of vector fields on a sphere https://zbmath.org/1472.17024 2021-11-25T18:46:10.358925Z "Billig, Yuly" https://zbmath.org/authors/?q=ai:billig.yuly "Nilsson, Jonathan" https://zbmath.org/authors/?q=ai:nilsson.jonathan Summary: For an affine algebraic variety $$X$$ we study a category of modules that admit compatible actions of both the algebra $$A$$ of functions on $$X$$ and the Lie algebra of vector fields on $$X$$. In particular, for the case when $$X$$ is the sphere $$\mathbb{S}^2$$, we construct a set of simple modules that are finitely generated over $$A$$. In addition, we prove that the monoidal category that these modules generate is equivalent to the category of finite-dimensional rational $$\mathrm{GL}_2$$-modules. On the action of the Koszul map on the enveloping algebra of the general linear Lie algebra https://zbmath.org/1472.17025 2021-11-25T18:46:10.358925Z "Brini, Andrea" https://zbmath.org/authors/?q=ai:brini.andrea "Teolis, Antonio" https://zbmath.org/authors/?q=ai:teolis.antonio-g-b Let $$gl_n$$ be the general linear Lie algebra (over the complex numbers $$\mathbf{C}$$), and let $$U(gl_n)$$ be its universal enveloping algebra. Denote by $$\mathbf{C}(M_{n,n})$$ the commutative polynomial algebra in the $$n^2$$ variables $$(i\mid j)$$, $$1\le i,j\le n$$ where $$(i\mid j)$$ are the corresponding entries of the generic $$n\times n$$ matrix $$M_{n,n}$$. Clearly the latter commutative algebra is isomorphic to Sym$$(gl_n)$$, and a natural isomorphism is given by the correspondence $$(i\mid j)\mapsto e_{ij}$$ where $$e_{ij}$$ is the matrix with 1 at position $$(i,j)$$, and zeros elsewhere. The main result of the paper under review is the construction of an equivariant isomorphism $$K\colon U(gl_n)\to \mathbf{C}(M_{n,n})$$. Furthermore the isomorphism $$K$$ sends the Capelli bitableaux from $$U(gl_n)$$, denoted by $$[S\mid T]$$ to the corresponding determinantal bitableaux $$(S\mid T)\in \mathbf{C}(M_{n,n})$$. Additionally, $$K$$ sends the $$*$$-bitableaux in $$U(gl_n)$$ to the corresponding permanental $$*$$-bitableaux in $$\mathbf{C}(M_{n,n})$$. \par An interesting corollary of the construction of the isomorphism $$K$$ isn deduced. Since $$gl_n$$ acts on $$U(gl_n)$$ by the adjoint representation, and also on $$\mathbf{C}(M_{n,n})$$ (by polarization), one gets that $$Z$$, the centre of $$U(gl_n)$$ is in fact the algebra of invariants. Hence it maps via $$K$$ to $$\mathbf{C}(M_{n,n}){(ad \, gl_n})$$. Young-Capelli bitableaux, Capelli immanants in $$\mathbb{U}(\mathrm{gl}(n))$$ and the Okounkov quantum immanants https://zbmath.org/1472.17026 2021-11-25T18:46:10.358925Z "Brini, A." https://zbmath.org/authors/?q=ai:brini.andrea "Teolis, A." https://zbmath.org/authors/?q=ai:teolis.antonio-g-b Let $$gl_n$$ be the general linear Lie algebra over the complex numbers $$\mathbb{C}$$. Its universal enveloping algebra $$U=U(gl_n)$$ admits a basis consisting of the standard Capelli bitableaux and another basis consisting of the standard Young-Capelli bitableaux. The authors of the paper under review introduce another spanning set of $$U$$, the so-called Capelli immanants. A similar notion already appeared in the study of the centre of $$U$$, see fro example [\textit{A. Okounkov}, Transform. Groups 1, No. 1--2, 99--126 (1996; Zbl 0864.17014)]. Let $$\mathbb{C}[M_{n,n}]$$ be the polynomial algebra in the $$n^2$$ variables $$x_{ij}$$, $$1\le i,j\le n$$; one can view it as the polynomial algebra for one generic $$n\times n$$ matrix. The latter algebra is, in fact, the symmetric algebra for $$gl_n$$, and is closely related to $$U$$. There is a natural isomorphism $$K$$ of $$\mathbb{C}[M_{n,n}]$$ and Sym$$(gl_n$$, called the Bitableax correspondence isomorphism or Koszul map (BCK for short). The authors transfer the action of the Capelli immanants to $$\mathbb{C}[M_{n,n}]$$. They consider the inverse isomorphism $$K^{-1}$$ and show that the images of the YoungCapelli bitableaux are exactly the standard right symmetrized bitableaux. They prove that the Capelli immanants are combinations of standard Young-Capelli bitableaux of the same shape. \par It turns out that the quantum immanants introduced by Okounkov in the above cited paper are simple combinations of diagonal Capelli immanants; the corresponding coefficients are given explicitly. A canonical presentations of quantum immanants in terms of double Young-Capelli bitableaux is given as well. Whittaker coinvariants for $$\mathrm{GL}(m|n)$$ https://zbmath.org/1472.17027 2021-11-25T18:46:10.358925Z "Brundan, Jonathan" https://zbmath.org/authors/?q=ai:brundan.jonathan "Goodwin, Simon M." https://zbmath.org/authors/?q=ai:goodwin.simon-m Summary: Let $$W_{m|n}$$ be the (finite) $$W$$-algebra attached to the principal nilpotent orbit in the general linear Lie superalgebra $$\mathfrak{gl}_{m | n}(\mathbb{C})$$. In this paper we study the Whittaker coinvariants functor, which is an exact functor from category $$\mathcal{O}$$ for $$\mathfrak{gl}_{m | n}(\mathbb{C})$$ to a certain category of finite-dimensional modules over $$W_{m | n}$$. We show that this functor has properties similar to Soergel's functor $$\mathbb{V}$$ in the setting of category $$\mathcal{O}$$ for a semisimple Lie algebra. We also use it to compute the center of $$W_{m | n}$$ explicitly, and deduce consequences for the classification of blocks of $$\mathcal{O}$$ up to Morita/derived equivalence. Irreducible twisted Heisenberg-Virasoro modules from tensor products https://zbmath.org/1472.17028 2021-11-25T18:46:10.358925Z "Chen, Haibo" https://zbmath.org/authors/?q=ai:chen.haibo.2 "Su, Yucai" https://zbmath.org/authors/?q=ai:su.yucai Summary: In this paper, we realize polynomial $$\mathcal{H}$$-modules $$\Omega (\lambda ,\alpha ,\beta )$$ from irreducible twisted Heisenberg-Virasoro modules $$\mathcal{A}_{\alpha ,\beta }$$. It follows from $$\mathcal{H}$$-modules $$\Omega (\lambda ,\alpha ,\beta )$$ and $$\text{Ind}(M)$$ that we obtain a class of tensor product modules $$\big (\bigotimes_{i=1}^m\Omega (\lambda_i,\alpha_i,\beta_i)\big )\otimes \text{Ind}(M)$$. We give the necessary and sufficient conditions under which these modules are irreducible and isomorphic, and also give that the irreducible modules in this class are new. Indecomposable manipulations with simple modules in category $$\mathcal{O}$$ https://zbmath.org/1472.17029 2021-11-25T18:46:10.358925Z "Coulembier, Kevin" https://zbmath.org/authors/?q=ai:coulembier.kevin "Mazorchuk, Volodymyr" https://zbmath.org/authors/?q=ai:mazorchuk.volodymyr "Zhang, Xiaoting" https://zbmath.org/authors/?q=ai:zhang.xiaoting Summary: We study the problem of indecomposability of translations of simple modules in the principal block of BGG category $$\mathcal{O}$$ for $$\mathfrak{sl}_n$$, as conjectured in [\textit{T. Kildetoft} and \textit{V. Mazorchuk}, Adv. Math. 301, 785--803 (2016; Zbl 1365.17007)]. We describe some general techniques and prove a few general results which may be applied to study various special cases of this problem. We apply our results to verify indecomposability for $$n \leq 6$$. We also study the problem of indecomposability of shufflings and twistings of simple modules and obtain some partial results. Minimal matrix representations of decomposable Lie algebras of dimension less than or equal to five https://zbmath.org/1472.17030 2021-11-25T18:46:10.358925Z "Ghanam, Ryad" https://zbmath.org/authors/?q=ai:ghanam.ryad-a "Lamichhane, Manoj" https://zbmath.org/authors/?q=ai:lamichhane.manoj "Thompson, Gerard" https://zbmath.org/authors/?q=ai:thompson.gerard Summary: We obtain minimal dimension matrix representations for each decomposable five-dimensional Lie algebra over $$\mathbb R$$ and justify in each case that they are minimal. This article is a continuation of the authors' papers in [Bull. Malays. Math. Sci. Soc. (2) 36, No. 2, 343--349 (2013; Zbl 1276.17006); Extr. Math. 30, No. 1, 95--133 (2015; Zbl 1366.17006)]. Simple Witt modules that are finitely generated over the Cartan subalgebra https://zbmath.org/1472.17031 2021-11-25T18:46:10.358925Z "Guo, Xiangqian" https://zbmath.org/authors/?q=ai:guo.xiangqian "Liu, Genqiang" https://zbmath.org/authors/?q=ai:liu.genqiang "Lu, Rencai" https://zbmath.org/authors/?q=ai:lu.rencai "Zhao, Kaiming" https://zbmath.org/authors/?q=ai:zhao.kaiming Summary: Let $$d\geq 1$$ be an integer, $$W_d$$ and $$\mathcal{K}_d$$ be the Witt algebra and the Weyl algebra over the Laurent polynomial algebra $$A_d=\mathbb{C} [x_1^{\pm 1}, x_2^{\pm 1}, \dots, x_d^{\pm1}]$$, respectively. For any $$\mathfrak{gl}_d$$-module $$V$$ and any admissible module $$P$$ over the extended Witt algebra $$\widetilde{W}_d$$, we define a $$W_d$$-module structure on the tensor product $$P\otimes V$$. In this paper, we classify all simple $$W_d$$-modules that are finitely generated over the Cartan subalgebra. They are actually the $$W_d$$-modules $$P \otimes V$$ for a finite-dimensional simple $$\mathfrak{gl}_d$$-module $$V$$ and a simple $$\mathcal{K}_d$$-module $$P$$ that is a finite-rank free module over the polynomial algebra in the variables $$x_1\frac{\partial}{\partial x_1},\dots,x_d\frac{\partial}{\partial x_d}$$, except for a few cases which are also clearly described. We also characterize all simple $$\mathcal{K}_d$$-modules and all simple admissible $$\widetilde{W}_d$$-modules that are finitely generated over the Cartan subalgebra. Representations of Lie superalgebras with fusion flags https://zbmath.org/1472.17032 2021-11-25T18:46:10.358925Z "Kus, Deniz" https://zbmath.org/authors/?q=ai:kus.deniz Summary: We study the category of finite-dimensional representations for a basic classical Lie superalgebra $$\mathfrak g=\mathfrak g_0\oplus \mathfrak g_1$$. For the orthosymplectic Lie superalgebra $$\mathfrak g=\mathfrak{osp}(1,2n),$$ we show that various objects in that category admit a fusion flag, that is, a sequence of graded $$\mathfrak g_0[t]$$-modules such that the successive quotients are isomorphic to fusion products. Among these objects we find fusion products of finite-dimensional irreducible $$\mathfrak g$$-modules, truncated Weyl modules and Demazure type modules. This result shows that the character of these types of representations can be expressed in terms of characters of fusion products and we prove that the graded multiplicities are given by products of $$q$$-binomial coefficents. Moreover, we establish a presentation for these types of fusion products in terms of generators and relations of the enveloping algebra. FFLV-type monomial bases for type B https://zbmath.org/1472.17033 2021-11-25T18:46:10.358925Z "Makhlin, Igor" https://zbmath.org/authors/?q=ai:makhlin.igor Summary: We present a combinatorial monomial basis (or, more precisely, a family of monomial bases) in every finite-dimensional irreducible $\mathfrak{so}_{2n+1}$-module. These bases are in many ways similar to the FFLV bases for types $A$ and $C$. They are also defined combinatorially via sums over Dyck paths in certain triangular grids. Our sums, however, involve weights depending on the length of the corresponding root. Accordingly, our bases also induce bases in certain degenerations of the modules but these degenerations are obtained not from the filtration by PBW degree but by a weighted version thereof. Supergroup $$OSP(2,2n)$$ and super Jacobi polynomials https://zbmath.org/1472.17034 2021-11-25T18:46:10.358925Z "Movsisyan, G. S." https://zbmath.org/authors/?q=ai:movsisyan.gevorg-s "Sergeev, A. N." https://zbmath.org/authors/?q=ai:sergeev.alexander-n Summary: The coefficients of the super Jacobi polynomials of $$B C_{1, n}$$-type are rational functions in three parameters $$k, p, q$$. At the point $$(- 1, 0, 0)$$ these coefficients may have poles. Let us set $$q = 0$$ and consider a pair $$(k, p)$$ as the point of $$\mathbb{A}^2$$. If we apply blow up procedure at the point $$(- 1, 0)$$ then we get a new family of polynomials. This family depends on one parameter $$t \in \mathbb{P}$$. If $$t = \infty$$ then we get Euler supercharacters for Lie supergroup $$OSP(2, 2 n)$$. The supercharacters of finite dimensional irreducible modules can be also obtained by a specialization of the parameter $$t$$. But in such a case the specialization depends on the highest weight of the corresponding irreducible module. On multiplicities of irreducibles in large tensor product of representations of simple Lie algebras https://zbmath.org/1472.17035 2021-11-25T18:46:10.358925Z "Postnova, Olga" https://zbmath.org/authors/?q=ai:postnova.olga-v "Reshetikhin, Nicolai" https://zbmath.org/authors/?q=ai:reshetikhin.nikolai-yu Summary: In this paper, we study the asymptotics of multiplicities of irreducible representations in large tensor products of finite dimensional representations of simple Lie algebras and their statistics with respect to Plancherel and character probability measures. We derive the asymptotic distribution of irreducible components for the Plancherel measure, generalizing results of \textit{P. Biane} [C. R. Acad. Sci., Paris, Sér. I 316, No. 8, 849--852 (1993; Zbl 0805.17005)] and \textit{T. Tate} and \textit{S. Zelditch} [J. Funct. Anal. 217, No. 2, 402--447 (2004; Zbl 1062.22026)]. We also derive the asymptotics of the character measure for generic parameters and an intermediate scaling in the vicinity of the Plancherel measure. It is interesting that the asymptotic measure is universal and after suitable renormalization does not depend on which representations were multiplied but depends significantly on the degeneracy of the parameter in the character distribution. A relationship between Gelfand-Tsetlin bases and Chari-Loktev bases for irreducible finite dimensional representations of special linear Lie algebras https://zbmath.org/1472.17036 2021-11-25T18:46:10.358925Z "Raghavan, K. N." https://zbmath.org/authors/?q=ai:raghavan.komaranapuram-n "Ravinder, B." https://zbmath.org/authors/?q=ai:ravinder.bhimarthi "Viswanath, Sankaran" https://zbmath.org/authors/?q=ai:viswanath.sankaran Summary: We consider two bases for an arbitrary finite dimensional irreducible representation of a complex special linear Lie algebra: the classical Gelfand-Tsetlin basis and the relatively new Chari-Loktev basis. Both are parametrized by the set of (integral Gelfand-Tsetlin) patterns with a fixed bounding sequence determined by the highest weight of the representation. We define the \textit{row-wise dominance} partial order on this set of patterns, and prove that the transition matrix between the two bases is triangular with respect to this partial order. We write down explicit expressions for the diagonal elements of the transition matrix. For the entire collection see [Zbl 1428.13001]. Pieri formulae and specialisation of super Jacobi polynomials https://zbmath.org/1472.17037 2021-11-25T18:46:10.358925Z "Sergeev, Aleksandr Nikolaevich" https://zbmath.org/authors/?q=ai:sergeev.alexander-n "Zharinov, Egor Dmitrievich" https://zbmath.org/authors/?q=ai:zharinov.egor-dmitrievich Summary: We give a new proof of the fact that the Euler supercharacters of the Lie superalgebra $$\mathfrak{osp}(2m+1,2n)$$ can be obtained as a certain limit of the super Jacobi polynomials. The known proof was not direct one and it was mostly based on calculations. In this paper we propose more simple and more conceptional proof. The main idea is to use the Pieri formulae from the beginning. It turns out that the super Jacobi polynomials and their specialisations can be uniquely characterised by two properties. The first one is that they are eigenfunctions of CMS operator and the second one is that they satisfy the Pieri formulae. As by product we get some interesting identities involving a Young diagram and rational functions. We hope that our approach can be useful in many similar cases. Non-weight representations of Cartan type H Lie algebras https://zbmath.org/1472.17038 2021-11-25T18:46:10.358925Z "Zhang, Juanjuan" https://zbmath.org/authors/?q=ai:zhang.juanjuan From the text: This paper continues the author's previous one [Commun. Algebra 46, No. 10, 4243--4264 (2018; Zbl 1444.17004)] on the study of rank-1 $$U(\mathfrak h)$$ free modules for the Cartan type Lie algebras of type S. In this paper, he studies the same question for the Cartan H type Lie algebras and finds the only irreducible submodules of them. Local derivations on Borel subalgebras of finite-dimensional simple Lie algebras https://zbmath.org/1472.17039 2021-11-25T18:46:10.358925Z "Yu, Yalong" https://zbmath.org/authors/?q=ai:yu.yalong "Chen, Zhengxin" https://zbmath.org/authors/?q=ai:chen.zhengxin Summary: Let $$\mathbb{F}$$ be an algebraically closed field of characteristic 0, let $$\mathcal{L}$$ be a finite-dimensional simple Lie algebra over $$\mathbb{F}$$, and let $$\mathcal{B}$$ be the standard Borel subalgebra of $$\mathcal{L}$$. In this paper we prove that every local derivation of $$\mathcal{B}$$ is a derivation. On rigid 2-step nilpotent Lie superalgebras https://zbmath.org/1472.17040 2021-11-25T18:46:10.358925Z "Alvarez, María Alejandra" https://zbmath.org/authors/?q=ai:alvarez.maria-alejandra "Anza, Yerko" https://zbmath.org/authors/?q=ai:anza.yerko In this work, the authors consider the rigidity problem of Lie superalgebras within the variety $$\mathcal{N}^{2}_{(m|n)}$$ of 2-step nilpotent superalgebras of dimension $$(m|n)$$. The approach is based on cohomological tools, for which a criterion that ensures rigidity within the variety is obtained. Several examples are given, in low dimensions as well as for arbitrary dimensions, such as the Heisenberg Lie superalgebras and 2-step nilpotent Lie superalgebras associated to graphs, for which some additional properties are obtained. Some properties of $$c$$-covers of a pair of Lie algebras https://zbmath.org/1472.17041 2021-11-25T18:46:10.358925Z "Arabyani, Homayoon" https://zbmath.org/authors/?q=ai:arabyani.homayoon "Safa, Hesam" https://zbmath.org/authors/?q=ai:safa.hesam Summary: In [Commun. Algebra 45, No. 10, 4429--4434 (2017; Zbl 1427.17021)], we characterized the structure of the $$c$$-nilpotent multiplier of a pair of Lie algebras in terms of its $$c$$-covering pairs and discussed some results on the existence of $$c$$-covers of a pair of Lie algebras. In the present paper, it is shown under some conditions that a relative $$c$$-central extension of a pair of Lie algebras is a homomorphic image of a $$c$$-covering pair. Moreover, we prove that a $$c$$-cover of a pair of finite dimensional Lie algebras, under some assumptions, has a unique domain up to isomorphism and also that every perfect pair of Lie algebras admits at least one $$c$$-cover. Finally, we discuss a result concerning the isoclinism of $$c$$-covering pairs. Almost inner derivations of 2-step nilpotent Lie algebras of genus 2 https://zbmath.org/1472.17042 2021-11-25T18:46:10.358925Z "Burde, Dietrich" https://zbmath.org/authors/?q=ai:burde.dietrich "Dekimpe, Karel" https://zbmath.org/authors/?q=ai:dekimpe.karel "Verbeke, Bert" https://zbmath.org/authors/?q=ai:verbeke.bert The authors study the algebra of almost inner derivations of a Lie algebra over any algebraically closed field of characteristic not two, and also over the real numbers. They work with 2-step nilpotent Lie algebras with 2-dimensional derived algebra. They use that these algebras can be described by skew-symmetric matrix pencils and their invariant polynomials and elementary divisors. Explicit formulas are found for the dimension for these algebras. Decompositions of algebras and post-associative algebra structures https://zbmath.org/1472.17043 2021-11-25T18:46:10.358925Z "Burde, Dietrich" https://zbmath.org/authors/?q=ai:burde.dietrich "Gubarev, Vsevolod" https://zbmath.org/authors/?q=ai:gubarev.vsevolod-yurevich In the paper the post-associative algebras are considered and their relationship to post-Lie algebra structures, Rota-Baxter operators and decompositions of associative algebras and Lie algebras'' are studied. Before some results in this direction were obtained [\textit{D. Burde} and \textit{V. Gubarev}, Commun. Algebra 47, No. 5, 2280--2296 (2019; Zbl 1455.17007)]. In the present paper, the authors obtain further results on post-Lie algebra structures and correct the proof of Propositions 3.7 and 3.8 form the paper cired above, which relied on a decomposition theorem of Lie algebras, namely Proposition 3.6 from that paper, which unfortunately is in error''. Computation of positively graded filiform nilpotent Lie algebras in low dimensions https://zbmath.org/1472.17044 2021-11-25T18:46:10.358925Z "Edwards, John" https://zbmath.org/authors/?q=ai:edwards.john-s|edwards.john-b|edwards.john-a|edwards.john-w|edwards.john-m|edwards.john-t "Krome, Cameron" https://zbmath.org/authors/?q=ai:krome.cameron "Payne, Tracy L." https://zbmath.org/authors/?q=ai:payne.tracy-l Let $$\mathfrak{g}$$ be a Lie algebra of finite dimension $$n$$ over the rational, real, or complex numbers. Suppose $$\mathfrak{g}$$ has a grading $$\oplus \mathfrak{g}_{i}$$ over the positive integers, so that $$\mathfrak{g}$$ is nilpotent, with the Lie power $$\mathfrak{g}^{n}$$ vanishing. $$\mathfrak{g}$$ is said to be positively graded filiform if it has the largest possible nilpotence class, that is, the Lie power $$\mathfrak{g}^{n-1}$$ is different from $$0$$. Positively graded filiform Lie algebras up to dimension $$8$$ have been classified, see [\textit{R. M. Arroyo}, Rocky Mt. J. Math. 41, No. 4, 1025--1043 (2011; Zbl 1222.53045)]. In the paper under review an algorithm to enumerate, via successive central extensions, positively graded filiform Lie algebras by dimension is described. The resulting classification for $$n \le 10$$, over the complex numbers, is reported in the paper, although the authors have been able to go up to $$n = 15$$. Capable Lie algebras with the derived subalgebra of dimension 2 over an arbitrary field https://zbmath.org/1472.17045 2021-11-25T18:46:10.358925Z "Niroomand, Peyman" https://zbmath.org/authors/?q=ai:niroomand.peyman "Johari, Farangis" https://zbmath.org/authors/?q=ai:johari.farangis "Parvizi, Mohsen" https://zbmath.org/authors/?q=ai:parvizi.mohsen Summary: In this paper, we classify all capable nilpotent Lie algebras with the derived subalgebra of dimension 2 over an arbitrary field. Moreover, the explicit structure of such Lie algebras of class 3 is given. On a homotopy version of the Duflo isomorphism https://zbmath.org/1472.17046 2021-11-25T18:46:10.358925Z "Felder, Matteo" https://zbmath.org/authors/?q=ai:felder.matteo Summary: For a finite-dimensional Lie algebra $$\mathfrak{g}$$, the Duflo map $$S\mathfrak{g}\rightarrow U\mathfrak{g}$$ defines an isomorphism of $$\mathfrak{g}$$-modules. On $$\mathfrak{g}$$-invariant elements, it gives an isomorphism of algebras. Moreover, it induces an isomorphism of algebras on the level of Lie algebra cohomology $$H(\mathfrak{g},S\mathfrak{g})\rightarrow H(\mathfrak{g}, U\mathfrak{g})$$. However, as shown by \textit{J. Alm} and \textit{S. A. Merkulov} [J. Noncommut. Geom. 9, No. 1, 185--214 (2015; Zbl 1344.53075)], it cannot be extended in a universal way to an $$A_\infty$$-isomorphism between the corresponding Chevalley-Eilenberg complexes. In this paper, we give an elementary and self-contained proof of this fact using a version of M. Kontsevich's graph complex. Universal enveloping Lie Rota-Baxter algebras of pre-Lie and post-Lie algebras https://zbmath.org/1472.17047 2021-11-25T18:46:10.358925Z "Gubarev, V. Yu." https://zbmath.org/authors/?q=ai:gubarev.vsevolod-yurevich Let $$L$$ be a vector space over a field $$F$$. Define a bilinear product $$xy$$ in $$L$$ such that for every $$x_1$$, $$x_2$$, $$x_3\in L$$ one has $$(x_1x_2)x_3 - x_1(x_2x_3) = (x_2x_1)x_3 - x_2(x_1x_3)$$, then $$L$$ is a \textit{pre-Lie algebra}. The name comes from the fact that substituting the product in $$L$$ by $$[x,y]=xy-yx$$ one gets a Lie algebra. \par A \textit{post-Lie algebra} is a vector space $$L$$ together with two bilinear operations $$[x,y]$$ and $$xy$$ such that $$L$$ is a Lie algebra considered with $$[x,y]$$, and $$(x_1x_2)x_3 - x_1(x_2x_3) -(x_2x_1)x_3 + x_2(x_1x_3) = [x_2,x_1]x_3$$, and moreover $$x_1[x_2,x_3] = [x_1x_2,x_3]+ [x_2,x_1x_3]$$, for every $$x_1$$, $$x_2$$, $$x_3\in L$$. \par The author studies Rota-Baxter algebras (that is algebras equipped with a Rota-Baxter operator). It is proved in the paper that the pre-Lie and the post-Lie algebras have universal enveloping Lie Rota-Baxter algebras, moreover the authors gives a direct construction of these universal enveloping algebras. Furthermore he proves that the variety of all Lie Rota-Baxter algebras is not a Schreier variety. Additionally he proves that the variety of Lie Rota-Baxter algebras and the one of the pre-Lie algebras are related as in the Poincaré-Birkhoff-Witt theorem (that is they form a PBW-pair). A similar result is proved for the varieties of all $$\lambda$$-Lie Rota-Baxter algebras and all post-Lie algebras. Quantized enveloping superalgebra of type $$P$$ https://zbmath.org/1472.17048 2021-11-25T18:46:10.358925Z "Ahmed, Saber" https://zbmath.org/authors/?q=ai:ahmed.saber "Grantcharov, Dimitar" https://zbmath.org/authors/?q=ai:grantcharov.dimitar "Guay, Nicolas" https://zbmath.org/authors/?q=ai:guay.nicolas The authors introduce a quantized version of the periplectic Lie superalgebra $$\mathfrak{p}$$ of type $$P$$. Until this paper under review, it has been neither nowhere found in the mathematics literature nor has it been constructed. The Lie superalgebra $$U_q(\mathfrak{p}_n)$$ is a quantization of a Lie bisuperalgebra structure on $$\mathfrak{p}_n$$, where the authors study some of its basic properties in this manuscript. The authors also introduce the periplectic $$q$$-Brauer algebra and prove that it is the centralizer of the $$U_q(\mathfrak{p}_n)$$-module structure on $$C_q(n|n)^{\otimes \ell}$$ (Theorem 5.5, page 84). They end by proposing a definition for a new periplectic $$q$$-Schur superalgebra of type $$P$$ (Definition 5.7, page 84). Ribbon braided module categories, quantum symmetric pairs and Knizhnik-Zamolodchikov equations https://zbmath.org/1472.17049 2021-11-25T18:46:10.358925Z "De Commer, Kenny" https://zbmath.org/authors/?q=ai:de-commer.kenny "Neshveyev, Sergey" https://zbmath.org/authors/?q=ai:neshveyev.sergey-v "Tuset, Lars" https://zbmath.org/authors/?q=ai:tuset.lars "Yamashita, Makoto" https://zbmath.org/authors/?q=ai:yamashita.makoto According to the Tannaka-Krein principle for quantum groups there is a duality between Hopf algebras and tensor categories with duals. This has been constructed by Drinfeld and verified for type $$A$$ quantum groups. The current paper formulates a framework for type $$B$$ quantum groups in terms of quantum symmetric pairs, the Knizhnik-Zamolodchikov equation and the Drinfeld twisting. Explicitly they construct three tensor categories for modules of quantum groups in type $$B$$ using the aforementioned structures, and show their equivalence for the rank one case. The key is to understand the twist in all three constructions and the authors conjecture that the equivalence holds in general. Polynomial super representations of $$U_q^{\mathrm{res}}(\mathfrak{gl}_{m\mid n})$$ at roots of unity https://zbmath.org/1472.17050 2021-11-25T18:46:10.358925Z "Du, Jie" https://zbmath.org/authors/?q=ai:du.jie "Lin, Yanan" https://zbmath.org/authors/?q=ai:lin.yanan "Zhou, Zhongguo" https://zbmath.org/authors/?q=ai:zhou.zhongguo Summary: As a homomorphic image of the hyperalgebra $$U_{q,R}(m|n)$$ associated with the quantum linear supergroup $$U_{\upsilon}(\mathfrak{gl}_{m|n})$$, we first give a presentation for the $$q$$-Schur superalgebra $$S_{q,R}(m|n,r)$$ over a commutative ring $$R$$. We then develop a criterion for polynomial supermodules of $$U_{q,F}(m|n)$$ over a field $$F$$ and use this to determine a classification of polynomial irreducible supermodules at roots of unity. This also gives classifications of irreducible $$S_{q,F}(m|n,r)$$-supermodules for all $$r$$. As an application when $$m=n\geq r$$ and motivated by the beautiful work of \textit{J. Brundan} and \textit{J. Kujawa} [J. Algebr. Comb. 18, No. 1, 13--39 (2003; Zbl 1043.20006)] in the classical (non-quantum) case, we provide a new proof for the Mullineux conjecture related to the irreducible modules over the Hecke algebra $$H_{q^2,F}({\mathfrak{S}}_r)$$; see \textit{J. Brundan} [Proc. Lond. Math. Soc., III. Ser. 77, No. 3, 551--581 (1998; Zbl 0904.20007)] for a proof without using the super theory. Equivalences between three presentations of orthogonal and symplectic Yangians https://zbmath.org/1472.17051 2021-11-25T18:46:10.358925Z "Guay, Nicolas" https://zbmath.org/authors/?q=ai:guay.nicolas "Regelskis, Vidas" https://zbmath.org/authors/?q=ai:regelskis.vidas "Wendlandt, Curtis" https://zbmath.org/authors/?q=ai:wendlandt.curtis Summary: We prove the equivalence of two presentations of the Yangian $$Y(\mathfrak{g})$$ of a simple Lie algebra $$\mathfrak{g}$$, and we also show the equivalence with a third presentation when $$\mathfrak{g}$$ is either an orthogonal or a symplectic Lie algebra. As an application, we obtain an explicit correspondence between two versions of the classification theorem of finite-dimensional irreducible modules for orthogonal and symplectic Yangians. Representations of twisted Yangians of types B, C, D. II https://zbmath.org/1472.17052 2021-11-25T18:46:10.358925Z "Guay, N." https://zbmath.org/authors/?q=ai:guay.nicolas "Regelskis, V." https://zbmath.org/authors/?q=ai:regelskis.vidas "Wendlandt, C." https://zbmath.org/authors/?q=ai:wendlandt.curtis This paper is a continuation of the work [\textit{N. Guay} et al., Sel. Math., New Ser. 23, No. 3, 2071--2156 (2017; Zbl 1375.81126)], in which the structure and representation theory of twisted Yangians were analyzed in detail. In this paper, the authors study the classification of irreducible finite-dimensional representations of twisted Yangians for certain symmetric pairs considered in the previous work, obtaining a full classification for the pairs $$(\mathfrak{so}(n),\mathfrak{so}(n-2)\oplus \mathfrak{so}(2))$$ and $$(\mathfrak{so}(2n+1),\mathfrak{so}(2n))$$. Relevant partial results for the classification of other pairs, that will be the subject of future work, are also discussed. Isomorphism between the $$R$$-matrix and Drinfeld presentations of Yangian in types $$B$$, $$C$$ and $$D$$ https://zbmath.org/1472.17053 2021-11-25T18:46:10.358925Z "Jing, Naihuan" https://zbmath.org/authors/?q=ai:jing.naihuan "Liu, Ming" https://zbmath.org/authors/?q=ai:liu.ming.2 "Molev, Alexander" https://zbmath.org/authors/?q=ai:molev.alexander-i Summary: It is well-known that the Gauss decomposition of the generator matrix in the $$R$$-matrix presentation of the Yangian in type $$A$$ yields generators of its Drinfeld presentation. Defining relations between these generators are known in an explicit form, thus providing an isomorphism between the presentations. It has been an open problem since the pioneering work of Drinfeld to extend this result to the remaining types. We give a solution for the classical types $$B$$, $$C$$ and $$D$$ by constructing an explicit isomorphism between the $$R$$-matrix and Drinfeld presentations of the Yangian. It is based on an embedding theorem which allows us to consider the Yangian of rank $${n-1}$$ as a subalgebra of the Yangian of rank $$n$$ of the same type. Monoidal categories of modules over quantum affine algebras of type $$A$$ and $$B$$ https://zbmath.org/1472.17054 2021-11-25T18:46:10.358925Z "Kashiwara, Masaki" https://zbmath.org/authors/?q=ai:kashiwara.masaki "Kim, Myungho" https://zbmath.org/authors/?q=ai:kim.myungho "Oh, Se-jin" https://zbmath.org/authors/?q=ai:oh.se-jin In this paper the authors apply the general procedure of the Khovanov-Lauda-Rouquier type quantum affine Schur-Weyl duality in order to obtain an exact tensor functor from the category $$\mathcal A$$ of finite-dimensional graded modules over the symmetric quiver Hecke algebra of type $$A_\infty$$ to the category $$\mathcal C_{B_n^{(1)}}$$ of finite-dimensional integrable modules over the quantum affine algebra of type $$B_n^{(1)}$$. This functor is shown to factor through a certain localization $$\mathcal T_{2n}$$ of the category $$\mathcal A$$, which is related to the categories of finite-dimensional integrable modules over the quantum affine algebra of type $$A^{(t)}_{2n-1}$$, $$t = 1, 2$$. This gives rise to a ring isomorphism between the Grothendieck rings of appropriate categories of modules over quantum affine algebras of types $$A$$ and $$B$$, which induces a bijection between the sets of classes of simple objects. Finite dimensional simple modules of $$(Q$$,\textbf{Q})-current algebras https://zbmath.org/1472.17055 2021-11-25T18:46:10.358925Z "Kodera, Ryosuke" https://zbmath.org/authors/?q=ai:kodera.ryosuke "Wada, Kentaro" https://zbmath.org/authors/?q=ai:wada.kentaro In [Publ. Res. Inst. Math. Sci. 52, No. 4, 497--555 (2016; Zbl 1396.20053)] the second author introduced the so-called $$(q, \mathbf{Q})$$-current algebra to study representation theory of cyclotomic $$q$$-Schur algebras. Here $$q$$ is a nonzero complex number and $$\mathbf{Q} = (Q_1, Q_2, \cdots, Q_n)$$ is a tuple of complex numbers. The present paper aims to classify finite-dimensional irreducible representations of the $$(q, \mathbf{Q})$$-current algebra under the condition that $$q$$ is not a root of unity. For this purpose, the authors make connections with the shifted quantum loop algebras of \textit{M. Finkelberg} and \textit{A. Tsymbaliuk} [Prog. Math. 330, 133--304 (2019; Zbl 1436.17021)]: \begin{itemize} \item The $$(q, \mathbf{Q})$$-current algebra is a subalgebra of a dominantly shifted quantum loop algebra of type $$A_n$$, with the shift determined by the tuple $$\mathbf{Q}$$. It is also a subalgebra of the non-shifted quantum loop algebra $$U_q(L\mathfrak{sl}_{n+1})$$. \end{itemize} Based on the coproducts for shifted quantum loop algebras, the authors reduce the classification problem to the rank-one case $$n = 1$$. The main result of this paper can be stated as follows: \begin{itemize} \item Each finite-dimensional irreducible representation of the $$(q, \mathbf{Q})$$-current algebra of $$\mathfrak{sl}_2$$ is the restriction of a tensor product module $$D \otimes V$$, where $$D$$ is a one-dimensional module over a shifted quantum loop algebra, and $$V$$ is a finite-dimensional irreducible $$U_q(L\mathfrak{sl}_2)$$-module. \end{itemize} Its proof is in the spirit of \textit{V. Chari} and \textit{A. Pressley} [Commun. Math. Phys. 142, No. 2, 261--283 (1991; Zbl 0739.17004)] on a similar classification problem for $$U_q(L\mathfrak{sl}_2)$$. We remark that $$U_q(L\mathfrak{sl}_2)$$ admits precisely two one-dimensional representations, while in the shifted case there exist infinitely many one-dimensional representations. The Steinberg-Lusztig tensor product theorem, Casselman-Shalika, and LLT polynomials https://zbmath.org/1472.17056 2021-11-25T18:46:10.358925Z "Lanini, Martina" https://zbmath.org/authors/?q=ai:lanini.martina "Ram, Arun" https://zbmath.org/authors/?q=ai:ram.arun Summary: In this paper we establish a Steinberg-Lusztig tensor product theorem for abstract Fock space. This is a generalization of the type A result of Leclerc-Thibon and a Grothendieck group version of the Steinberg-Lusztig tensor product theorem for representations of quantum groups at roots of unity. Although the statement can be phrased in terms of parabolic affine Kazhdan-Lusztig polynomials and thus has geometric content, our proof is combinatorial, using the theory of crystals (Littelmann paths). We derive the Casselman-Shalika formula as a consequence of the Steinberg-Lusztig tensor product theorem for abstract Fock space. Cartan subalgebras for quantum symmetric pair coideals https://zbmath.org/1472.17057 2021-11-25T18:46:10.358925Z "Letzter, Gail" https://zbmath.org/authors/?q=ai:letzter.gail Summary: There is renewed interest in the coideal subalgebras used to form quantum symmetric pairs because of recent discoveries showing that they play a fundamental role in the representation theory of quantized enveloping algebras. However, there is still no general theory of finite-dimensional modules for these coideals. In this paper, we establish an important step in this direction: we show that every quantum symmetric pair coideal subalgebra admits a quantum Cartan subalgebra which is a polynomial ring that specializes to its classical counterpart. The construction builds on Kostant and Sugiura's classification of Cartan subalgebras for real semisimple Lie algebras via strongly orthogonal systems of positive roots. We show that these quantum Cartan subalgebras act semisimply on finite-dimensional unitary modules and identify particularly nice generators of the quantum Cartan subalgebra for a family of examples. $$\mathfrak{sl}_N$$-web categories and categorified skew Howe duality https://zbmath.org/1472.17058 2021-11-25T18:46:10.358925Z "Mackaay, Marco" https://zbmath.org/authors/?q=ai:mackaay.marco "Yonezawa, Yasuyoshi" https://zbmath.org/authors/?q=ai:yonezawa.yasuyoshi Summary: In this paper we show how the colored Khovanov-Rozansky $$\mathfrak{sl}_N$$-matrix factorizations, due to \textit{Hao Wu} [Diss. Math. 499, 215 p. (2014; Zbl 1320.57020)] and \textit{Y. Yonezawa} [Nagoya Math. J. 204, 69--123 (2011; Zbl 1271.57033)], can be used to categorify the type $$A$$ quantum skew Howe duality defined by Cautis, Kamnitzer and Morrison in [\textit{S. Cautis} et al., Math. Ann. 360, No. 1--2, 351--390 (2014; Zbl 1387.17027)]. In particular, we define $$\mathfrak{sl}_N$$-web categories and 2-representations of Khovanov and Lauda's categorical quantum $$\mathfrak{sl}_m$$ on them. We also show that this implies that each such web category is equivalent to the category of finite-dimensional graded projective modules over a certain type $$A$$ cyclotomic KLR-algebra. Folding KLR algebras https://zbmath.org/1472.17059 2021-11-25T18:46:10.358925Z "McNamara, Peter J." https://zbmath.org/authors/?q=ai:mcnamara.peter-j From the introduction: In Lie theory, the process of folding by a Dynkin diagram automorphism is a technique that can be used to extend theorems originally proved for symmetric Cartan data to all Lie types. This paper develops the theory of folding for KLR algebras. KLR algebras (named after Khovanov, Lauda and Rouquier) are a family of graded algebras introduced in [\textit{M. Khovanov} and \textit{A. D. Lauda}, Represent. Theory 13, 309--347 (2009; Zbl 1188.81117); \textit{R. Rouquier}, 2-Kac-Moody algebras. \url{arXiv:0812.5023}] for the purposes of categorifying quantised enveloping algebras. They also appear in the literature under the name of quiver Hecke algebras. While KLR algebras exist for arbitrary symmetrisable Cartan data, it is known that the KLR algebras in symmetric types have a richer theory with more desirable properties. This is usually a consequence of the geometric interpretation of symmetric KLR algebras or through the theory of R-matrices of [\textit{S.-J. Kang} et al., Invent. Math. 211, No. 2, 591--685 (2018; Zbl 1407.81108); correction ibid. 216, No. 2, 597--599 (2019)]. Thus we believe that incorporating a diagram automorphism into the narrative and using the technique of folding is an important way to think about categorified quantum groups in nonsymmetric types, as an alternative to working with nonsymmetric KLR algebras. The folding constructions performed in this paper are modelled on those of [\textit{G. Lusztig}, Introduction to quantum groups. Boston: Birkhäuser (1993; Zbl 0788.17010), Chapter 2], where Lusztig constructs the canonical basis using perverse sheaves and a diagram automorphism. In fact when one considers the geometric interpretation of KLR algebras as extension algebras and works over the field $$\overline{\mathcal Q}_l$$, the category $$\mathcal P_\nu$$ which we study is equivalent to Lusztig's $$\tilde{\mathcal Q}_V$$. Our main aim in this paper is to develop the theory of folded KLR algebras, to a depth comparable to that of [Khovanov-Lauda, loc. cit.]. Our main theorems are the categorification theorems, Theorems 6.1 and 6.2. Our proof is different from Khovanov-Lauda in that we do not rely on the quantum Gabber-Kac theorem. Instead, we first identify an appropriate class of simple objects with the crystal $$B(\infty)$$, using the Kashiwara-Saito characterisation of $$B(\infty)$$. This identification is Theorem 10.8, and generalises [\textit{A. D. Lauda} and \textit{M. Vazirani}, Adv. Math. 228, No. 2, 803--861 (2011; Zbl 1246.17017)]. We also conclude with a section proving that this categorification provides us with a basis of canonical type. This concept of a basis of canonical type is motivated from the definition given in [\textit{Pierre Baumann}, The canonical basis and the quantum Frobenius morphism. \url{arXiv:1201.0303}] and is a strengthening of the notion of a perfect basis. An application of this work to the KLR categorification of cluster algebras will appear in a forthcoming paper of the author [Cluster monomials are dual canonical. Preprint]. Existence of Kirillov-Reshetikhin crystals of type $$G_2^{(1)}$$ and $$D_4^{(3)}$$ https://zbmath.org/1472.17060 2021-11-25T18:46:10.358925Z "Naoi, Katsuyuki" https://zbmath.org/authors/?q=ai:naoi.katsuyuki Let $$U_q'(\mathfrak{g})$$ be a quantum affine algebra without the degree operator. We denote by $$W^{r,\ell}$$ the Kirillov-Reshetikhin module over $$U_q'(\mathfrak{g})$$, where $$r$$ corresponds to a node of the Dynkin diagram of $$U_q'(\mathfrak{g})$$ except the node 0, and $$\ell$$ is a positive integer. The Kirillov-Reshetikhin modules $$W^{r,\ell}$$ are central and important in the research area of quantum affine algebras. It was conjectured that a Kirillov-Reshetikhin module $$W^{r,\ell}$$ has a crystal base in the sense of Kashiwara. It was proved that $$W^{r,\ell}$$ have crystal (pseudo)bases in various types but it is still open in some other cases. In the paper under review, the author proves that every Kirillov-Reshetikhin module of type $$G_2^{(1)}$$ and $$D_4^{(3)}$$ has a crystal pseudobase. Note that the existence of a crystal pseudobase of $$W^{1,\ell}$$ of type type $$G_2^{(1)}$$ and $$D_4^{(3)}$$ was proved in [\textit{S. Yamane}, J. Algebra 210, No. 2, 440--486 (1998; Zbl 0929.17013)] for type $$G_2^{(1)}$$ and [\textit{M. Kashiwara} et al., J. Algebra 317, No. 1, 392--423 (2007; Zbl 1140.17012)] for type $$D_4^{(3)}$$. Thus the author focuses on the case for the remaining node $$2$$, and proves the existence of a crystal pseudobase of $$W^{2, \ell}$$ by investigating the embedding $$W^{2,\ell} \hookrightarrow W^{2, \ell-1}_{q^{-1}} \otimes W^{2,1}_{q^{\ell-1}}$$ for induction. Webs and $$q$$-Howe dualities in types BCD https://zbmath.org/1472.17061 2021-11-25T18:46:10.358925Z "Sartori, Antonio" https://zbmath.org/authors/?q=ai:sartori.antonio "Tubbenhauer, Daniel" https://zbmath.org/authors/?q=ai:tubbenhauer.daniel Summary: We define web categories describing intertwiners for the orthogonal and symplectic Lie algebras and, in the quantized setup, for certain orthogonal and symplectic coideal subalgebras. They generalize the Brauer category and allow us to prove quantum versions of some classical type BCD Howe dualities. Description of 2-local and local derivations on some Lie rings of skew-adjoint matrices https://zbmath.org/1472.17062 2021-11-25T18:46:10.358925Z "Ayupov, Sh. A." https://zbmath.org/authors/?q=ai:ayupov.sh-a "Arzikulov, F. N." https://zbmath.org/authors/?q=ai:arzikulov.farhodjon-nematjonovich Summary: In the present paper, we prove that every 2-local inner derivation on the Lie ring of skew-symmetric matrices over a commutative ring is an inner derivation. We also apply our technique to various Lie algebras of infinite-dimensional skew-adjoint matrix-valued maps on a set and prove that every 2-local spatial derivation on such algebras is a spatial derivation. A similar technique is applied to the same Lie algebras and proved that every local spatial derivation on such algebras is a spatial derivation. Biderivations and commutative post-Lie algebra structures on the Lie algebra $$\mathcal{W}(a,b)$$ https://zbmath.org/1472.17063 2021-11-25T18:46:10.358925Z "Tang, Xiaomin" https://zbmath.org/authors/?q=ai:tang.xiaomin.1 Summary: For $$a,b \in \mathbb{C}$$, the Lie algebra $$\mathcal{W}(a,b)$$ is the semidirect product of the Witt algebra and a module of the intermediate series. In this paper, all biderivations of $$\mathcal{W}(a,b)$$ are determined. Surprisingly, these Lie algebras have symmetric (and skew-symmetric) non-inner biderivations. As an application, commutative post-Lie algebra structures on $$\mathcal{W}(a,b)$$ are obtained. 2-local automorphisms on basic classical Lie superalgebras https://zbmath.org/1472.17064 2021-11-25T18:46:10.358925Z "Yu, Li" https://zbmath.org/authors/?q=ai:yu.li|yu.li.1 "Wang, Ying" https://zbmath.org/authors/?q=ai:wang.ying.5 "Chen, Hai Xian" https://zbmath.org/authors/?q=ai:chen.haixian "Nan, Ji Zhu" https://zbmath.org/authors/?q=ai:nan.jizhu Summary: Let $$G$$ be a basic classical Lie superalgebra except $$A(n, n)$$ and $$D(2, 1, \alpha)$$ over the complex number field $$\mathbb{C}$$. Using existence of a non-degenerate invariant bilinear form and root space decomposition, we prove that every 2-local automorphism on $$G$$ is an automorphism. Furthermore, we give an example of a 2-local automorphism which is not an automorphism on a subalgebra of Lie superalgebra $$\mathrm{spl}(3, 3)$$. On self-similar Lie algebras and virtual endomorphisms https://zbmath.org/1472.17065 2021-11-25T18:46:10.358925Z "Futorny, Vyacheslav" https://zbmath.org/authors/?q=ai:futorny.vyacheslav-m "Kochloukova, Dessislava H." https://zbmath.org/authors/?q=ai:kochloukova.dessislava-hristova "Sidki, Said N." https://zbmath.org/authors/?q=ai:sidki.said-n|sidki.said-najati The notion of self-similar groups plays an important role in modern group theory. Nekrashevich and Sidki showed that self-similarity of groups is closely related to the notion of a \textit{virtual endomorphism}, which is a homomorphism $$\psi:H\to G$$, where $$H$$ is a subgroup of finite index in the group $$G$$, see e.g. [\textit{V. Nekrashevych}, Self-similar groups. Providence, RI: AMS (2005; Zbl 1087.20032)] A Lie algebra $$L$$ over a field $$k$$ is called \textit{self-similar} if there exists an embedding $\psi: L \to (X\otimes_k L )\leftthreetimes \mathrm{Der} X,$ where $$X$$ is a commutative algebra, $$\mathrm{Der} X$$ being the Lie algebra of derivations [\textit{L. Bartholdi}, J. Eur. Math. Soc. (JEMS) 17, No. 12, 3113--3151 (2015; Zbl 1350.17005).] Now, the authors start to study self-similarity of Lie algebras in terms of relation to \textit{virtual endomorphisms}, which are defined similarly as homomorphisms $$H\to L$$, where $$H$$ is a subalgebra of finite codimension in a Lie algebra $$L$$. In some particular but important cases, the authors suggest a relation of self-similar Lie algebras with virtual endomorphisms, but the respective subalgebras $$H\subset L$$ are ideals; and a complete correspondence between self-similarity and virtual endomorphisms is not established. In case of group theory, a similar relation is easily established using geometric methods. The authors discover an unexpected fact that the simple Lie algebra $$\mathfrak{sl}_{n+1}$$, where the positive characteristic is not dividing $$n+1$$, is self-similar. They also construct examples of self-similar metabelian Lie algebras of the homological type $$FP_n$$. The first cohomology of $$\mathfrak{sl}(2, 1)$$ with coefficients in $$\chi$$-reduced Kac modules and simple modules https://zbmath.org/1472.17066 2021-11-25T18:46:10.358925Z "Wang, Shujuan" https://zbmath.org/authors/?q=ai:wang.shujuan "Liu, Wende" https://zbmath.org/authors/?q=ai:liu.wende Summary: Over a field of characteristic $$p > 2$$, the first cohomology of the special linear Lie superalgebra $$\mathfrak{sl}(2, 1)$$ with coefficients in all $$\chi$$-reduced Kac modules and simple modules is determined by use of the weight space decompositions of these modules relative to the standard Cartan subalgebra of $$\mathfrak{sl}(2, 1)$$. Character formulas for a class of simple restricted modules over the simple Lie superalgebras of Witt type https://zbmath.org/1472.17067 2021-11-25T18:46:10.358925Z "Yao, Yu-Feng" https://zbmath.org/authors/?q=ai:yao.yufeng.2|yao.yufeng.1 Summary: Let $$F$$ be an algebraically closed field of prime characteristic, and $$W(m, n, \mathbf{1})$$ be the simple restricted Lie superalgebra of Witt type over $$F$$, which is the Lie superalgebra of superderivations of the superalgebra $$\mathfrak{A}(m;\mathbf{1}) \otimes \wedge(n)$$, where $$\mathfrak{A}(m;\mathbf{1})$$ is the truncated polynomial algebra with $$m$$ indeterminants and $$\wedge (n)$$ is the Grassmann algebra with $$n$$ indeterminants. In this paper, the author determines the character formulas for a class of simple restricted modules of $$W(m, n, \mathbf{1})$$ with atypical weights of type I. Minimal $$W$$-superalgebras and the modular representations of basic Lie superalgebras https://zbmath.org/1472.17068 2021-11-25T18:46:10.358925Z "Zeng, Yang" https://zbmath.org/authors/?q=ai:zeng.yang "Shu, Bin" https://zbmath.org/authors/?q=ai:shu.bin Summary: Let $$\mathfrak g = \mathfrak g_{\bar 0} + \mathfrak g_{\bar 1}$$ be a basic Lie superalgebra over $$\mathbb{C}$$, and $$e$$ a minimal nilpotent element in $$\mathfrak g_{\bar 0}$$. Set $$W_\chi'$$ to be the refined $$W$$-superalgebra associated with the pair $$\mathfrak g,e)$$, which is called a minimal $$W$$-superalgebra. In this paper we present a set of explicit generators of minimal $$W$$-superalgebras and the commutators between them. In virtue of this, we show that over an algebraically closed field $$\mathbb k$$ of characteristic $$p \gg 0$$, the lower bounds of dimensions in the modular representations of basic Lie superalgebras with minimal nilpotent $$p$$-characters are attainable. Such lower bounds are indicated in [\textit{W.-Q. Wang} and \textit{L. Zhao}, Proc. Lond. Math. Soc. (3) 99, No. 1, 145--167 (2009; Zbl 1176.17013)] as the super Kac-Weisfeiler property. On support varieties for Lie superalgebras and finite supergroup schemes https://zbmath.org/1472.17069 2021-11-25T18:46:10.358925Z "Drupieski, Christopher M." https://zbmath.org/authors/?q=ai:drupieski.christopher-m "Kujawa, Jonathan R." https://zbmath.org/authors/?q=ai:kujawa.jonathan-r|kujawa.jonathan-robert Summary: We study the spectrum of the cohomology rings of cocommutative Hopf superalgebras, restricted and non-restricted Lie superalgebras, and finite supergroup schemes. We also investigate support varieties in these settings and demonstrate that they have the desirable properties of such a theory. We completely characterize support varieties for finite supergroup schemes over algebraically closed fields of characteristic zero, while for non-restricted Lie superalgebras in positive characteristic we obtain descriptions that are suggestive of varieties previously defined by Duflo and Serganova in characteristic zero but which have no known cohomological interpretation. Our computations for restricted Lie superalgebras and infinitesimal supergroup schemes provide natural generalizations of foundational results of Friedlander and Parshall and of Bendel, Friedlander, and Suslin in the classical setting. Deformations on the twisted Heisenberg-Virasoro algebra https://zbmath.org/1472.17070 2021-11-25T18:46:10.358925Z "Liu, Dong" https://zbmath.org/authors/?q=ai:liu.dong "Pei, Yufeng" https://zbmath.org/authors/?q=ai:pei.yufeng Summary: With the cohomology results on the Virasoro algebra, the authors determine the second cohomology group on the twisted Heisenberg-Virasoro algebra, which gives all deformations on the twisted Heisenberg-Virasoro algebra. On the finite generation of relative cohomology for Lie superalgebras https://zbmath.org/1472.17071 2021-11-25T18:46:10.358925Z "Maurer, Andrew" https://zbmath.org/authors/?q=ai:maurer.andrew Summary: The author establishes the finite generation of the cohomology ring of a classical Lie superalgebra relative to an even subsuperalgebra. A spectral sequence is constructed to provide conditions for when this relative cohomology ring is Cohen-Macaulay. With finite generation established, support varieties for modules are defined via the relative cohomology, which generalize those of \textit {B. D. Boe} et al. [Trans. Am. Math. Soc. 362, No. 12, 6551--6590 (2010; Zbl 1253.17012)]. The associated Lie algebra of a right-angled Coxeter group https://zbmath.org/1472.17072 2021-11-25T18:46:10.358925Z "Verëvkin, Ya. A." https://zbmath.org/authors/?q=ai:verevkin.ya-a Summary: We study the lower central series of a right-angled Coxeter group $$\mathrm{RC}_{\mathcal{K}}$$ and the associated graded Lie algebra $$L\left(\mathrm{RC}_{\mathcal{K}}\right)$$. The latter is related to the graph Lie algebra $$L_{\mathcal{K}}$$. We give an explicit combinatorial description of the first three consecutive factors of the lower central series of the group $$\mathrm{RC}_{\mathcal{K}}$$. On Hom-Lie superbialgebras https://zbmath.org/1472.17073 2021-11-25T18:46:10.358925Z "Fadous, Mohamed" https://zbmath.org/authors/?q=ai:fadous.mohamed "Mabrouk, Sami" https://zbmath.org/authors/?q=ai:mabrouk.sami "Makhlouf, Abdenacer" https://zbmath.org/authors/?q=ai:makhlouf.abdenacer Summary: The purpose of this paper is to generalize to $$\mathbb{Z}_2$$-graded case the study of Hom-Lie bialgebras which were discussed first by D. Yau, then by Y. Sheng and C. Bai. We first provide various constructions of Hom-Lie superbialgebras and a classification of 3-dimensional Hom-Lie superbialgebras with 2-dimensional even part. Moreover, we study coboundary and triangular Hom-Lie superbialgebras, as well as infinitesimal deformations of the cobracket. Furthermore, following Sheng and Bai we define matched pairs, Manin supertriples and discuss their relationships. On Hom-pre-Lie bialgebras https://zbmath.org/1472.17074 2021-11-25T18:46:10.358925Z "Liu, Shanshan" https://zbmath.org/authors/?q=ai:liu.shanshan "Makhlouf, Abdenacer" https://zbmath.org/authors/?q=ai:makhlouf.abdenacer "Song, Lina" https://zbmath.org/authors/?q=ai:song.lina In [\textit{Q. Sun} and \textit{H. Li}, Commun. Algebra 45, No. 1, 105--120 (2017; Zbl 1418.17068)], a notion of a Hom-pre-Lie bialgebra was introduced under the terminology of Hom-left-symmetric bialgebra, but did not enjoy a coboundary theory. This, in part, motivated the authors of the paper under review to study Hom-pre-Lie bialgebras that enjoy a rich structure theory with the goal of giving a systematic study of the bialgebra theory for Hom-pre-Lie algebras. The goal was achieved since their Hom-pre-Lie bialgebra structure has the following properties (1) equivalent to a Manin triple for Hom-pre-Lie algebras as well as certain matched pair of Hom-pre-Lie algebras (this is done in Section 3); (2) Hom-$${\mathfrak{s}}$$-matrices (see Section 4 for the precise definitions) were defined to produce Hom-pre-Lie bialgebras; and (3) $$\mathcal{O}$$-operators on Hom-pre-Lie algebras were defined to give Hom-$${\mathfrak{s}}$$-matrices in the semidirect product Hom-pre-Lie algebras (this is done in Section 5). Another approach to Hom-Lie bialgebras via Manin triples https://zbmath.org/1472.17075 2021-11-25T18:46:10.358925Z "Tao, Yi" https://zbmath.org/authors/?q=ai:tao.yi "Bai, Chengming" https://zbmath.org/authors/?q=ai:bai.chengming "Guo, Li" https://zbmath.org/authors/?q=ai:guo.li A Hom-Lie algebra $$(\mathfrak{g},[-,-],\phi)$$ consists of a vector space $$\mathfrak{g}$$, a skew-symmetric linear map $$[-,-]:\mathfrak{g}\otimes\mathfrak{g} \to\mathfrak{g}$$, and a linear endomorphism $$\phi:\mathfrak{g}\to\mathfrak{g}$$ such that the Hom-Jacobi identity $$[\phi(x),[y,z]]+[\phi(y),[z,x]]+[\phi(z),[x,y]]=0$$ is satisfied for any elements $$x,y,z\in\mathfrak{g}$$. A representation of such a Hom-Lie algebra is a triple $$(V,\beta,\rho)$$ consisting of a vector space $$V$$ (over the same ground field as $$\mathfrak{g}$$), a linear endomorphism $$\beta:V\to V$$, and a linear transformation $$\rho:\mathfrak{g}\to\mathrm{End}(V)$$ such that the identities $\rho(\phi(x))\circ\beta=\beta\circ\rho(x)$ and $\rho([x,y])\circ\beta =\rho(\phi(x))\circ\rho(y)-\rho(\phi(y))\circ\rho(x)$ are satisfied for any elements $$x,y\in\mathfrak{g}$$. \textit{S. Benayadi} and \textit{A. Makhlouf} [J. Geom. Phys. 76, 38--60 (2014; Zbl 1331.17028)] observed that the triple $$(V^*,\beta^*,\rho^*)$$ is not always a representation on the linear dual $$V^*$$ of $$V$$. In the paper under review this definition is modified by replacing $$\rho^*$$ by $$\rho^0:=\rho^*\circ\phi$$, and then it is characterized exactly when the triple $$(V^*,\beta^*,\rho^0)$$ is a representation. In particular, it is shown that the condition $$\rho(\phi^2(x))=\rho(x)$$ for every element $$x\in\mathfrak{g}$$ is sufficient for $$(V^*,\beta^*,\rho^0)$$ being a representation (such representations are called weakly involutive), and conversely, if $$\beta$$ is invertible, then every representation $$(V^*,\beta^*,\rho^0)$$ is weakly involutive. Moreover, a Hom-Lie algebra is called weakly involutive if its adjoint representation $$\mathrm{ad}$$ is weakly involutive. Then the authors prove that if a Hom-Lie algebra $$(\mathfrak{g}, [-,-],\phi)$$ admits a non-degenerate invariant symmetric bilinear form, then $$(\mathfrak{g},[-,-],\phi)$$ is weakly involutive and its adjoint representation $$(\mathfrak{g},\phi,\mathrm{ad})$$ and its coadjoint representation $$(\mathfrak{g}^*,\phi^*,\mathrm{ad}^0)$$ are equivalent, and conversely, if $$(\mathfrak{g},[-,-],\phi)$$ is weakly involutive and its adjoint representation $$(\mathfrak{g},\phi,\mathrm{ad})$$ and its coadjoint representation $$(\mathfrak{g}^*,\phi^*,\mathrm{ad}^0)$$ are equivalent, then $$(\mathfrak{g}, [-,-],\phi)$$ admits a non-degenerate invariant bilinear form. \textit{Y. Sheng} and the second author of the paper under review [J. Algebra 399, 232--250 (2014; Zbl 1345.17002)] introduced a matched pair of Hom-Lie algebras $$(\mathfrak{g},\mathfrak{g}^\prime;\rho,\rho^\prime)$$ consisting of two Hom-Lie algebras $$(\mathfrak{g},[-,-],\phi)$$ and $$(\mathfrak{g}^\prime, [-,-]^\prime,\phi^\prime)$$ with representations $$(\mathfrak{g}^\prime, \phi^\prime,\rho)$$ of $$(\mathfrak{g},[-,-],\phi)$$ and $$(\mathfrak{g},\phi, \rho^\prime)$$ of $$(\mathfrak{g}^\prime,[-,-]^\prime,\phi^\prime)$$, respectively, that satisfy two additional compatibility conditions. The authors define then a Manin triple of Hom-Lie algebras as a triple $$(\mathfrak{k};\mathfrak{g},\mathfrak{g}')$$ of Hom-Lie algebras such that $$\mathfrak{k}=\mathfrak{g}\oplus\mathfrak{g}'$$ as vector spaces and $$\mathfrak{k}$$ admits a non-degenerate invariant symmetric bilinear form for which $$\mathfrak{g}$$ and $$\mathfrak{g}'$$ are isotropic Hom-Lie subalgebras. Finally, a Hom-Lie bialgebra is defined as a pair of two weakly involutive Hom-Lie algebras $$(\mathfrak{g},\mathfrak{g}^*)$$ that satisfy a compatibility condition, and Hom-Lie bialgebras are characterized in terms of matched pairs and Manin triples. Note that the author's definition of a Hom-Lie bialgebra is different from other definitions in the literature (see [\textit{D. Yau}, Int. Electron. J. Algebra 17, 11--45 (2015; Zbl 1323.16027); \textit{Y. Sheng} and \textit{C. Bai}, J. Algebra 399, 232--250 (2014; Zbl 1345.17002); \textit{L. Cai} and \textit{Y. Sheng}, Sci. China, Math. 61, No. 9, 1553--1566 (2018; Zbl 1398.16032)]. As an application the authors study boundary Hom-Lie bialgebras, quasi-triangular Hom-Lie bialgebras, and solutions of the classical Hom-Yang-Baxter equation. In the final section of the paper solutions of the classical Hom-Yang-Baxter equation are constructed by using $$\mathcal{O}$$-operators associated to a weakly involutive representation of a Hom-Lie algebra and Hom-left-symmetric algebras. Nambu structures and associated bialgebroids https://zbmath.org/1472.17076 2021-11-25T18:46:10.358925Z "Basu, Samik" https://zbmath.org/authors/?q=ai:basu.samik "Basu, Somnath" https://zbmath.org/authors/?q=ai:basu.somnath "Das, Apurba" https://zbmath.org/authors/?q=ai:das.apurba "Mukherjee, Goutam" https://zbmath.org/authors/?q=ai:mukherjee.goutam Summary: We investigate higher-order generalizations of well known results for Lie algebroids and bialgebroids. It is proved that $$n$$-Lie algebroid structures correspond to $$n$$-ary generalization of Gerstenhaber algebras and are implied by $$n$$-ary generalization of linear Poisson structures on the dual bundle. A Nambu-Poisson manifold (of order $$n>2$$) gives rise to a special bialgebroid structure which is referred to as a weak Lie-Filippov bialgebroid (of order $$n$$). It is further demonstrated that such bialgebroids canonically induce a Nambu-Poisson structure on the base manifold. Finally, the tangent space of a Nambu Lie group gives an example of a weak Lie-Filippov bialgebroid over a point. Conformal classical Yang-Baxter equation, $$S$$-equation and $${\mathcal{O}}$$-operators https://zbmath.org/1472.17077 2021-11-25T18:46:10.358925Z "Hong, Yanyong" https://zbmath.org/authors/?q=ai:hong.yanyong "Bai, Chengming" https://zbmath.org/authors/?q=ai:bai.chengming Summary: Conformal classical Yang-Baxter equation and $$S$$-equation naturally appear in the study of Lie conformal bialgebras and left-symmetric conformal bialgebras. In this paper, they are interpreted in terms of a kind of operators, namely $$\mathcal{O}$$-operators in the conformal sense. Explicitly, the skew-symmetric part of a conformal linear map $$T$$ where $$T_0=T_\lambda \mid_{\lambda =0}$$ is an $${\mathcal{O}}$$-operator in the conformal sense is a skew-symmetric solution of conformal classical Yang-Baxter equation, whereas the symmetric part is a symmetric solution of conformal $$S$$-equation. One by-product is that a finite left-symmetric conformal algebra which is a free $${\mathbb{C}}[\partial]$$-module gives a natural $${\mathcal{O}}$$-operator, and hence, there is a construction of solutions of conformal classical Yang-Baxter equation and conformal $$S$$-equation from the former. Another by-product is that the non-degenerate solutions of these two equations correspond to 2-cocycles of Lie conformal algebras and left-symmetric conformal algebras, respectively. We also give a further study on a special class of $${\mathcal{O}}$$-operators called Rota-Baxter operators on Lie conformal algebras, and some explicit examples are presented. The generalized Weyl Poisson algebras and their Poisson simplicity criterion https://zbmath.org/1472.17078 2021-11-25T18:46:10.358925Z "Bavula, V. V." https://zbmath.org/authors/?q=ai:bavula.vladimir-v Summary: A new large class of Poisson algebras, the class of generalized Weyl Poisson algebras, is introduced. It can be seen as Poisson algebra analogue of generalized Weyl algebras or as giving a Poisson structure to (certain) generalized Weyl algebras. A Poisson simplicity criterion is given for generalized Weyl Poisson algebras, and an explicit description of the Poisson centre is obtained. Many examples are considered (e.g. the classical polynomial Poisson algebra in $$2n$$ variables is a generalized Weyl Poisson algebra). Classical $$N$$-reflection equation and Gaudin models https://zbmath.org/1472.17079 2021-11-25T18:46:10.358925Z "Caudrelier, Vincent" https://zbmath.org/authors/?q=ai:caudrelier.vincent "Crampé, Nicolas" https://zbmath.org/authors/?q=ai:crampe.nicolas Summary: We introduce the notion of $$N$$-reflection equation which provides a generalization of the usual classical reflection equation describing integrable boundary conditions. The latter is recovered as a special example of the $$N=2$$ case. The basic theory is established and illustrated with several examples of solutions of the $$N$$-reflection equation associated with the rational and trigonometric $$r$$-matrices. A central result is the construction of a Poisson algebra associated with a non-skew-symmetric $$r$$-matrix whose form is specified by a solution of the $$N$$-reflection equation. Generating functions of quantities in involution can be identified within this Poisson algebra. As an application, we construct new classical Gaudin-type Hamiltonians, particular cases of which are Gaudin Hamiltonians of $$BC_L$$-type. On the Dixmier-Moeglin equivalence for Poisson-Hopf algebras https://zbmath.org/1472.17080 2021-11-25T18:46:10.358925Z "Launois, Stéphane" https://zbmath.org/authors/?q=ai:launois.stephane "León Sánchez, Omar" https://zbmath.org/authors/?q=ai:leon-sanchez.omar Summary: We prove that the Poisson version of the Dixmier-Moeglin equivalence holds for cocommutative affine Poisson-Hopf algebras. This is a first step towards understanding the symplectic foliation and the representation theory of (cocommutative) affine Poisson-Hopf algebras. Our proof makes substantial use of the model theory of fields equipped with finitely many possibly noncommuting derivations. As an application, we show that the symmetric algebra of a finite dimensional Lie algebra, equipped with its natural Poisson structure, satisfies the Poisson Dixmier-Moeglin equivalence. Linear Poisson structures and Hom-Lie algebroids https://zbmath.org/1472.17081 2021-11-25T18:46:10.358925Z "Peyghan, Esmaeil" https://zbmath.org/authors/?q=ai:peyghan.esmaeil "Baghban, Amir" https://zbmath.org/authors/?q=ai:baghban.amir "Sharahi, Esa" https://zbmath.org/authors/?q=ai:sharahi.esa Summary: Considering Hom-Lie algebroids in some special cases, we obtain some results of Lie algebroids for Hom-Lie algebroids. In particular, we introduce the local splitting theorem for Hom-Lie algebroids. Moreover, linear Hom-Poisson structure on the dual Hom-bundle will be introduced and a one-to-one correspondence between Hom-Poisson structures and Hom-Lie algebroids will be presented. Also, we introduce Hamiltonian vector fields by using linear Poisson structures and show that there exists a relation between these vector fields and the anchor map of a Hom-Lie algebroid. Numerical characteristics of varieties of Poisson algebras https://zbmath.org/1472.17082 2021-11-25T18:46:10.358925Z "Ratseev, S. M." https://zbmath.org/authors/?q=ai:ratseev.sergej-mikhaijlovich The author presents a survey of recent results on varieties of Poisson algebras. In particular, he gives constructions of varieties of Poisson algebras with extremal properties. Also, equivalent conditions for the polynomial codimension growth of a variety of Poisson algebras are presented. Next, the author studies varieties of Poisson algebras satisfying the identity $$\{x,y\}\cdot \{z,t\} = 0$$, and connections between such varieties and varieties of Lie algebras. Solvability of Poisson algebras https://zbmath.org/1472.17083 2021-11-25T18:46:10.358925Z "Siciliano, Salvatore" https://zbmath.org/authors/?q=ai:siciliano.salvatore "Usefi, Hamid" https://zbmath.org/authors/?q=ai:usefi.hamid Let $$P$$ be a Poisson algebra with a Lie bracket $$\{\ ,\ \}$$ over a field of characteristic $$p\ge 0$$. The authors study the Lie structure of $$P$$. The main result is as follows. Let $$P$$ be solvable with respect to the Lie bracket. Then the Poisson ideal of $$P$$ generated by all elements $$\{\{\{x_1, x_2\}, \{x_3, x_4\}\}, x_5 \}$$, where $$x_i\in P$$, is associative nilpotent of class bounded by a function of the derived length of $$P$$. The authors deduce applications, one of them is as follows. Let $$P$$ be Lie solvable and $$p\ne 2$$, then the Poisson ideal $$\{P, P\}P$$ is nil. If additionally $$p\ge 3$$, then $$\{P, P\}P$$ is nil of bounded index. If $$P$$ is Lie nilpotent and $$p>0$$, then $$\{P, P\}P$$ is nil of bounded index. Lie conformal superalgebras and duality of modules over linearly compact Lie superalgebras https://zbmath.org/1472.17084 2021-11-25T18:46:10.358925Z "Cantarini, Nicoletta" https://zbmath.org/authors/?q=ai:cantarini.nicoletta "Caselli, Fabrizio" https://zbmath.org/authors/?q=ai:caselli.fabrizio "Kac, Victor" https://zbmath.org/authors/?q=ai:kac.victor-g A Lie superalgebra $$\mathfrak{g}$$ is said to be linearly compact when it is endowed with a topology such that: as a topological space, $$\mathfrak{g}$$ is complete, and there exists a fundamental system of neighborhoods of $$0$$ which consists of finite-codimensional subspaces. The classification of infinite-dimensional simple linearly compact Lie superalgebras was given by \textit{V. G. Kac} [Adv. Math. 139, No. 1, 1--55 (1998; Zbl 0929.17026)] It is known that linearly compact Lie superalgebras are closely related to the so-called Lie conformal superalgebras. One of the goals of the authors of this paper is to further the development of the representation theory of linearly compact Lie superalgebras by studying their conformal counterparts. The authors define Lie conformal superalgebras of type $$(r,s)$$ in Definition~2.1 and their associated annihilation Lie superalgebras in Definition~2.4. These definitions generalize the usual definitions of Lie conformal superalgebras (which are of type $$(1,0)$$) and their associated annihilation Lie superalgebras. Then, in Definition~3.1, the authors define conformal modules for Lie conformal superalgebras of type $$(r,s)$$. Under a few assumptions (see Assumptions~3.3), they prove that every conformal module over a Lie conformal superalgebra of type $$(r,s)$$ admits a structure of a continuous module for a certain extension of the associated annihilation Lie superalgebra (see Proposition~3.4). The main result of this article is Theorem~3.17, where the authors prove that: the conformal dual (see Definition~3.6) of the generalized Verma module (see page~16) associated to a finite-dimensional module $$F$$ is isomorphic to the generalized Verma module associated to the shifted dual $$F^\vee$$. The authors finish the paper by proving that the linearly compact Lie superalgebras $$W(r, s)$$, $$K(1, n)$$, $$E(5, 10)$$, $$E(3, 6)$$ and $$E(3, 8)$$ are annihilation Lie superalgebras associated to certain Lie conformal superalgebras of type $$(r,s)$$ in Sections 4--8. Lie algebras of derivations with large abelian ideals https://zbmath.org/1472.17085 2021-11-25T18:46:10.358925Z "Klymenko, I. S." https://zbmath.org/authors/?q=ai:klymenko.i-s "Lysenko, S. V." https://zbmath.org/authors/?q=ai:lysenko.s-v "Petravchuk, Anatoliy" https://zbmath.org/authors/?q=ai:petravchuk.anatolii-p Summary: Let $$\mathbb{K}$$ be a field of characteristic zero, $$A=\mathbb{K}[x_1,\ldots,x_n]$$ the polynomial ring and $$R=\mathbb{K}(x_1,\dots,x_n)$$ the field of rational functions. The Lie algebra $$\widetilde{W}_n(\mathbb{K}):=\operatorname{Der}_{\mathbb{K}}R$$ of all $$\mathbb{K}$$-derivation on $$R$$ is a vector space (of dimension n) over $$R$$ and every its subalgebra $$L$$ has rank $$\operatorname{rk}_RL=\dim_RRL$$. We study subalgebras $$L$$ of rank $$m$$ over $$R$$ of the Lie algebra $$\widetilde{W}_n(\mathbb{K})$$ with an abelian ideal $$I\subset L$$ of the same rank $$m$$ over $$R$$. Let $$F$$ be the field of constants of $$L$$ in $$R$$. It is proved that there exist a basis $$D_1, \ldots, D_m$$ of $$FI$$ over $$F$$, elements $$a_1, \ldots, a_k\in R$$ such that $$D_i(a_j)=\delta_{ij}$$, $$i=1, \ldots, m$$, $$j=1,\ldots, k$$, and every element $$D\in FL$$ is of the form $$D=\sum_{i=1}^mf_i(a_1, \ldots, a_k)D_i$$ for some $$f_i\in F[t_1, \ldots t_k], \deg f_i\leqslant 1$$. As a consequence it is proved that $$L$$ is isomorphic to a subalgebra (of a very special type) of the general affine Lie algebra $$\mathrm{aff}_m(F)$$. Higher level Zhu algebras and modules for vertex operator algebras https://zbmath.org/1472.17086 2021-11-25T18:46:10.358925Z "Barron, Katrina" https://zbmath.org/authors/?q=ai:barron.katrina "Vander Werf, Nathan" https://zbmath.org/authors/?q=ai:vander-werf.nathan "Yang, Jinwei" https://zbmath.org/authors/?q=ai:yang.jinwei Summary: Motivated by the study of indecomposable, nonsimple modules for a vertex operator algebra $$V$$, we study the relationship between various types of $$V$$-modules and modules for the higher level Zhu algebras for $$V$$, denoted $$A_n(V)$$, for $$n \in \mathbb{N}$$, first introduced by Dong, Li, and Mason in 1998 [\textit{C. Dong} et al., J. Algebra 206, No. 1, 67--96 (1998; Zbl 0911.17017)]. We resolve some issues that arise in a few theorems previously presented when these algebras were first introduced, and give examples illustrating the need for certain modifications of the statements of those theorems. We establish that whether or not $$A_{n - 1}(V)$$ is isomorphic to a direct summand of $$A_n(V)$$ affects the types of indecomposable $$V$$-modules which can be constructed by inducing from an $$A_n(V)$$-module, and in particular whether there are $$V$$-modules induced from $$A_n(V)$$-modules that were not already induced by $$A_0(V)$$. We give some characterizations of the $$V$$-modules that can be constructed from such inducings, in particular as regards their singular vectors. To illustrate these results, we discuss two examples of $$A_1(V)$$: when $$V$$ is the vertex operator algebra associated to either the Heisenberg algebra or the Virasoro algebra. For these two examples, we show how the structure of $$A_1(V)$$ in relationship to $$A_0(V)$$ determines what types of indecomposable $$V$$-modules can be induced from a module for the level zero versus level one Zhu algebras. We construct a family of indecomposable modules for the Virasoro vertex operator algebra that are logarithmic modules and are not highest weight modules. Harish-Chandra modules of the intermediate series over the topological $$N=2$$ superconformal algebra https://zbmath.org/1472.17087 2021-11-25T18:46:10.358925Z "Yang, Hengyun" https://zbmath.org/authors/?q=ai:yang.hengyun "Xu, Ying" https://zbmath.org/authors/?q=ai:xu.ying "Sun, Jiancai" https://zbmath.org/authors/?q=ai:sun.jiancai Parafermion vertex operator algebras and $$W$$-algebras https://zbmath.org/1472.17088 2021-11-25T18:46:10.358925Z "Arakawa, Tomoyuki" https://zbmath.org/authors/?q=ai:arakawa.tomoyuki "Lam, Ching Hung" https://zbmath.org/authors/?q=ai:lam.ching-hung "Yamada, Hiromichi" https://zbmath.org/authors/?q=ai:yamada.hiromichi Summary: We prove the conjectural isomorphism between the level $$k$$ $$\widehat {\mathfrak{sl}}_2$$-parafermion vertex operator algebra and the $$(k+1,k+2)$$-minimal series $$W_k$$-algebra for all $$k\geq 2$$. As a consequence, we obtain the conjectural isomorphism between the $$(k+1,k+2)$$-minimal series $$W_k$$-algebra and the coset vertex operator algebra $$\mathrm{SU}(k)_1\otimes \mathrm{SU}(k)_1/\mathrm{SU}(k)_2$$. Fusion categories for affine vertex algebras at admissible levels https://zbmath.org/1472.17089 2021-11-25T18:46:10.358925Z "Creutzig, Thomas" https://zbmath.org/authors/?q=ai:creutzig.thomas Summary: The main result is that the category of ordinary modules of an affine vertex operator algebra of a simply laced Lie algebra at admissible level is rigid and thus a braided fusion category. If the level satisfies a certain coprime property then it is even a modular tensor category. In all cases open Hopf links coincide with the corresponding normalized S-matrix entries of torus one-point functions. This is interpreted as a Verlinde formula beyond rational vertex operator algebras. A preparatory Theorem is a convenient formula for the fusion rules of rational principal W-algebras of any type. Simple current extensions beyond semi-simplicity https://zbmath.org/1472.17090 2021-11-25T18:46:10.358925Z "Creutzig, Thomas" https://zbmath.org/authors/?q=ai:creutzig.thomas "Kanade, Shashank" https://zbmath.org/authors/?q=ai:kanade.shashank "Linshaw, Andrew R." https://zbmath.org/authors/?q=ai:linshaw.andrew-r Modules over axial algebras https://zbmath.org/1472.17091 2021-11-25T18:46:10.358925Z "De Medts, Tom" https://zbmath.org/authors/?q=ai:de-medts.tom "Van Couwenberghe, Michiel" https://zbmath.org/authors/?q=ai:van-couwenberghe.michiel Summary: We introduce axial representations and modules over axial algebras as new tools to study axial algebras. All known interesting examples of axial algebras fall into this setting, in particular the Griess algebra whose automorphism group is the Monster group. Our results become especially interesting for Matsuo algebras. We vitalize the connection between Matsuo algebras and 3-transposition groups by relating modules over Matsuo algebras with representations of 3-transposition groups. As a by-product, we define, given a Fischer space, a group that can fulfill the role of a universal 3-transposition group. Orbifolds of lattice vertex operator algebras at $$d = 48$$ and $$d = 72$$ https://zbmath.org/1472.17092 2021-11-25T18:46:10.358925Z "Gemünden, Thomas" https://zbmath.org/authors/?q=ai:gemunden.thomas "Keller, Christoph A." https://zbmath.org/authors/?q=ai:keller.christoph-a Summary: Motivated by the notion of extremal vertex operator algebras, we investigate cyclic orbifolds of vertex operator algebras coming from extremal even self-dual lattices in $$d = 48$$ and $$d = 72$$. In this way we construct about one hundred new examples of holomorphic VOAs with a small number of low weight states. A finite presentation of Jordan algebras https://zbmath.org/1472.17093 2021-11-25T18:46:10.358925Z "Shestakov, Ivan" https://zbmath.org/authors/?q=ai:shestakov.ivan-p "Zelmanov, Efim" https://zbmath.org/authors/?q=ai:zelmanov.efim-i Some majorization inequalities induced by Schur products in Euclidean Jordan algebras https://zbmath.org/1472.17094 2021-11-25T18:46:10.358925Z "Gowda, M. Seetharama" https://zbmath.org/authors/?q=ai:gowda.muddappa-seetharama Summary: In a Euclidean Jordan algebra $$\mathcal{V}$$ of rank $$n$$, an element $$x$$ is said to be majorized by an element $$y$$, symbolically $$x \prec y$$, if the corresponding eigenvalue vector $$\lambda(x)$$ is majorized by $$\lambda(y)$$ in $$\mathcal{R}^n$$. In this article, we describe pointwise majorization inequalities of the form $$T(x) \prec S(x)$$, where $$T$$ and $$S$$ are linear transformations induced by Schur products. Specializing, we recover analogs of majorization inequalities of Schur, Hadamard, and Oppenheimer stated in the setting of Euclidean Jordan algebras, as well as majorization inequalities connecting quadratic and Lyapunov transformations on $$\mathcal{V}$$. We also show how Schur products induced by certain scalar means (such as arithmetic, geometric, harmonic, and logarithmic means) naturally lead to majorization inequalities. Jordan superalgebra of type $$\mathrm{JP}_n$$, $$n \ge 3$$ and the Wedderburn principal theorem https://zbmath.org/1472.17095 2021-11-25T18:46:10.358925Z "Gómez-González, F. A." https://zbmath.org/authors/?q=ai:gomez-gonzalez.f-a "Ramírez Bermúdez, J. A." https://zbmath.org/authors/?q=ai:ramirez-bermudez.j-a Let $$\mathbb F$$ be a field of characteristic zero and consider the Jordan superalgebra $$H(A,*)$$ where $$A={\mathcal M}_{n\vert n}(\mathbb F)$$ and $\begin{pmatrix} a & b\\ c & d\end{pmatrix}^*=\begin{pmatrix}d^t & -b^t\\ c^t & a^t\end{pmatrix}$ denoting $$x^t$$ the transposition matrix of $$x$$. This Jordan superalgebra will be denoted $$\mathrm{JP}_n$$. The authors investigate an analog to the Wedderburn Principal Theorem for finite-dimensional Jordan superalgebras $$J$$ with solvable radical $$N$$ such that $$N^2=0$$ and $$J/N\cong \mathrm{JP}_n$$ with $$n\ge3$$: they consider $$N$$ as an irreducible $$\mathrm{JP}_n$$-bimodule and we prove that the Wedderburn Principal Theorem holds for $$J$$. Simple asymmetric doubles, their automorphisms and derivations https://zbmath.org/1472.17096 2021-11-25T18:46:10.358925Z "Pchelintsev, S. V." https://zbmath.org/authors/?q=ai:pchelintsev.sergei-valentinovich "Shashkov, O. V." https://zbmath.org/authors/?q=ai:shashkov.oleg-vladimirovich Summary: A simple right-alternative, but not alternative, superalgebra whose even part coincides with an algebra of second-order matrices is called an asymmetric double. It is known that such superalgebras are eight-dimensional. We give a solution to the isomorphism problem for asymmetric doubles, point out their automorphism groups and derivation superalgebras. Rota-Baxter operators on BiHom-associative algebras and related structures https://zbmath.org/1472.17097 2021-11-25T18:46:10.358925Z "Liu, Ling" https://zbmath.org/authors/?q=ai:liu.ling "Makhlouf, Abdenacer" https://zbmath.org/authors/?q=ai:makhlouf.abdenacer "Menini, Claudia" https://zbmath.org/authors/?q=ai:menini.claudia "Panaite, Florin" https://zbmath.org/authors/?q=ai:panaite.florin Following the work of Makhlouf and Yau on Rota-Baxter Hom-algebras, Rota-Baxter on BiHom associative algebras are introduced. A BiHom associative algebra is an associative algebra with two commuting algebra endomorphisms $$\alpha$$ and $$\beta$$, with the following axiom: $\alpha(x)yz=xy\beta(z).$ Firstly, BiHom dendriform, Zinbiel, tridendriform and quadri algebras are introduced, and classical relations between these objects are extended to the BiHom context. It is also proved that a Yau twist exists for all of them. Secondly, a theory of Rota-Baxter operators on these objects is developed. It is proved that if the Rota-Baxter operator $$R$$ commutes with both algebra endomorphisms, then the Yau twist is also a Rota-Baxter algebra. Free Rota-Baxter BiHom associative algebras are described with the help of planar trees and with considerations of functors related to Rota-Baxter structures. The paper ends with considerations on weak pseudotwistors. Tensor products and perturbations of BiHom-Novikov-Poisson algebras https://zbmath.org/1472.17098 2021-11-25T18:46:10.358925Z "Liu, Ling" https://zbmath.org/authors/?q=ai:liu.ling "Makhlouf, Abdenacer" https://zbmath.org/authors/?q=ai:makhlouf.abdenacer "Menini, Claudia" https://zbmath.org/authors/?q=ai:menini.claudia "Panaite, Florin" https://zbmath.org/authors/?q=ai:panaite.florin A Novikov-Poisson algebra is given an associative product $$\cdot$$ and a pre-Lie product $$*$$, with the complementary axioms $x*(y*z)=(x*z)*y,$ $(x*y)\cdot z-x*(y\cdot z)=(y*x)\cdot z-y*(x\cdot z),$ $(x\cdot y)*z=(x*z)\cdot y.$ A biHom Novikov-Poisson algebra is a Novikov-Poisson algebra with two twocommuting algebra endomorphisms, satisfying complementary conditions of compatibilities with the products $$\cdot$$ and $$*$$. In this paper, a structure of biHom Novikov-Poisson algebra on the tensor product of two biHom Novikov-Poisson algebras. Moreover, it is shown that these objects can de deformed according to certain of their elements, generalizing a result due to Xu in the classical case and to Yau in the Hom case. Finally, a classe of biHom Novikov-Poisson algebras is introduced, which gives rise to biHom Poisson algebras. It is shown that this class is preserved under the tensor product, the Yau twist and the deformations mentioned above. Endowing evolution algebras with properties of discrete structures https://zbmath.org/1472.17099 2021-11-25T18:46:10.358925Z "González-López, Rafael" https://zbmath.org/authors/?q=ai:gonzalez-lopez.rafael "Núñez, Juan" https://zbmath.org/authors/?q=ai:nunez-valdes.juan In this paper, the authors delve into the basic properties characterizing the directed graph that is uniquely associated to an evolution algebra, and which was introduced in [\textit{J. P. Tian}, Evolution algebras and their applications. Berlin: Springer (2008; Zbl 1136.17001)]. These properties are conveniently translated to algebraic concepts and results concerning the evolution algebra under consideration. In particular, the authors focus on the adjacency of graphs, whose immediate translation into algebraic language enables them to introduce the concepts of adjacency, walk, trail, circuit, path and cycle of an evolution algebra. Also the notions of strongly and weakly connected evolution algebras are introduced as the algebraic equivalences of the same concepts in graph theory. It enables the authors to introduce the notions of distance, girth, circumference, eccentricity, center, radio, diameter and geodesic of an evolution algebra, together with the concepts of Eulerian and Hamiltonian evolution algebras. Some basic results on these topics are then described. The relationship among all of these notions and their analogous in graph theory are visually illustrated throughout the paper. Degenerations of Zinbiel and nilpotent Leibniz algebras https://zbmath.org/1472.17100 2021-11-25T18:46:10.358925Z "Kaygorodov, Ivan" https://zbmath.org/authors/?q=ai:kaigorodov.i-b "Popov, Yury" https://zbmath.org/authors/?q=ai:popov.yury "Pozhidaev, Alexandre" https://zbmath.org/authors/?q=ai:pozhidaev.aleksandr-petrovich "Volkov, Yury" https://zbmath.org/authors/?q=ai:volkov.yu-s A \emph{Leibniz algebra} [\textit{J.-L. Loday}, Enseign. Math. (2) 39, No. 3--4, 269--293 (1993; Zbl 0806.55009)] is an algebra satisfying the \emph{Leibniz identity}: $(xy)z=(xz)y+x(yz), \quad x, y, z \in A.$ A \emph{Zinbiel algebra} [\textit{J.-L. Loday}, Math. Scand. 77, No. 2, 189--196 (1995; Zbl 0859.17015)] is an algebra satisfying the \emph{identity}: $(xy)z=x(yz+zy), \quad x, y, z \in A.$ Leibniz algebras were introduced by \textit{A. Blokh} as D-algebras [On a generalization of the concept of Lie algebra'', Dokl. Akad. Nauk SSSR 165, 471--473 (1965; Zbl 0139.25702); translated in Sov. Math., Dokl. 6, 1450--1452 (1965)], then rediscovered by J. L. Loday when he obtained a new chain complex, called the Loday complex, by lifting the Chevalley-Eilenberg boundary map in the exterior module of a Lie algebra to the tensor module. Leibniz algebras are often considered as non commutative Lie algebras, since the Leibniz identity is equivalent to the Jacobi identity when the two-sided ideal $$\{a\in L~|~[a,a]=0\}$$ coincides with $$L$$. For this reason, a significant amount of research attempts to extend results on groups and Lie algebras to Leibniz algebras. Degenerations is an important concept with many applications in physics and other fields of mathematics. It is closely related to the notion of deformation and is often used as a tool for finding rigid algebras. Although degenerations have been studied in many varieties of algebras, especially in low dimensional algebras, results are still fragmentary. In this article, the authors fully describe degenerations of Zinbiel and nilpotent Leibniz algebras of dimension four over complex numbers. They also describe rigid algebras and all irreducible components in the variety of nilpotent four-dimensional Leibniz algebras. These descriptions are done using the same methods applied in the case of lie algebras (see [\textit{C. Seeley}, Commun. Algebra 18, No. 10, 3493--3505 (1990; Zbl 0709.17006)] for example). Dendriform algebras relative to a semigroup https://zbmath.org/1472.18012 2021-11-25T18:46:10.358925Z "Aguiar, Marcelo" https://zbmath.org/authors/?q=ai:aguiar.marcelo Recently, numerous generalizations of Rota-Baxter algebras and dendriform algebras appear, where one or two parameters belonging to a semigroup are used: these are family algebras. This paper first shows that similar objects exist for associative, associative commutative, Lie, and many other types of algebras. It then gives a categorical interpretation of these objects: it is shown that they are in fact classical objects in a monoidal category of graded objects over the chosen monoid, satisfying a uniformity condition. Deligne categories and the periplectic Lie superalgebra https://zbmath.org/1472.18014 2021-11-25T18:46:10.358925Z "Entova-Aizenbud, Inna" https://zbmath.org/authors/?q=ai:aizenbud.inna-entova "Serganova, Vera" https://zbmath.org/authors/?q=ai:serganova.vera-v The authors give a construction of the tensor category $$\text{Rep}(\underline{P})$$, possessing nice universal properties among tensor categories over the category $$\textsf{sVect}$$ of finite-dimensional complex vector superspaces with parity-preserving maps. One way that the category $$\textsf{sVect}$$ is constructed is through an explicit limit of the tensor categories $$\text{Rep}(\mathfrak{p}(n))$$ under Duflo-Serganova functors $$DS_x:\text{Rep}(\mathfrak{p}(n))\rightarrow \text{Rep}(\mathfrak{p}(n-2))$$, where $$n\geq 1$$. That is, let $$x$$ be an odd element in a Lie superalgebra $$\mathfrak{g}$$ satisfying $$[x,x]=0$$. Let $$\mathfrak{g}_x =\mathsf{ker} \:\text{ad}_x/\mathsf{im}\: \text{ad}_x$$, a Lie superalgebra. Define the functor $$DS_x:\text{Rep}(\mathfrak{g})\rightarrow \text{Rep}(\mathfrak{g}_x)$$ to be $$M\mapsto M_x =\mathsf{ker}\: x/\mathsf{im} \: x$$, which is symmetric monoidal. Take $$\mathfrak{g}=\mathfrak{p}(n)$$ and $$x$$ of rank $$2$$ to obtain $$\mathfrak{g}_x\cong \mathfrak{p}(n-2)$$. The second way that one constructs $$\textsf{sVect}$$, inspired by P. Etingof, is by describing $$\text{Rep}(\underline{P})$$ as the category of representations of a periplectic Lie supergroup in the Deligne category $$\textsf{sVect}\boxtimes \text{Rep}(\underline{\text{GL}}_t)$$, where $$\text{Rep}(\underline{\text{GL}}_t)$$ is an example of a non-Tannakian tensor category for $$t\not\in\mathbb{Z}$$; it is not equivalent to the category of representations of any affine algebraic group, nor supergroup. The category $$\text{Rep}(\underline{P})$$ is a lower highest weight category, i.e., there is a filtration $$\text{Rep}(\underline{P})=\bigcup_{k\geq 0}\text{Rep}^k(\underline{P})$$ by full highest weight subcategories, whose standard, costandard and tilting objects play the same role in each $$\text{Rep}^{k'}(\underline{P})$$, where $$k'\geq k$$. The full subcategory of tilting objects in $$\text{Rep}(\underline{P})$$ is precisely a Karoubian additive Deligne category $$\mathfrak{P}$$ in the periplectic case, which is a module category over $$\mathsf{sVect}$$. One nice property of the limit category'' $$\text{Rep}(\underline{P})$$ includes that it is the abelian envelope of the Deligne category corresponding to the periplectic Lie superalgebra. Another nice property is that given a tensor category $$\mathcal{C}$$ over $$\textsf{sVect}$$, exact tensor functors $$\text{Rep}(\underline{P})\rightarrow \mathcal{C}$$ classify pairs $$(X,\omega)$$ in $$\mathcal{C}$$, where $$\omega:X\otimes X\rightarrow \prod \mathbf{1}$$ is a nondegenerate odd symmetric form and $$X$$ cannot be annihilated by any Schur functor (Theorems 1 and 2, page 510). The authors study stabilization properties of finite-dimensional integrable representations of the periplectic Lie superalgebras $$\mathfrak{p}(n)$$ as $$n\rightarrow \infty$$ (Section 5, page 529). Verbal width in the Nottingham group and related Lie algebras https://zbmath.org/1472.20063 2021-11-25T18:46:10.358925Z "Martínez Carracedo, Jorge" https://zbmath.org/authors/?q=ai:carracedo.jorge-martinez "Martínez López, Consuelo" https://zbmath.org/authors/?q=ai:martinez.consuelo \textit{M. A. Virasoro} introduced his celebrated algebraic structure (known as Virasoro algebra) in order to describe some physical phenomena, which are related to the harmonic oscillator in [Subsidiary conditions and ghosts in dual-resonance models'', Phys. Rev. D. 1, No. 10, 2933--2936 (1970; \url{doi:10.1103/PhysRevD.1.2933})]. Virasoro algebras appeared in different ways in many dynamical systems, because their structure is very interesting and deserves a separate interest, not only in mathematical physics, but also in group theory and Lie theory. It should be mentioned (from the mathematical perspective) that a well-known result of \textit{M. Lazard} [Ann. Sci. Éc. Norm. Supér. (3) 71, 101--190 (1954; Zbl 0055.25103)] explores in fact a correspondence between nilpotent groups and nilpotent Lie rings. The present contribution shows some peculiarities of Virasoro algebras in connection with a profinite group, known as Nottingham group, and introduced by \textit{D. L. Johnson} [J. Aust. Math. Soc., Ser. A 45, No. 3, 296--302 (1988; Zbl 0666.20016)]. Virasoro algebras and some of their subalgebras are in fact related to the Nottingham group. Theorem 3.7 of the paper under review shows that the Nottingham group in zero characteristic is verbally elliptic, that is, all the words of such a group have finite width. Theorems 2.2 and 2.3 show that arbitrary multilinear polynomials have corresponding properties of width in the context of Virasoro algebras. Theorem 2.4 describes additional properties of structural nature for Virasoro algebras. $$\mathrm{SO}(N)_2$$ braid group representations are Gaussian https://zbmath.org/1472.20075 2021-11-25T18:46:10.358925Z "Rowell, Eric C." https://zbmath.org/authors/?q=ai:rowell.eric-c "Wenzl, Hans" https://zbmath.org/authors/?q=ai:wenzl.hans Summary: We give a description of the centralizer algebras for tensor powers of spin objects in the pre-modular categories $$\mathrm{SO}(N)_2$$ (for $$N$$ odd) and $$\mathrm{O}(N)_2$$ (for $$N$$ even) in terms of quantum $$(n-1)$$-tori, via non-standard deformations of $$\mathrm{U}\mathfrak{so}_N$$. As a consequence we show that the corresponding braid group representations are Gaussian representations, the images of which are finite groups. This verifies special cases of a conjecture that braid group representations coming from weakly integral braided fusion categories have finite image. Highest weight vectors and transmutation https://zbmath.org/1472.20100 2021-11-25T18:46:10.358925Z "Tange, Rudolf" https://zbmath.org/authors/?q=ai:tange.rudolf Summary: Let $$G = GL_{n}$$ be the general linear group over an algebraically closed field $$k$$, let $$\mathfrak{g}={\mathfrak{gl}}_n$$ be its Lie algebra and let $$U$$ be the subgroup of $$G$$ which consists of the upper uni-triangular matrices. Let $$k[\mathfrak{g}]$$be the algebra of polynomial functions on $$\mathfrak{g}$$ and let $$k{[\mathfrak{g}]}^G$$ be the algebra of invariants under the conjugation action of $$G$$. We consider the problem of giving finite homogeneous spanning sets for the $$k{[\mathfrak{g}]}^G$$-modules of highest weight vectors for the conjugation action on $$k[\mathfrak{g}]$$. We prove a general result in arbitrary characteristic which reduces the problem to giving spanning sets for the vector spaces of highest weight vectors for the action of $$GL_{r} \times GL_s$$ on tuples of $$r \times s$$ matrices. This requires the technique called transmutation'' by \textit{R. K. Brylinski} [Adv. Math. 100, No. 1, 28--52 (1993; Zbl 0819.20047)] which is based on an instance of Howe duality. In characteristic zero, we give for all dominant weights $$\chi \in \mathbb{Z}^n$$ finite homogeneous spanning sets for the $$k{[\mathfrak{g}]}^G$$-modules $$k[\mathfrak{g}]_{\chi}^U$$ of highest weight vectors. This result was already stated by J. F. Donin, but he only gave proofs for his related results on skew representations for the symmetric group. We do the same for tuples of $$n \times n$$-matrices under the diagonal conjugation action. Low degree cohomology of Frobenius kernels https://zbmath.org/1472.20104 2021-11-25T18:46:10.358925Z "Ngo, Nham V." https://zbmath.org/authors/?q=ai:ngo.nham-vo Summary: Let $$G$$ be a simple algebraic group defined over an algebraically closed field of characteristic $$p>0$$. For a positive integer $$r$$, let $$G_r$$ be the $$r$$-th Frobenius kernel of $$G$$. We determine in this paper a number $$m$$ such that the cohomology $$\mathrm{H}^n(G_r,k)$$ is isomorphic to $$\mathrm{H}^n(G_1,k)$$ for all $$n\le m$$ where $$m$$ depends on $$p$$ and the type of $$G$$. For the entire collection see [Zbl 1411.13002]. Some tables of right set properties in affine Weyl groups of type A https://zbmath.org/1472.20106 2021-11-25T18:46:10.358925Z "Scott, Leonard L." https://zbmath.org/authors/?q=ai:scott.leonard-l "Zell, Ethan C." https://zbmath.org/authors/?q=ai:zell.ethan-c Summary: The tables of the title are a first attempt to understand empirically the sizes of certain distinguished sets, introduced by Hankyung Ko, of elements in affine Weyl groups. The distinguished sets themselves each have a largest element $$w$$, and all other elements are constructible combinatorially from that largest element. The combinatorics are given in the language of right sets, in the sense of Kazhdan-Lusztig. Collectively, the elements in a given distinguished set parameterize highest weights of possible modular composition factors of the reduction modulo $$p$$ of a $$p$$th root of unity irreducible characteristic 0 quantum group module. Here, $$p$$ is a prime, subject to conditions discussed below, in some cases known to be quite mild. Thus, the sizes of the distinguished sets in question are relevant to estimating how much time might be saved in any future direct approach to computing irreducible modular characters of algebraic groups from larger irreducible characters of quantum groups. Actually, Ko has described two methods for obtaining potentially effective systems of such sets. She has proved one method to work at least for all primes $$p$$ as large as the Coxeter number $$h$$, in a context she indicates largely generalizes to smaller $$p$$. The other method, which produces smaller distinguished sets, is known for primes $$p\ge h$$ for which the Lusztig character formula holds, but is currently unknown to be valid without the latter condition. In the tables of this paper, we calculate, for all $$w$$ indexing a ($$p$$-)regular highest weight in the ($$p$$-)restricted parallelotope, distinguished set sizes for both methods, for affine types $$\mathrm{A}_3$$, $$\mathrm{A}_4$$, and $$\mathrm{A}_5$$. To keep the printed version of this paper sufficiently small, we only use those $$w$$ indexing actual restricted weights in the $$\mathrm{A}_5$$ case. The sizes corresponding to the two methods of Ko are listed in columns (6) and (5), respectively, of the tables. We also make calculations in column (7) for a third, more obvious'' system of distinguished sets (see part (1) of Proposition 1), to indicate how much of an improvement each of the first two systems provides. Finally, all calculations have been recently completed for affine type $$\mathrm{A}_6$$, and the restricted cases are listed in this paper as a final table. For the entire collection see [Zbl 1411.13002]. Trace versions of Young inequality and its applications https://zbmath.org/1472.26014 2021-11-25T18:46:10.358925Z "Huang, Chien-Hao" https://zbmath.org/authors/?q=ai:huang.chien-hao "Chen, Jein-Shan" https://zbmath.org/authors/?q=ai:chen.jein-shan "Hu, Chu-Chin" https://zbmath.org/authors/?q=ai:hu.chu-chin Summary: In this paper, we derive a few type of trace versions of Young inequality associated with second-order cone, which can be applied to derive the Hölder inequality, Minkowski inequality. Moreover, the triangular inequality is also considered. The dispersionless Veselov-Novikov equation: symmetries, exact solutions, and conservation laws https://zbmath.org/1472.35020 2021-11-25T18:46:10.358925Z "Morozov, Oleg I." https://zbmath.org/authors/?q=ai:morozov.oleg-i "Chang, Jen-Hsu" https://zbmath.org/authors/?q=ai:chang.jen-hsu Summary: We study symmetries, invariant solutions, and conservation laws for the dispersionless Veselov-Novikov equation. The emphasis is placed on cases when the odes involved in description of the invariant solutions are integrable by quadratures. Then we find some non-invariant solutions, in particular, solutions that are polynomials of an arbitrary degree $$N \ge 3$$ with respect to the spatial variables. Finally we compute all conservation laws that are associated to cosymmetries of second order. Integrable systems, multicomponent twisted Heisenberg-Virasoro algebra and its central extensions https://zbmath.org/1472.35281 2021-11-25T18:46:10.358925Z "Wu, Yemo" https://zbmath.org/authors/?q=ai:wu.yemo "Xu, Xiurong" https://zbmath.org/authors/?q=ai:xu.xiurong "Zuo, Dafeng" https://zbmath.org/authors/?q=ai:zuo.dafeng Summary: Let $$\mathscr{D}_N$$ be the multicomponent twisted Heisenberg-Virasoro algebra. We compute the second continuous cohomology group with coefficients in $$\mathbb{C}$$ and study the bihamiltonian Euler equations associated to $$\mathscr{D}_N$$ and its central extensions. Hamiltonian operators and related differential-algebraic Balinsky-Novikov, Riemann and Leibniz type structures on nonassociative noncommutative algebras https://zbmath.org/1472.37061 2021-11-25T18:46:10.358925Z "Artemovych, Orest D." https://zbmath.org/authors/?q=ai:artemovych.orest-d "Balinsky, Alexander A." https://zbmath.org/authors/?q=ai:balinsky.alexander-a "Prykarpatski, Anatolij K." https://zbmath.org/authors/?q=ai:prykarpatsky.anatoliy-karolevych Summary: We review main differential-algebraic structures lying in background of analytical constructing multi-component Hamiltonian operators as derivatives on suitably constructed loop Lie algebras, generated by nonassociative noncommutative algebras. The related Balinsky-Novikov and Leibniz type algebraic structures are derived, a new nonassociative Riemann'' algebra is constructed, deeply related with infinite multi-component Riemann type integrable hierarchies. An approach, based on the classical Lie-Poisson structure on coadjoint orbits, closely related with those, analyzed in the present work and allowing effectively enough construction of Hamiltonian operators, is also briefly revisited. As the compatible Hamiltonian operators are constructed by means of suitable central extentions of the adjacent weak Lie algebras, generated by the right Leibniz and Riemann type nonassociative and noncommutative algebras, the problem of their description requires a detailed investigation both of their structural properties and finite-dimensional representations of the right Leibniz algebras defined by the corresponding structural constraints. Subject to these important aspects we stop in the work mostly on the structural properties of the right Leibniz algebras, especially on their derivation algebras and their generalizations. We have also added a short Supplement within which we revisited the classical Poisson manifold approach, closely related to our construction of Hamiltonian operators, generated by nonassociative and noncommutative algebras. In particular, we presented its natural and simple generalization allowing effectively to describe a wide class of Lax type integrable nonlinear Kontsevich type Hamiltonian systems on associative noncommutative algebras. Supersymmetric Euler equations associated to the $$N \leq 3$$ Neveu-Schwarz algebra https://zbmath.org/1472.37063 2021-11-25T18:46:10.358925Z "Zuo, Dafeng" https://zbmath.org/authors/?q=ai:zuo.dafeng Summary: We give a systematic study about supersymmetric Euler equations on the smooth dual $$\mathcal{N}_{\mathcal{S}_{reg}}^*(N)$$ of the Neveu-Schwarz algebra $$\mathcal{N}_{\mathcal{S}}(N)$$ for $$N \leq 3$$. Let $$\mathcal{A}$$ be the inertia operator and $$c_1, c_2 \in \mathbb{R}$$, we will show that the $$N = 2$$ supersymmetric Euler equation with $$\mathcal{A} = c_1 + c_2 \mathfrak{D}_2$$ is local bi-super-Hamiltonian with the freezing point $$(c_1, c_2) \in \mathcal{N}_{\mathcal{S}_{reg}}^*(2)$$, which is similar to that of the Virasoro algebra $$\mathcal{N}_{\mathcal{S}}(0)$$, and the $$N = 3$$ supersymmetric Euler equation with $$\mathcal{A} = c_2 \mathfrak{D}_3 \partial^{- 1}$$ is local bi-super-Hamiltonian with the freezing point $$(0, c_2) \in \mathcal{N}_{\mathcal{S}_{reg}}^*(3)$$, which is similar to that of the $$N = 1$$ Neveu-Schwarz algebra $$\mathcal{N}_{\mathcal{S}}(1)$$.\par{\copyright 2019 American Institute of Physics} Super-Hamiltonian structures and conservation laws of a new six-component super-Ablowitz-Kaup-Newell-Segur hierarchy https://zbmath.org/1472.37076 2021-11-25T18:46:10.358925Z "You, Fucai" https://zbmath.org/authors/?q=ai:you.fucai "Zhang, Jiao" https://zbmath.org/authors/?q=ai:zhang.jiao "Zhao, Yan" https://zbmath.org/authors/?q=ai:zhao.yan Summary: A six-component super-Ablowitz-Kaup-Newell-Segur (-AKNS) hierarchy is proposed by the zero curvature equation associated with Lie superalgebras. Supertrace identity is used to furnish the super-Hamiltonian structures for the resulting nonlinear superintegrable hierarchy. Furthermore, we derive the infinite conservation laws of the first two nonlinear super-AKNS equations in the hierarchy by utilizing spectral parameter expansions. A generalized super AKNS hierarchy associated with orthosymplectic Lie superalgebra $$OSP(2,2)$$ and its super bi-Hamiltonian structures https://zbmath.org/1472.37077 2021-11-25T18:46:10.358925Z "Yu, Jing" https://zbmath.org/authors/?q=ai:yu.jing.1 "Han, Jingwei" https://zbmath.org/authors/?q=ai:han.jingwei "Li, Chuanzhong" https://zbmath.org/authors/?q=ai:li.chuanzhong.1|li.chuanzhong Summary: For the orthosymplectic Lie superalgebra $$OSP(2,2)$$, we choose a set of basis matrices. A linear combination of those basis matrices presents a spatial spectral matrix. The compatible condition of the spatial part and the corresponding temporal parts of the spectral problem leads to a generalized super AKNS (GSAKNS) hierarchy. By making use of the supertrace identity, the obtained GSAKNS hierarchy can be written as the super bi-Hamiltonian structures. Periodic one-point rank one commuting difference operators https://zbmath.org/1472.39037 2021-11-25T18:46:10.358925Z "Dobrogowska, Alina" https://zbmath.org/authors/?q=ai:dobrogowska.alina "Mironov, Andrey E." https://zbmath.org/authors/?q=ai:mironov.andrei-evgenevich Summary: In this paper we study one-point rank one commutative rings of difference operators. We find conditions on spectral data which specify such operators with periodic coefficients. For the entire collection see [Zbl 1472.53006]. Lyapunov convexity theorem for von Neumann algebras and Jordan operator structures https://zbmath.org/1472.46065 2021-11-25T18:46:10.358925Z "Hamhalter, Jan" https://zbmath.org/authors/?q=ai:hamhalter.jan By noncommutative Lyapunov type theorem for von Neumann algebras, the author means sufficient conditions on a von Neumann algebra $$M$$, a linear space $$X$$ and a finitely additive measure $$\mu$$ on the projection lattice $$P(M)$$ with values in $$X$$, ensuring that the range $$\mu(P(M))$$ is a convex subset of $$X$$. The starting point is a reformulation of the classical Lyapunov theorem, where $$M$$ is abelian and non-atomic, $$X=\mathbb{C}^n$$ and $$\mu$$ is bounded. In case $$\mu$$ has an affine extension $$\hat\mu$$ to the positive part $$M_1^+$$ of the unit ball of $$M$$, a stronger formulation is the claim that the range is identical to $$\hat\mu(M_1^+)$$. The author proves a theorem of this kind for large' von Neumann algebras $$M$$, i.e., algebras without a non-zero $$\sigma$$-finite direct summand: If $$X$$ is a normed space with weak$$^*$$ separable dual, then the ranges of $$M_1^+$$ and $$P(M)$$ coincide under a norm continuous linear map from $$M$$ into $$X$$. Theorems of this kind are also proved in various Jordan structures, such as JBW$$^*$$ triples. Worpitzky-compatible subarrangements of braid arrangements and cocomparability graphs https://zbmath.org/1472.52037 2021-11-25T18:46:10.358925Z "Tran, Tan Nhat" https://zbmath.org/authors/?q=ai:tran.tan-nhat "Tsuchiya, Akiyoshi" https://zbmath.org/authors/?q=ai:tsuchiya.akiyoshi Summary: The class of Worpitzky-compatible subarrangements of a Weyl arrangement together with an associated Eulerian polynomial was recently introduced by \textit{A. U. Ashraf} et al. [Adv. Appl. Math. 120, Article ID 102064, 24 p. (2020; Zbl 1447.52026)], which brings the characteristic and Ehrhart quasi-polynomials into one formula. The subarrangements of the braid arrangement, the Weyl arrangement of type $$A$$, are known as the graphic arrangements. We prove that the Worpitzky-compatible graphic arrangements are characterized by cocomparability graphs. This can be regarded as a counterpart of the characterization by Stanley and Edelman-Reiner of free and supersolvable graphic arrangements in terms of chordal graphs. Our main result yields new formulas for the chromatic and graphic Eulerian polynomials of cocomparability graphs. Diagonalizing the Ricci tensor https://zbmath.org/1472.53025 2021-11-25T18:46:10.358925Z "Krishnan, Anusha M." https://zbmath.org/authors/?q=ai:krishnan.anusha-m Summary: We show that a basis of a semisimple Lie algebra of compact type, for which any diagonal left-invariant metric has a diagonal Ricci tensor, is characterized by the Lie algebraic condition of being nice''. Namely, the bracket of any two basis elements is a multiple of another basis element. This extends the work of \textit{J. Lauret} and \textit{C. Will} [Proc. Am. Math. Soc. 141, No. 10, 3651--3663 (2013; Zbl 1279.53065)] on nilpotent Lie algebras. The result follows from a more general characterization for diagonalizing the Ricci tensor for homogeneous spaces. Finally, we also study the Ricci flow behavior of diagonal metrics on cohomogeneity one manifolds. A compact $$\mathrm G_2$$-calibrated manifold with first Betti number $$b_1 = 1$$ https://zbmath.org/1472.53061 2021-11-25T18:46:10.358925Z "Fernández, Marisa" https://zbmath.org/authors/?q=ai:fernandez.marisa "Fino, Anna" https://zbmath.org/authors/?q=ai:fino.anna "Kovalev, Alexei" https://zbmath.org/authors/?q=ai:kovalev.aleksei-viktorovich|kovalev.alexei-g "Muñoz, Vicente" https://zbmath.org/authors/?q=ai:munoz.vicente The authors construct a compact formal 7-manifold with a closed $$\mathrm{G}_2$$-structure and with first Betti number $$b_1 = 1$$ not admitting any torsion-free $$\mathrm{G}_2$$-structure. This manifold is not a product. To construct such a manifold, they start with a compact 7-manifold $$M$$ equipped with a closed $$\mathrm{G}_2$$ form $$\varphi$$ and with first Betti number $$b_1(M) = 3$$. In fact $$M$$ is a nilmanifold, that is, the coset space of a nilpotent Lie group by a cocompact lattice. Then they quotient $$M$$ by $$\mathbb Z_2$$ preserving $$\varphi$$ to obtain an orbifold $$\widehat{M}$$ with a closed orbifold $$\mathrm{G}_2$$ form $$\widehat{\varphi}$$ and with first Betti number $$b_1(\widehat{M}) = 1$$. The authors resolve the singularities of the 7-orbifold $$\widehat{M}$$ to produce a smooth 7-manifold $$\widetilde{M}$$ with a closed $$\mathrm{G}_2$$ form $$\widetilde{\varphi}$$, with first Betti number $$b_1(\widetilde{M}) = 1$$ and such that $$(\widetilde{M},\widetilde{\varphi})$$ is isomorphic to $$(\widehat{M},\widehat{\varphi})$$ outside the singular locus of $$\widehat{M}$$. Then they prove the properties following: the 7-manifold $$\widetilde{M}$$ is formal, with fundamental group $$\pi_1(\widetilde{M}) = \mathbb Z$$ and $$\widetilde{M}$$ does not admit any torsion-free $$\mathrm{G}_2$$-structure. For the compact 7-manifold $$M$$ with the closed $$\mathrm{G}_2$$ form $$\varphi$$ mentioned above, the authors consider a non-trivial involution of $$M$$ preserving $$\varphi$$, and they construct an example of a 3-dimensional family of associative volume-minimizing 3-tori in $$\widetilde{M}$$. This deformation family is maximal''. Finally the authors construct a smooth fibration map $$\widetilde{M}\to S^2 \times S^1$$ with generic fiber a coassociative torus and some singular fibers, with both smooth and singular fibers forming maximal deformation families. Ribbon 2-knots, $$1+1=2$$ and Duflo's theorem for arbitrary Lie algebras https://zbmath.org/1472.57001 2021-11-25T18:46:10.358925Z "Bar-Natan, Dror" https://zbmath.org/authors/?q=ai:bar-natan.dror "Dancso, Zsuzsanna" https://zbmath.org/authors/?q=ai:dancso.zsuzsanna "Scherich, Nancy" https://zbmath.org/authors/?q=ai:scherich.nancy-c Let $$\mathfrak{g}$$ be a finite-dimensional Lie algebra. The Duflo isomorphism is an algebra isomorphism $$\mathcal{D}: S(\mathfrak{g})^\mathfrak{g} \to U(\mathfrak{g})^\mathfrak{g}$$, given by an explicit formula, where $$S(\mathfrak{g})^\mathfrak{g}$$ and $$U(\mathfrak{g})^\mathfrak{g}$$ are the $$\mathfrak{g}$$-invariant subspaces for the adjoint action of $$\mathfrak{g}$$ on the symmetric algebra and the universal enveloping algebra of $$\mathfrak{g}$$, respectively. To show Duflo's theorem that the map $$\mathcal{D}$$ is an algebra isomorphism, the difficulty is the part to show that $$\mathcal{D}$$ is an algebra homomorphism, namely that $$\mathcal{D}$$ is multiplicative. There have been many proofs of Duflo's theorem. A topological proof was given by the first author, Le and Thurston, for metrized Lie algebras, using the Kontsevich integral and giving an interpretation of $$1+1=2$$ on an abacus'' in terms of knots in 3-dimensional space [\textit{D. Bar-Natan} et al., Geom. Topol. 7, 1--31 (2003; Zbl 1032.57008)]. In this paper, the authors give a proof of Duflo's theorem, for arbitrary finite-dimensional Lie algebras, using a 4-dimensional abacus''. The proof is given in such a process as follows. They use the setup and results due to the first and second authors [Math. Ann. 367, No. 3-4, 1517--1586 (2017; Zbl 1362.57005)]. They consider a certain circuit algebra called the space of w-foams such that each generator and relation is interpreted in terms of certain knotted objects in $$\mathbb{R}^4$$ that are ribbon knotted tubes with foam vertices and strings in $$\mathbb{R}^4$$. A 4-dimensional abacus bead is an element of the space of w-foams. They give an interpretation of $$1+1=2$$'' in terms of w-foams, and using a certain filtered linear map called the homomorphic expansion, they describe $$1+1=2$$'' diagrammatically in terms of arrow diagrams. Using the tensor interpretation map, they give an equality in $$\hat{S}(\mathfrak{g}^*)_\mathfrak{g} \otimes \hat{U}(\mathfrak{g})$$. Here $$S(\mathfrak{g}^*)$$ denotes the symmetric algebra of the linear dual of $$\mathfrak{g}$$, and $$\hat{}$$ denotes the degree completion where elements of $$\mathfrak{g}^*$$ and $$\mathfrak{g}$$ are defined to be degree 1 and degree 0, respectively, and the subscript $$\mathfrak{g}$$ denotes co-invariants under the co-adjoint action of $$\mathfrak{g}$$, and $$U(\mathfrak{g})$$ and $$\hat{}$$ denote the universal enveloping algebra of $$\mathfrak{g}$$ and the degree completion, respectively. Then they show the Duflo's theorem. Further, they derive the explicit formula for the Duflo map from the homomorphic expansion. Poisson principal bundles https://zbmath.org/1472.58004 2021-11-25T18:46:10.358925Z "Majid, Shahn" https://zbmath.org/authors/?q=ai:majid.shahn "Williams, Liam" https://zbmath.org/authors/?q=ai:williams.liam Authors' abstract: We semiclassicalise the theory of quantum group principal bundles to the level of Poisson geometry. The total space $X$ is a Poisson manifold with Poisson-compatible contravariant connection, the fibre is a Poisson-Lie group in the sense of Drinfeld with bicovariant Poisson-compatible contravariant connection, and the base has an inherited Poisson structure and Poisson-compatible contravariant connection. The latter are known to be the semiclassical data for a quantum differential calculus. The theory is illustrated by the Poisson level of the $q$-Hopf fibration on the standard $q$-sphere. We also construct the Poisson level of the spin connection on a principal bundle. Local tomography and the role of the complex numbers in quantum mechanics https://zbmath.org/1472.81020 2021-11-25T18:46:10.358925Z "Niestegge, Gerd" https://zbmath.org/authors/?q=ai:niestegge.gerd Summary: Various reconstructions of finite-dimensional quantum mechanics result in a formally real Jordan algebra $$A$$ and a last step remains to conclude that $$A$$ is the self-adjoint part of a C*-algebra. Using a quantum logical setting, it is shown that this can be achieved by postulating that there is a locally tomographic model for a composite system consisting of two copies of the same system. Local tomography is a feature of classical probability theory and quantum mechanics; it means that state tomography for a multipartite system can be performed by simultaneous measurements in all subsystems. The quantum logical definition of local tomography is sufficient, but it is less restrictive than the prevalent definition in the literature and involves some subtleties concerning the so-called spin factors. Staggered and affine Kac modules over $$A_1^{(1)}$$ https://zbmath.org/1472.81115 2021-11-25T18:46:10.358925Z "Rasmussen, Jørgen" https://zbmath.org/authors/?q=ai:rasmussen.jorgen-born|rasmussen.jorgen|rasmussen.jorgen-h Summary: This work concerns the representation theory of the affine Lie algebra $$A_1^{(1)}$$ at fractional level and its links to the representation theory of the Virasoro algebra. We introduce affine Kac modules as certain finitely generated submodules of Wakimoto modules. We conjecture the existence of several classes of staggered $$A_1^{(1)}$$-modules and provide evidence in the form of detailed examples. We extend the applicability of the Goddard-Kent-Olive coset construction to include the affine Kac and staggered modules. We introduce an exact functor between the associated category of $$A_1^{(1)}$$-modules and the corresponding category of Virasoro modules. At the level of characters, its action generalises the Mukhi-Panda residue formula. We also obtain explicit expressions for all irreducible $$A_1^{(1)}$$-characters appearing in the decomposition of Verma modules, re-examine the construction of Malikov-Feigin-Fuchs vectors, and extend the Fuchs-Astashkevich theorem from the Virasoro algebra to $$A_1^{(1)}$$. Multiplicity-free $$U_q(sl_N)$$ 6-j symbols: relations, asymptotics, symmetries https://zbmath.org/1472.81122 2021-11-25T18:46:10.358925Z "Alekseev, Victor" https://zbmath.org/authors/?q=ai:alekseev.victor "Morozov, Andrey" https://zbmath.org/authors/?q=ai:morozov.andrei-alekseevich "Sleptsov, Alexey" https://zbmath.org/authors/?q=ai:sleptsov.alexey Summary: A closed form expression for multiplicity-free quantum 6-j symbols (MFS) was proposed in [\textit{S. Nawata} et al., Lett. Math. Phys. 103, No. 12, 1389--1398 (2013; Zbl 1330.17020)] for symmetric representations of $$U_q(sl_N)$$, which are the simplest class of multiplicity-free representations. In this paper we rewrite this expression in terms of q-hypergeometric series $$_4\Phi_3$$. We claim that it is possible to express any MFS through the 6-j symbol for $$U_q(sl_2)$$ with a certain factor. It gives us a universal tool for the extension of various properties of the quantum 6-j symbols for $$U_q(sl_2)$$ to the MFS. We demonstrate this idea by deriving the asymptotics of the MFS in terms of associated tetrahedron for classical algebra $$U(sl_N)$$. Next we study MFS symmetries using known hypergeometric identities such as argument permutations and Sears' transformation. We describe symmetry groups of MFS. As a result we get new symmetries, which are a generalization of the tetrahedral symmetries and the Regge symmetries for $$N = 2$$. Antisymmetric characters and Fourier duality https://zbmath.org/1472.81124 2021-11-25T18:46:10.358925Z "Liu, Zhengwei" https://zbmath.org/authors/?q=ai:liu.zhengwei "Wu, Jinsong" https://zbmath.org/authors/?q=ai:wu.jinsong Based on the quantum McKay correspondence (c.f. section 2 of this paper for an elementary and self-contained expiation), the authors consider the \textit{ADE} classification related to quantum $\mathfrak{sl}_2$. In the paper under review, a quantum group is considered as a simple Lie algebra $\mathfrak{g}$ with a natural number $k$ (the level). The authors introduced anti-symmetric characters for $\mathfrak{g}$ at level $k$ and constructed a $C^{\ast}$-algebra of these characters, which is the Verlinde algebra of the corresponding quantum group. Moreover, they investigate the Fourier duality to study the spectral theory. Finally, in the \textit{ADE} Lie theory, there is a correspondence between the eigenvalues of the Coxeter element and the eigenvalues of the adjacency matrix. A generalization of the adjacency matrices and the Coxeter elements for Verlinde algebras of simple Lie algebras at any level is studied, including generalized Dynkin diagrams and higher Dynkin diagrams as special cases. Nilpotence varieties https://zbmath.org/1472.81238 2021-11-25T18:46:10.358925Z "Eager, Richard" https://zbmath.org/authors/?q=ai:eager.richard "Saberi, Ingmar" https://zbmath.org/authors/?q=ai:saberi.ingmar-a "Walcher, Johannes" https://zbmath.org/authors/?q=ai:walcher.johannes Algebraic varieties coming from orbits of nilpotent elements of Lie algebras is an important subject, which has also recently found deep connections with theoretical physics. The current paper is a wonderful monograph on this subject, focusing on varieties canonically associated with any Lie superalgebra, and treating them from a systematic and new perspective of natural moduli spaces parameterizing twists of a super-Poincare-invariant physical theory. Non-relativistic limits and three-dimensional coadjoint Poincaré gravity https://zbmath.org/1472.83067 2021-11-25T18:46:10.358925Z "Bergshoeff, Eric" https://zbmath.org/authors/?q=ai:bergshoeff.eric-a "Gomis, Joaquim" https://zbmath.org/authors/?q=ai:gomis.joaquim "Salgado-Rebolledo, Patricio" https://zbmath.org/authors/?q=ai:salgado-rebolledo.patricio Summary: We show that a recently proposed action for three-dimensional non-relativistic gravity can be obtained by taking the limit of a relativistic Lagrangian that involves the coadjoint Poincaré algebra. We point out the similarity of our construction with the way that three-dimensional Galilei gravity and extended Bargmann gravity can be obtained by taking the limit of a relativistic Lagrangian that involves the Poincaré algebra. We extend our results to the anti-de Sitter case and we will see that there is a chiral decomposition at both the relativistic and non-relativistic level. We comment on possible further generalizations. Generalised holonomies and $$K(E_9)$$ https://zbmath.org/1472.83106 2021-11-25T18:46:10.358925Z "Kleinschmidt, Axel" https://zbmath.org/authors/?q=ai:kleinschmidt.axel "Nicolai, Hermann" https://zbmath.org/authors/?q=ai:nicolai.hermann Summary: The involutory subalgebra $$K( \mathfrak{e}_9)$$ of the affine Kac-Moody algebra $$\mathfrak{e}_9$$ was recently shown to admit an infinite sequence of unfaithful representations of ever increasing dimensions [\textit{A. Kleinschmidt} et al., Representations of involutory subalgebras of affine Kac-Moody algebras'', Preprint, \url{arXiv:2102.00870}]. We revisit these representations and describe their associated ideals in more detail, with particular emphasis on two chiral versions that can be constructed for each such representation. For every such unfaithful representation we show that the action of $$K( \mathfrak{e}_9)$$ decomposes into a direct sum of two mutually commuting (chiral' and anti-chiral') parabolic algebras with Levi subalgebra $$\mathfrak{so} (16)_+ \oplus \mathfrak{so} (16)_-$$. We also spell out the consistency conditions for uplifting such representations to unfaithful representations of $$K( \mathfrak{e}_{10} )$$. From these results it is evident that the holonomy groups so far discussed in the literature are mere shadows (in a Platonic sense) of a much larger structure.