Recent zbMATH articles in MSC 17https://zbmath.org/atom/cc/172023-01-20T17:58:23.823708ZWerkzeugPoincaré polynomial at \(-1\) associated with a Young diagram of three rowshttps://zbmath.org/1500.050052023-01-20T17:58:23.823708Z"Mansour, Ronit"https://zbmath.org/authors/?q=ai:mansour.ronitSummary: In this paper, we give an explicit formula for the Poincaré polynomial \(P_\lambda (x)\) for the Betti numbers of the Springer fibers over nilpotent elements in \(gl_n(\mathbb{C})\) of Jordan form \(\lambda =abc\) with \(a\ge b\ge c\ge 0\) at \(x=-1\). In particular, we introduce \(\lambda \)-vacillating diagrams and show that \(P_{ab}(-1)\) is equal to the number of restricted Dyck paths.Symmetrisation and the Feigin-Frenkel centrehttps://zbmath.org/1500.160292023-01-20T17:58:23.823708Z"Yakimova, Oksana"https://zbmath.org/authors/?q=ai:yakimova.oksana-sThe Feigin-Frenkel centre \(\mathfrak{z}(\hat{g})\) is a remarkable commutative subalgebra of the enveloping algebra \(U(t^{-1}g[t^{-1}])\).
Unlike the situation with the central elements of \(U(g)\) in general, an element of \(\mathfrak{z}(\hat{g})\) cannot be obtained by the symmetrisation from a homogeneous \(g[t]\)-invariant in \(S(t^{-1}g[t^{-1}])\).
Nevertheless, there are explicit constructions of so called complete sets of Segal-Sugawara vectors from which the generators of \(\mathfrak{z}(\hat{g})\) can be easily obtained. These vectors are known in an explicit form for the series \(A\) [\textit{A. Chervov} and \textit{D. Talalaev}, ``Quantum spectral curves, quantum integrable systems and the geometric Langlands correspondence'', Preprint, \url{arXiv:hep-th/0604128}; \textit{A. V. Chervov} and \textit{A. I. Molev}, Int. Math. Res. Not. 2009, No. 9, 1612--1635 (2009; Zbl 1225.17031)], for \(B\), \(C\), \(D\) [\textit{A. I. Molev}, Invent. Math. 191, No. 1, 1--34 (2013; Zbl 1266.17016)], for \(G_2\) [\textit{A. I. Molev} et al., J. Algebra 455, 386--401 (2016; Zbl 1338.17023)].
The considerations in these papers are type-dependent. In the paper under review a type-free construction of Segal-Sugawara vectors is given.
Reviewer: Dmitry Artamonov (Moskva)Lie nilpotency index of a modular group algebrahttps://zbmath.org/1500.160312023-01-20T17:58:23.823708Z"Sahai, Meena"https://zbmath.org/authors/?q=ai:sahai.meena"Sharan, Bhagwat"https://zbmath.org/authors/?q=ai:sharan.bhagwatThe paper under review considers the classification of a modular group algebra \(KG\) of a (possibly non-commutative) group \(G\) over a field \(K\) of characteristic \(p\not= 0\) having special upper Lie nilpotency index. The main results are the quite technical Theorems 2.1 and 2.2 in which the authors classified such group algebras up to an isomorphism.
Reviewer: Peter Danchev (Sofia)Post-Lie Magnus expansion and BCH-recursionhttps://zbmath.org/1500.160342023-01-20T17:58:23.823708Z"Al-Kaabi, Mahdi J. Hasan"https://zbmath.org/authors/?q=ai:al-kaabi.mahdi-j-hasan"Ebrahimi-Fard, Kurusch"https://zbmath.org/authors/?q=ai:ebrahimi-fard.kurusch"Manchon, Dominique"https://zbmath.org/authors/?q=ai:manchon.dominiqueThe paper essentially consists into an interesting survey of results about combinatorial and algebraic structures in relation with some well-known formal power series with non-commutative variables, together with a final section with new results. Section 2 deals with post-Lie algebras which are Lie algebras with another bilinear product satisfying some compatibility relations with the Lie product. Section 3 is devoted to the Baxter-Campbell-Hausdorff (BCH) recursion, and its relation with (complete) Rota-Baxter algebras. The authors then describe the Magnus expansion on the free post-Lie algebra on a set, in Section 4. Finally in Section 5, they show that the aforementioned expansion described in Section 4 is the same as the BCH-recursion from Section 3.
Reviewer: Laurent Poinsot (Villetaneuse)Finite Cartan graphs attached to Nichols algebras of diagonal typehttps://zbmath.org/1500.160352023-01-20T17:58:23.823708Z"Qian, Chen"https://zbmath.org/authors/?q=ai:qian.chen"Wang, Jing"https://zbmath.org/authors/?q=ai:wang.jingIn the paper under review the authors ``illustrate the main properties of the finite Cartan graphs of rank 6 attached to the Nichols algebras of diagonal type''.
Authors' abstract: Nichols algebras are fundamental objects in the construction of quantized enveloping algebras and in the classification of pointed Hopf algebras by the lifting method of \textit{N. Andruskiewitsch} and \textit{H. J. Schneider} [J. Algebra 209, No. 2, 658--691 (1998; Zbl 0919.16027)]. The structure of Cartan graphs can be attached to any Nichols algebras of diagonal type and plays an important role in the classification of Nichols algebras of diagonal type with a finite root system. In this paper, the main properties of all simply connected Cartan graphs attached to rank 6 Nichols algebras of diagonal type are determined. As an application, we obtain a subclass of rank 6 finite dimensional Nichols algebras of diagonal type.
Reviewer: Dmitry Artamonov (Moskva)Embedding of \(\mathfrak{sl}_2(\mathbb{C})\)-modules into four-dimensional power-associative zero-algebra moduleshttps://zbmath.org/1500.170012023-01-20T17:58:23.823708Z"Quintero Vanegas, Elkin Oveimar"https://zbmath.org/authors/?q=ai:vanegas.elkin-oveimar-quinteroMotivated by the goal of obtaining information on the finite-dimensional irreducible modules over commutative power-associative algebras, the main result of this paper,Theorem 1, establishes that there exists an embedding (depending of certain parameter \(\Omega\)) \({\mathcal F}_{\Omega}\colon {\mathfrak M}od _{\mathfrak{sl}_2}\to {\mathfrak M}od _4\), where \({\mathfrak M}od _{\mathfrak{sl}_2}\) denotes the category of finite-dimensional \(\mathfrak{sl}_2({\mathbb C})\)-modules and \({\mathfrak M}od _4\) the category of finite-dimensional modules over \({\mathcal A}_4\) (the zero-algebra of dimension \(4\) over a field \(k\) algebraically closed of characteristic other than two, three and five). So, this paper uses the infinitely many non isomorphic finite-dimensional irreducible (Lie) modules over \(\mathfrak{sl}_2({\mathbb C})\) to construct an infinite collection of non isomorphic finite-dimensional irreducible \({\mathcal A}_4\)-modules. As a consequence of the existence of an infinite number of irreducible modules for \({\mathcal A}_4\) and from Lemma 6 of \textit{J. C. Gutierrez Fernandez} et al.[``On power-associative modules'', J. Algebra Appl. (to appear)], the author obtains that if \(\mathcal A\) is a commutative power-associative algebra over \(k\) such that \(\mathrm{codim}\,{\mathcal A}^2 \ge 4\), then \(\mathcal A\) has an infinite number of non isomorphic irreducible modules. In addition to other corollaries of Theorem 1, the author shows (in Section 4) that for any positive integer \(n\ge 2\) there exist two non isomorphic families of irreducible \({\mathcal A}_4\)-modules of dimension \(3n\) which are \(n\) and \(n+1\)-parametrized.
Reviewer: Dolores Martín Barquero (Málaga)Relative (pre-)anti-flexible algebras and associated algebraic structureshttps://zbmath.org/1500.170022023-01-20T17:58:23.823708Z"Dassoundo, Mafoya Landry"https://zbmath.org/authors/?q=ai:dassoundo.mafoya-landryIn this paper the notion of pre-anti-flexible family algebras has been introduced. This notion is used to define and describe the family of \(\Omega _{C}\)-relative anti-flexible algebras, left and right pre-Lie family algebras and \(\Omega _{C}\)-relative Lie algebras, where \(\Omega \) denote an associative semi-group and \(\Omega _{C}\) denote a commutative associative semi-group. Moreover, the Rota-Baxter operators are defined and studied on an \(\Omega _{C}\)-relative anti-flexible algebra. They study their properties and prove that Rota-Baxter operators and its generalization provide \(\Omega _{C}\)-relative pre-anti-flexible algebras.
Reviewer: Cristina Flaut (Constanta)Some properties of \(\mathrm{ID}_\ast\)-\(n\)-Lie-derivations of Leibniz algebrashttps://zbmath.org/1500.170032023-01-20T17:58:23.823708Z"Biyogmam, G. R."https://zbmath.org/authors/?q=ai:biyogmam.guy-roger"Tcheka, C."https://zbmath.org/authors/?q=ai:tcheka.calvin"Kamgam, D. A."https://zbmath.org/authors/?q=ai:kamgam.d-aHall introduced the notion of isoclinism for the classification of finite \(p\)-groups (\(p\) prime) long time ago in [\textit{P. Hall}, J. Reine Angew. Math. 182, 130--141 (1940; JFM 66.0081.01)]. Roughly speaking, the idea of Hall was to classify finite \(p\)-groups whenever their central quotients were isomorphic and their derived subgroups as well. There is of course a formal definition in Hall's contribution, but this is a first intuitive idea of the concept of isoclinism. The classical example is given by the dihedral group \(D\) of order eight and by the quaternion group \(Q\) of order eight; even if \(D\) and \(Q\) are not isomorphic as groups of order eight, they have isomorphic central quotients \(D/Z(D)\) and \(Q/Z(Q)\) and cyclic derived subgroups \([D,D]\) and \([Q,Q]\) of order two. According to the terminology of Hall, \(D\) and \(Q\) are in fact isoclinic nonisomorphic \(2\)-groups of order eight. This motivated Hall to introduce the notion of ``isoclinism'' for arbitrary finite \(p\)-groups.
The original idea of Hall was applied with success to wider categories of objects with corresponding morphisms; for instance, finite dimensional nilpotent Lie algebras provide a first context where generalizations are possible. Another is given in the paper under review, where the authors study a notion of isoclinism in the context of Leibniz algebras (see Definition 1.7). Then Theorem 2.7 describes a result of classification up to isoclinisms (in the sense of Definition 1.7) for Leibniz algebras; in fact some types of derivations of Leibniz algebras (those in Definition 2.4) turn out be the same when isoclinisms are present. Consequences are discussed in Corollaries 2.8, 2.11 and 2.12.
Reviewer: Francesco G. Russo (Rondebosch)On the lengths of standard composition algebrashttps://zbmath.org/1500.170042023-01-20T17:58:23.823708Z"Guterman, A. E."https://zbmath.org/authors/?q=ai:guterman.alexander-e"Zhilina, S. A."https://zbmath.org/authors/?q=ai:zhilina.s-aFor a finite dimensional algebra, the study of various numerical invariants plays a very important role. One of these invariants is the length function which was first introduced for the algebra $\mathcal{M}_{3}\left( K\right)$, with $K$ an arbitrary field, char$K\neq 2$, in the papers: [\textit{A. J. M. Spencer} and \textit{R. S. Rivlin}, Arch. Ration. Mech. Anal. 2, 309--336 (1959; Zbl 0095.25101)] and [\textit{A. J. M. Spencer} and \textit{R. S. Rivlin}, Arch. Ration. Mech. Anal. 4, 214--230 (1960; Zbl 0095.25103)]. The problem was generalized for the algebra $\mathcal{M}_{n}\left( K\right),K$ an arbitrary field and $n\in \mathbb{N}$, an arbitrary natural number, in the paper: [\textit{A. Paz}, Linear Multilinear Algebra 15, 161--170 (1984; Zbl 0536.15007)].
In this paper, the authors studied the lengths in the non-associative case and focused on composition algebras. They proposed a method with which can be computed the lengths for the standard composition algebras. Since the known methods for the length computation cannot be used in this case, the authors solved the problem by introducing new conditions, as for example the condition of descending flexibility. The paper is well written and well organised.
Reviewer: Cristina Flaut (Constanta)Primary ideals of Lie algebrashttps://zbmath.org/1500.170052023-01-20T17:58:23.823708Z"Ashour, Arwa E."https://zbmath.org/authors/?q=ai:ashour.arwa-eid"Al-Ashker, Mohammed M."https://zbmath.org/authors/?q=ai:al-ashker.mohammed-mahmoud"Al-Aydi, Mohammed A."https://zbmath.org/authors/?q=ai:al-aydi.mohammed-aLet L be a not necessarily finite dimensional Lie algebra. The popular concepts of prime and primary ideals in ring theory were introduced in Lie algebras by \textit{N. Kawamoto} [Hiroshima Math. J. 4, 679--684 (1974; Zbl 0303.17008)] and \textit{F. Aldosray} [The ideal and subideal structure of Lie algebras. University of Warwick (PhD Thesis) (1984)]. The present paper contributes greatly to these ideas. Among the results they find that every prime ideal is primary, semi-prime and strongly irreducible, every maximal ideal is primary and irreducible and a semi-simple Lie algebra is a semi-prime Lie algebra. Examples are given that show when possible results fail; for instance, a semi-simple Lie algebra may be neither prime nor primary.
Reviewer: Ernest L. Stitzinger (Raleigh)The structure of almost abelian Lie algebrashttps://zbmath.org/1500.170062023-01-20T17:58:23.823708Z"Avetisyan, Zhirayr"https://zbmath.org/authors/?q=ai:avetisyan.zhirayr-gBy definition, an almost abelian Lie algebra (AALA) is a non-abelian Lie algebra with a codimension 1 abelian ideal. The class of all almost abelian Lie algebras (AALAs) plays the important role in differential geometry, theoretical physics, anisotropic media - cosmology, crystallography, etc. Therefore, the class of AALAs has been studied by many mathematicians in the past few decades. This paper can be considered as the first one to introduce a fairly complete and systematic study of the class of AALAs. First of all, the paper introduces the structure (Propositions 1, 2, 4, 5), gives the description of subalgebras and ideals (Propositions 3) of an AALA of arbitrary dimension over any field. Next, the paper introduces the automorphism group of an indecomposable AALA (Proposition 9) and gives a classification of the class of AALAs (Proposition 10). Finally, the paper introduces an explicit description of the derivatives or Lie orthogonal operators of an AALA (Propositions 11, 13, 15, 16) and gives the description of the center of the universal enveloping algebra of an AALA (Proposition 17).
Reviewer: Le Anh Vu (Ho Chi Minh City)Whittaker modules for classical Lie superalgebrashttps://zbmath.org/1500.170072023-01-20T17:58:23.823708Z"Chen, Chih-Whi"https://zbmath.org/authors/?q=ai:chen.chih-whiThis article studies the category of Whittaker modules for classical Lie superalgebras. A Lie superalgebra \(\mathfrak{g}=\mathfrak{g}_{\overline{0}}\oplus\mathfrak{g}_{\overline{1}}\) is called classical if \(\mathfrak{g}_{\overline{0}}\) is reductive and acts semisimply on \(\mathfrak{g}\) under the adjoint action. For such a Lie superalgebra, one has a Cartan subalgebra, along with triangular decompositions \(\mathfrak{g}=\mathfrak{n}^-\oplus\mathfrak{h}\oplus\mathfrak{n}\), which unlike in the purely even case, are not all conjugate. However, up to conjugacy one may assume that \(\mathfrak{n}_{\overline{0}}\) and \(\mathfrak{n}^-_{\overline{0}}\) are fixed, which the author does. Further, for technical reasons, the author assumes that \(\mathfrak{h}=\mathfrak{h}_{\overline{0}}\).
The category \(\widetilde{\mathcal{N}}\) of Whittaker modules consists of finitely generated \(\mathfrak{g}\)-modules which are finite over \(\mathfrak{n}_{\overline{0}}\) and \(Z(\mathcal{U}\mathfrak{g}_{\overline{0}})\). This in particular implies that the modules are finite over \(\mathfrak{n}\), as \(\mathcal{U}\mathfrak{n}\) is itself finitely generated over \(\mathcal{U}\mathfrak{n}_{\overline{0}}\). Further, these conditions have the pleasant property that modules from \(\widetilde{\mathcal{N}}\) lie in the category of Whittaker modules for \(\mathfrak{g}_{\overline{0}}\) upon restriction.
The study of the category of Whittaker modules for reductive Lie algebras is well-developed (see [\textit{E. McDowell}, Proc. Am. Math. Soc. 118, No. 2, 349--354 (1993; Zbl 0774.17009); \textit{D. Miličić} and \textit{W. Soergel}, Comment. Math. Helv. 72, No. 4, 503--520 (1997; Zbl 0956.17004)]). Simple modules are classified by pairs \((\lambda,\zeta)\) where \(\lambda\in\mathfrak{h}^*\) and \(\zeta:\mathfrak{n}_{\overline{0}}\to\mathbb{C}\) is a Lie algebra homomorphism, modulo a certain relation. Further, there are standard Whittaker modules indexed by the same set which play an analogous role to Verma modules in category \(\mathcal{O}\). The composition factors of standard Whittaker modules may be computed in terms of Kazhdan-Lusztig combinatorics.
One goal of this paper is to extend the above results to the super setting. This study was initiated in [\textit{I. Bagci} et al., Commun. Algebra 42, No. 11, 4932--4947 (2014; Zbl 1364.17008)], where the case of type I superalgebras was the main focus. A type I superalgebra \(\mathfrak{g}\) is one that admits a compatible \(\mathbb{Z}\)-grading \(\mathfrak{g}=\mathfrak{g}_{-1}\oplus\mathfrak{g}_0\oplus\mathfrak{g}_1\). In particular for type I superalgebras one has the Kac-induction functor \(\operatorname{Ind}_{\mathfrak{g}_0\oplus\mathfrak{g}_{1}}^{\mathfrak{g}}(-)\), which via inflation from \(\mathfrak{g}_0\) to \(\mathfrak{g}_0\oplus\mathfrak{g}_1\) defines a powerful functor from \(\mathfrak{g}_0\)-modules to \(\mathfrak{g}\)-modules.
A main technical challenge for the study of Whittaker modules in the super setting is that \(\mathfrak{n}\) may have finite-dimensional irreducibles that are not one-dimensional; this happens whenever \([\mathfrak{n}_{\overline{1}},\mathfrak{n}_{\overline{1}}]\) does not lie in \([\mathfrak{n}_{\overline{0}},\mathfrak{n}_{\overline{0}}]\). For type I superalgebras, one may skirt around this issue by considering triangular decompositions with \(\mathfrak{n}=\mathfrak{n}_{\overline{0}}\oplus\mathfrak{g}_1\); indeed here \(\mathfrak{n}_{\overline{1}}=\mathfrak{g}_1\) is supercommutative. In this paper, the author avoids the issue for an arbitrary Lie superalgebra by restricting consideration to the irreducibles of \(\mathfrak{n}\) that are one-dimensional, and allowing \(\mathfrak{n}\) to vary.
To explain, the irreducible representations of \(\mathfrak{n}\) are indexed (up to parity) by characters \(\zeta:\mathfrak{n}_{\overline{0}}\to\mathbb{C}\). Let us write \(C_{\mathfrak{n}}(\zeta)\) for an irreducible representation corresponding to \(\zeta\) (it is unique up to parity). Note that \(C_{\mathfrak{n}}(\zeta)\) is one-dimensional if and only if \(\zeta([\mathfrak{n}_{\overline{1}},\mathfrak{n}_{\overline{1}}])=0\). By the same argument as in the even setting, one can show that we have a block decomposition \(\widetilde{\mathcal{N}}=\bigoplus\limits_{\zeta}\widetilde{\mathcal{N}}(\zeta)\). In order to ensure that a triangular decomposition exists for which \(C_{\mathfrak{n}}(\zeta)\) is one-dimensional, the author assumes that \(\mathfrak{l}_{\zeta}\), which is a Levi subalgebra of \(\mathfrak{g}_{\overline{0}}\) constructed from \(\zeta\), is also a Levi subalgebra of \(\mathfrak{g}\). We refer to the paper for the construction of \(\mathfrak{l}_{\zeta}\); it is constructed exactly as in the even setting. The author refers to such \(\zeta\) as being admissible.
Then for admissible \(\zeta\) and for \(\lambda\in\mathfrak{h}^*\), the author constructs the standard Whittaker module \(\widetilde{M}(\lambda,\zeta)\). The author then proves that \(\widetilde{M}(\lambda,\zeta)\) has a simple top \(\widetilde{L}(\lambda,\zeta)\), that every simple module in \(\widetilde{\mathcal{N}}\) is isomorphic to \(\widetilde{L}(\lambda,\zeta)\) for some \(\lambda\), and gives an exact condition on when \(\widetilde{L}(\lambda,\zeta)\cong\widetilde{L}(\mu,\zeta)\). However, unlike in the type I case, this argument leaves unconsidered the blocks \(\widetilde{\mathcal{N}}(\zeta)\) for which \(\zeta\) is not admissible. The author goes through the list of basic classical Lie superalgebras and explains which \(\zeta\) are admissible in these cases.
After the construction of standard Whittaker modules and the classification of simple modules for admissible \(\zeta\), the author goes on to prove a Miličić-Soergel type equivalence between blocks of \(\widetilde{\mathcal{N}}(\zeta)\) (again for \(\zeta\) admissible) and a category of Harish Chandra \((\mathcal{U}\mathfrak{g},\mathcal{U}\mathfrak{g}_{\overline{0}})\) bimodules. This gives rise to nontrivial equivalences of blocks of the category of Whittaker modules.
Finally, the extension of the Backelin functor to the case of type I superalgebras is studied. It is shown to take Verma modules to standard Whittaker modules, simple modules to either simple Whittaker modules or 0, and Kac-inductions of simple modules to Kac-inductions of simple Whittaker modules or 0. Using this, one obtains that the calculation of composition factors in a standard Whittaker module for a type I superalgebra can be reduced to the computation of composition factors of Verma modules in category \(\mathcal{O}\) (see Theorem C for a precise result). For \(\mathfrak{gl}(m|n)\) and \(\mathfrak{osp}(2|2n)\) such computations are known.
Reviewer: Alexander Sherman (Berkeley)A basis theorem for the affine Kauffman category and its cyclotomic quotientshttps://zbmath.org/1500.170082023-01-20T17:58:23.823708Z"Gao, Mengmeng"https://zbmath.org/authors/?q=ai:gao.mengmeng"Rui, Hebing"https://zbmath.org/authors/?q=ai:rui.hebing"Song, Linliang"https://zbmath.org/authors/?q=ai:song.linliangLet \(\mathcal{K}\) denote the Kauffman category of tangle diagrams over an integral domain \(k\) as introduced by \textit{V. Turaev} [Math. USSR, Izv. 35, No. 2, 411--444 (1990; Zbl 0707.57003)]. Turaev showed that \(\mathcal{K}\) is a strict \(k\)-linear monoidal category generated by a single object and four elementary morphisms (subject to a set of 9 relations) and identified a basis for homomorphism spaces in \(\mathcal{K}\). The goal of this work is to introduce and study an affine Kauffman category \(\mathcal{AK}\) along with certain cyclotomic quotients in a manner that parallels earlier work of the second and third authors [Math. Z. 293, Nos. 1--2, 503--550 (2019; Zbl 1461.18013)], where they constructed an affine Brauer category. The category \(\mathcal{AK}\) is again a strict \(k\)-linear monoidal category generated by a single object with two additional elementary morphisms (and three additional relations).
\par With the objects in \(\mathcal{K}\) or \(\mathcal{AK}\) identified with the natural numbers \({\mathbb N}\), given \(m, s \in {\mathbb N}\), the first main result is an identification of \(\Hom_{\mathcal{AK}}(m,s)\). As for \(\mathcal{K}\), this Hom-space is zero if \(m + s\) is odd. When \(m + s\) is even, an infinite \(k\)-basis is given in terms of certain equivalence classes of normally ordered tangle diagrams. The authors then introduce a family of cyclotomic quotients of \(\mathcal{AK}\) determined by two functions (or, equivalently, collections of elements in \(k\)): \({\mathcal C}(\omega,{\mathbf u})\) for \(\omega : {\mathbb Z} \to k\) and \({\mathbf u} : \{1, 2, \dots, a\} \to k^{\times}\) (for some \(a\)). That is, \(\omega\) represents an infinite sequence of elements in \(k\), while \({\mathbf u}\) represents a finite set of units in \(k\). Further, one needs certain ``admissibility'' conditions on \(\omega\). When working with cyclotomic categories, the authors also add the assumption that \(k\) contains a unit \(q\) for which \(q - q^{-1}\) is also a unit. The second main result is an identification of a basis (this time finite) for the homomorphisms \(\Hom_{\mathcal{CK}}(\omega,\mathbf{u})(m,s)\).
To obtain the main results, the authors first identify an appropriate spanning set in the \(\mathcal{AK}\)-case. The more challenging task is showing linear independence. The problem is first reduced to identifying a basis for \(\Hom_{\mathcal{AK}}(2m,0)\) (and similarly for the cyclotomic case). To deal with this case, the authors make use of symplectic and orthogonal quantum groups. More precisely, one considers a Lusztig quantum group \(\mathbf{U}_v(\mathfrak{g})\) over a an extension of the complex numbers for an indeterminate \(v\) and simple complex symplectic or special orthogonal Lie algebra \(\mathfrak{g}\). Much of the paper is devoted to computations in \({\mathbf U}_v(\mathfrak{g})\) and its action on certain modules and tensor products. For the cyclotomic case, one also needs to consider the BGG category \(\mathcal{O}\) inside the category of \(\mathbf{U}_v(\mathfrak{g})\)-modules. Ultimately, the authors construct monoidal functors from \(\mathcal{K}\) to \(\mathbf{U}_v(\mathfrak{g})\)-mod and then from \(\mathcal{AK}\) to \(\mathrm{End}(\mathbf{U}_v(\mathfrak{g})\text{-mod})\) (the category of endofunctors). This allows them to translate the problem to one involving an action of \(\mathcal{AK}\) on certain \(U_v(\mathfrak{g})\)-modules.
\par As consequences of their main results, it is shown that endomorphism algebras for affine and cyclotomic Kauffman categories are isomorphic (as \(k\)-algebras) to affine and cyclotomic Birman-Murakami-Wenzl algebras, respectively. Such a connection was one of the motivations for this work, with results to be applied in more recent work of the authors [``Representations of weakly triangulated categories'', J. Algebra 614, 481--534 (2023)] in the realm of categorification.
Reviewer: Christopher P. Bendel (Menomonie)Snowflake modules and Enright functor for Kac-Moody superalgebrashttps://zbmath.org/1500.170092023-01-20T17:58:23.823708Z"Gorelik, Maria"https://zbmath.org/authors/?q=ai:gorelik.maria"Serganova, Vera"https://zbmath.org/authors/?q=ai:serganova.vera-vThe paper under review introduces a new class of modules over Kac-Moody superalgebras. The modules in this class are called snowflake modules, they are characterized by a certain invariance property of their characters with respect to a subgroup of the Weyl group. The main technical tool used in the paper is Enright's functor. The main application is a proof of Arakawa's Theorem for the Lie superalgebra \(\mathfrak{osp}(1\vert 2l)^{(1)}\). This theorem asserts that, for an admissible level \(k\), the corresponding simple vertex affine algebra is rational in category \(\mathcal{O}\). In fact, in this case all modules over this vertex affine algebra in category \(\mathcal{O}\) are certain snowflake modules.
Reviewer: Volodymyr Mazorchuk (Uppsala)Branching from the general linear group to the symmetric group and the principal embeddinghttps://zbmath.org/1500.170102023-01-20T17:58:23.823708Z"Heaton, Alexander"https://zbmath.org/authors/?q=ai:heaton.alexander"Sriwongsa, Songpon"https://zbmath.org/authors/?q=ai:sriwongsa.songpon"Willenbring, Jeb F."https://zbmath.org/authors/?q=ai:willenbring.jeb-fSummary: Let \(S\) be a principally embedded \(\mathfrak{sl}_2\)-subalgebra in \(\mathfrak{sl}_n\) for \(n\ge 3\). A special case of results of \textit{J. F. Willenbring} and \textit{G. J. Zuckerman} [Lect. Notes Ser., Inst. Math. Sci., Natl. Univ. Singap. 12, 403--429 (2007; Zbl 1390.17017)] implies that there exists a positive integer \(b(n)\) such that for any finite-dimensional irreducible \(\mathfrak{sl}_n\)-representation, \(V\), there exists an irreducible \(S\)-representation embedding in \(V\) with dimension at most \(b(n)\). In [\textit{H. Lhou} and \textit{J. F. Willenbring}, Represent. Theory 21, 20--34 (2017; Zbl 1419.17014)], they prove that \(b(n)=n\) is the sharpest possible bound, and also address embeddings other than the principal one.
These results concerning embeddings may be interpreted as statements about \textit{plethysm}. Then, in turn, a well known result about these plethysms can be interpreted as a ``branching rule''. Specifically, a finite dimensional irreducible representation of \(\mathrm{GL}(n,\mathbb{C})\) will decompose into irreducible representations of the symmetric group when it is restricted to the subgroup consisting of permutation matrices. The question of which irreducible representations of the symmetric group occur with positive multiplicity is the topic of this paper, applying the previous work of Lhou, Zuckerman, and the third author.Invariants of the Weyl group of type \(A^{(2)}_{2l}\)https://zbmath.org/1500.170112023-01-20T17:58:23.823708Z"Iohara, Kenji"https://zbmath.org/authors/?q=ai:iohara.kenji"Saito, Yoshihisa"https://zbmath.org/authors/?q=ai:saito.yoshihisaIn the paper under review the authors ``determine the Jacobian of the fundamental characters for the affine Lie algebra g of type \(A_{2l}^{(1)}\). As a corollary, it follows that the fundamental characters for this Lie algebra are algebraically independent.''
Reviewer: Dmitry Artamonov (Moskva)Global bases for quantum Borcherds-Bozec algebrashttps://zbmath.org/1500.170122023-01-20T17:58:23.823708Z"Fan, Zhaobing"https://zbmath.org/authors/?q=ai:fan.zhaobing"Kang, Seok-Jin"https://zbmath.org/authors/?q=ai:kang.seok-jin"Kim, Young Rock"https://zbmath.org/authors/?q=ai:kim.youngrock"Tong, Bolun"https://zbmath.org/authors/?q=ai:tong.bolunConsider a quiver with loops. The Grothendieck group arising from Lusztig sheaves on representation varieties is generated by the elementary simple perverse sheaves \(F_i^{(n)}\) with all vertices \(i\) and \(n\in \mathbb{N}\). Bozec showed that this Grothendieck group is isomorphic to \(U_q^-(\mathfrak{g})\), which gives a construction of its canonical basis.
Z. Fan, S.-J. Kang, Y. R. Kim and B. Tong give an explicit description of the radical of Lusztig's bilinear form (Theorem 4, page 3731). The authors also obtain higher order Serre relations, which have more general forms, resulting in a new presentation of the quantum Borcherds-Bozec algebra in terms of primitive generators. They prove the existence and uniqueness of global bases for quantum Borcherds-Bozec algebras and their integrable highest weight modules (Theorem 26, page 3751), conjecturing that their global bases coincide with a variation of Bozec's canonical bases.
Reviewer: Mee Seong Im (Annapolis)Representations of quantum affine algebras in their \(R\)-matrix realizationhttps://zbmath.org/1500.170132023-01-20T17:58:23.823708Z"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-iIn this paper the authors study finite-dimensional irreducible representations, in the R-matrix realization, of the quantum affine algebras in types B, C and D using the isomorphisms between the R-matrix and Drinfeld presentations. In the case of Yangians of types B, C and D, the authors use the Gauss decomposition to establish an equivalence of the descriptions of the representations in the R-matrix and Drinfeld presentations.
Reviewer: Nenad Manojlović (Faro)A note on odd reflections of super Yangian and Bethe ansatzhttps://zbmath.org/1500.170142023-01-20T17:58:23.823708Z"Lu, Kang"https://zbmath.org/authors/?q=ai:lu.kang.1|lu.kangThe classification of the finite-dimensional irreducible representations of the super Yangian \(Y(\mathfrak{gl}_{m|n})\) given in the work of \textit{ R.-B. Zhang} [Lett. Math. Phys. 37, 419--434 (1996; Zbl 0861.17020)] relies on the presentation of the Yangian corresponding to the standard parity sequence. \textit{A. Molev} in his work [Lett. Math. Phys. 112, No. 1, Paper No. 8, 15 p. (2022; Zbl 1485.17025)], by introducing analogues of the odd reflections for the Yangian \(Y(\mathfrak{gl}_{m|n})\), derived a transition rule for the change of the highest weight when the parity sequence is altered by switching two neighbors.
Author of this work reproduces a similar transition rule, described in terms of Drinfeld-type presentations for \(Y(\mathfrak{gl}_{m|n})\). This is done by obtaining the transition rule for the \(Y(\mathfrak{gl}_{1|1})\) case and reducing the case of \(Y(\mathfrak{gl}_{m|n})\) to the \(Y(\mathfrak{gl}_{1|1})\) case. Such transition rule can also be obtained using the fermionic reproduction procedure of the Bethe ansatz equation for XXX spin chains introduced in the work of the author et al. in [``Solutions of \({\mathfrak{gl}_{m\vert n}}\) XXX Bethe ansatz equation and rational difference operators'', J. Phys. A, Math. Theor. 52, No. 37, Article ID 375204, 27 p. (2019; \url{doi:10.1088/1751-8121/ab1960})]. The author also gave an algorithm how \(q\)-characters change under the odd reflections.
Reviewer: Marijana Butorac (Rijeka)\(\mathcal{O}\)-operators on Lie \(\infty\)-algebras with respect to Lie \(\infty\)-actionshttps://zbmath.org/1500.170152023-01-20T17:58:23.823708Z"Caseiro, Raquel"https://zbmath.org/authors/?q=ai:caseiro.raquel"Nunes da Costa, Joana"https://zbmath.org/authors/?q=ai:nunes-da-costa.joana-margarida-mThe \(\mathcal{O}\)-operators are relative version of Rota-Baxter operators. They appear in many algebraic and geometric settings. In the case of Lie algebras, they are defined as follows. For a Lie algebra \((E,[])\) and a representation \(\Phi: E \rightarrow \mathrm{End}(V)\), an \(\mathcal{O}\)-operator for \((E,\Phi)\) refers to a linear map \(T: V\rightarrow E\) with \([T(x),T(y)]=T(\Phi(T(x))(y)-\Phi(T(y))(x))\) for all \(x,y \in V\). The \(\mathcal{O}\)-operators for Lie \(\infty\)-algebras were introduced in [\textit{A. Lazarev} et al., Commun. Math. Phys. 383, No. 1, 595--631 (2021; Zbl 1476.17010); \textit{R. Tang} et al., ``Homotopy Rota-Baxter operators, homotopy O-operators and homotopy post-Lie algebras'', Preprint, \url{arXiv:1907.13504}] by the representations of Lie \(\infty\)-algebras on the graded vector spaces. In the paper under review, the authors study \(\mathcal{O}\)-operators on Lie \(\infty\)-algebras. They defined \(\mathcal{O}\)-operators using Lie \(\infty\)-algebra action on another Lie \(\infty\)-algebra instead of their representations as introduced above. Then they characterize those \(\mathcal{O}\)-operators as Maurer-Cartan elements of a certain Lie \(\infty\)-algebra obtained by Voronov's higher derived brackets construction. They also determined the Lie-\(\infty\) action controlling the deformations of \(\mathcal{O}\)-operators.
Reviewer: Husileng Xiao (Harbin)Representations and cohomologies of relative Rota-Baxter Lie algebras and applicationshttps://zbmath.org/1500.170162023-01-20T17:58:23.823708Z"Jiang, Jun"https://zbmath.org/authors/?q=ai:jiang.jun"Sheng, Yunhe"https://zbmath.org/authors/?q=ai:sheng.yunheA \textbf{Rota-Baxter Lie algebra of weight \(\lambda\)} is a triple \((\mathfrak{g},[\,,\,]_\mathfrak{g}, T)\), where \((\mathfrak{g},[\,,\,])\) is a Lie algebra and \(T\) is a linear operator on \(\mathfrak{g}\) satisfying \[[T(x),T(y)]_\mathfrak{g}=T[T(x),y]_\mathfrak{g}+T[x,T(y)]_\mathfrak{g}+\lambda T[x,y]_\mathfrak{g},\quad\forall x,y\in\mathfrak{g}.\] Such a linear operator \(T\) is called a \textbf{Rota-Baxter operator of weight \(\lambda\)}.
Let \(V\) be a vector space. A \textit{ relative Rota-Baxter Lie algebra} is a triple \(((\mathfrak{g},[\,,\,]_\mathfrak{g}),(V,\rho), T)\), where \((\mathfrak{g},[\,,\,]_\mathfrak{g})\) is a Lie algebra, \(\rho:\mathfrak{g}\longrightarrow \mathfrak{gl}(V)\) is a representation of \(\mathfrak{g}\) on \(V\) and \(T:V\longrightarrow \mathfrak{g}\) is a \textit{relative Rota-Baxter operator}, that is, \(T\) satisfies the following identity: \[[T(x),T(y)]_\mathfrak{g}=T(\rho(T(x))(y))-T(\rho(T(y))(x)),\quad\forall x,y\in V.\]
Taking the representation \(\rho\) of \(\mathfrak{g}\) to be the adjoint representation of \(\mathfrak{g}\), one can deduce that a relative Rota-Baxter operator is a Rota-Baxter operator of weight zero on \(\mathfrak{g}\). Thus a relative Rota-Baxter operator is a natural generalization of a Rota-Baxter operator of weight zero.
In this paper, the authors first introduce the representation of a relative Rota-Baxter Lie algebra \(((\mathfrak{g},[\,,\,]_\mathfrak{g}),(V,\rho), T)\) on a two-term complex of vector spaces \(W\stackrel{\mathcal{T}}{\longrightarrow}\mathfrak{h}\) and develop the cohomology theory of a relative Rota-Baxter Lie algebra. The relationship between the cohomology of a relative Rota-Baxter operator and the cohomology of the underlying pre-Lie algebra is given. At last, by using the obtained general framework of a relative Rota-Baxter Lie algebra, the representations and cohomologies of a Rota-Baxter Lie algebra are investigated. By applying the second cohomology group, the abelian extensions of a Rota-Baxter Lie algebra are classified.
Reviewer: Shanghua Zheng (Nanchang)On simple 15-dimensional Lie algebras in characteristic 2https://zbmath.org/1500.170172023-01-20T17:58:23.823708Z"Grishkov, Alexander"https://zbmath.org/authors/?q=ai:grishkov.alexander-n"Guzzo, Henrique jun."https://zbmath.org/authors/?q=ai:guzzo.henrique-jun"Rasskazova, Marina"https://zbmath.org/authors/?q=ai:rasskazova.marina"Zusmanovich, Pasha"https://zbmath.org/authors/?q=ai:zusmanovich.pashaIn a previous paper by two of the authors: [\textit{A. Grishkov} and \textit{P. Zusmanovich}, J. Algebra 473, 513--544 (2017; Zbl 1415.17017)], a two-parameter family \(\mathcal{L}(\beta,\delta)\) of simple Lie algebras over a field of characteristic \(2\) was considered, with \(\mathcal{L}=\mathcal{L}(0,0)\) being an algebra first considered by \textit{S. Skryabin} [J. Algebra 200, No. 2, 650--700 (1998; Zbl 0894.17018)]. The first main result of the paper under review shows that all the Lie algebras in this family are isomorphic to Skryabin's algebra \(\mathcal{L}\).
Then quite a number of properties of Skryabin's algebra are obtained:
\begin{itemize}
\item \(\mathcal{L}\) is not reduced and its \(2\)-envelope has dimension \(19\) and coincides with its Lie algebra of derivations.
\item The subalgebra spanned by the sandwich elements (\(\operatorname{ad}_x^2=0\), \([[\mathcal{L},x],[\mathcal{L},x]]=0\)) has dimension \(3\).
\item The absolute toral rank is \(4\).
\item The root space decomposition relative to a maximal torus gives a fine grading \(\mathcal{L}=\bigoplus_{0\neq \alpha\in\left(\mathbb{Z}/2\right)^4} \mathcal{L}_\alpha\), where all (nonzero) homogeneous components are one-dimensional. (These are called \textit{thin} gradings here.)
\item The automorphism group is solvable and eight-dimensional, and hence the automorphism group scheme is far from being smooth.
\item Over the field of two elements, \(\mathcal{L}\) does not appear in the list of simple Lie algebras obtained by \textit{B. Eick} [J. Symb. Comput. 45, No. 9, 943--951 (2010; Zbl 1229.17024)].
\end{itemize}
The paper finishes with an interesting list of open questions related to simple Lie algebras over fields of characteristic \(2\).
Reviewer: Alberto Elduque (Zaragoza)Crossed modules for Hom-Lie antialgebrashttps://zbmath.org/1500.170182023-01-20T17:58:23.823708Z"Zhang, Tao"https://zbmath.org/authors/?q=ai:zhang.tao.6|zhang.tao.2|zhang.tao.1|zhang.tao.4|zhang.tao.5"Zhang, Heyu"https://zbmath.org/authors/?q=ai:zhang.heyuLie antialgebras form a peculiar class of Jordan superalgebras introduced by \textit{V. Ovsienko} [J. Algebra 325, No. 1, 216--247 (2011; Zbl 1247.17006)]. Very roughly, these algebras relate the classes of Lie algebras and of associative commutative algebras via a \(\mathbb Z/2\mathbb Z\)-graded structure. Hom-Lie algebras are generalizations of Lie algebras, where the Jacobi identity is twisted by a linear map. Crossed modules of various algebraic structures (groups, Lie algebras, associative algebras, etc., and, more generally, of (semi)abelian categories) is a classical notion arising from the action of one structure on another. Here the authors combine these 3 notions together with a lot of complicated formulas.
The part describing crossed modules in terms of the respective third cohomology is problematic, as it is ultimately based on cohomology theory of Lie antialgebras developed in the preprint by \textit{P. B. A. Lecomte} and \textit{V. Ovsienko} [``Alternated Hochschild cohomology'', Preprint, \url{arXiv:1012.3885}]. The latter preprint has a fatal flaw, as the complex constructed there is not really a cochain complex (the square of differential does not vanish). (The reviewer is grateful to Valentin Ovsienko for this comment).
Reviewer: Pasha Zusmanovich (Ostrava)Finite W-algebras associated to truncated current Lie algebrashttps://zbmath.org/1500.170192023-01-20T17:58:23.823708Z"He, Xiao"https://zbmath.org/authors/?q=ai:he.xiaoTruncated current algebras are constructed in the following way. Let \(\mathfrak g\) be a Lie algebra over \(\mathbb C\), \(p\) be a non-negative integer, and \(A = \mathbb C[t]/\langle t^{p+1} \rangle\) be the quotient of the algebra of polynomials in a variable \(t\) by the ideal generated by \(t^{p+1}\). The corresponding \textit{truncated current algebra} is defined to be the vector space \(\mathfrak g \otimes_{\mathbb C} A\) endowed with the unique Lie bracket that satisfies:
\[
[x \otimes a, \, y \otimes b ] = [x, y]_{\mathfrak g} \otimes (a \cdot_{_A} b) \qquad \text{for all } x, y \in \mathfrak g \ \text{ and } \ a, b \in A.
\]
In this paper, the author defines finite \(W\)-algebras associated to truncated current algebras. Then, the author proves an analog of Kostant's Theorem, which in this case relates a finite \(W\)-algebra to the center of the universal enveloping algebra of the corresponding truncated current algebra. Finally, the author proves an analog of Skryabin's equivalence of categories, which in this case is an equivalence between the category of finitely-generated Whittaker modules for a truncated current algebra and the category of finitely-generated left modules for the corresponding finite \(W\)-algebra.
Reviewer: Tiago Macedo (São Paulo)On rigid 3-dimensional Hom-Lie algebrashttps://zbmath.org/1500.170202023-01-20T17:58:23.823708Z"Alvarez, María Alejandra"https://zbmath.org/authors/?q=ai:alvarez.maria-alejandra"Vera, Sonia"https://zbmath.org/authors/?q=ai:vera.soniaThe paper under review explores and determines all the rigid 3-dimensional multiplicative Hom-Lie algebras over the field of complex numbers. A key to reach the goal is to study deformations of multiplicative Hom-Lie algebras whose product is also a Lie bracket. A byproduct of the paper is to recover the well-known classification of 3-dimensional multiplicative (non-Lie) Hom-Lie algebras.
Reviewer: Yin Chen (Changchun)Coherent categorical structures for Lie bialgebras, Manin triples, classical \(r\)-matrices and pre-Lie algebrashttps://zbmath.org/1500.170212023-01-20T17:58:23.823708Z"Bai, Chengming"https://zbmath.org/authors/?q=ai:bai.chengming"Guo, Li"https://zbmath.org/authors/?q=ai:guo.li"Sheng, Yunhe"https://zbmath.org/authors/?q=ai:sheng.yunheThe paper under review presents a categorical structure to each of the classes of Lie bialgebras, Manin triples, classical \(r\)-matrices, \(\mathcal{O}\)-operators and pre-Lie algebras by introducing their morphisms that are compatible with natural correspondences among these classes, so that these correspondences become functors. First,the paper introduces the notion of an endo Lie algebra as a triple \((\mathfrak{g}, [\cdot,\cdot], \phi)\) where \((\mathfrak{g}, [\cdot,\cdot]\) is a Lie algebra and \(\phi:\mathfrak{g}\rightarrow\mathfrak{g}\) is a Lie algebra endomorphism. Then the authors give the equivalent structures of bialgebras, matched pairs and Manin triples for endo Lie algebras. As the next step, the authors extend the classical relations of Lie bialgebras with the classical Yang-Baxter equation as well as classical \(r\)-matrices to the context of endo Lie algebras. This naturally gives rise to a notion of coherent homomorphisms for all \(r\)-matrices, not just the skew-symmetric ones. This notion is shown to be compatible with the coherent homomorphisms of Lie bialgebras, leading to a functor of the corresponding categories. Then the paper introduces the notion of \(\mathcal{O}\)-operators on endo Lie algebras and apply it to define coherent homomorphisms of \(\mathcal{O}\)-operators in such a way that they are compatible with the previously considered coherent homomorphisms of classical \(r\)-matrices. The notion of coherent homomorphisms of \(\mathcal{O}\)-operators is moreover compatible with the natural notion of homomorphism of pre-Lie algebras, giving rise to a pair of adjoint functors between the corresponding two categories. The authors also consider a case where all constructions can be given explicitly, providing natural examples of coherent isomorphisms of Lie bialgebras that are not the previously defined isomorphisms. This further justifies the significance of the coherent homomorphisms of Lie bialgebras introduced in this paper.
Reviewer: Alexander B. Levin (Washington)A new class of irreducible modules over the affine-Virasoro algebra of type \(A_1\)https://zbmath.org/1500.170222023-01-20T17:58:23.823708Z"Chen, Qiu-Fan"https://zbmath.org/authors/?q=ai:chen.qiufan"Yao, Yu-Feng"https://zbmath.org/authors/?q=ai:yao.yufeng.2|yao.yufeng.1It is well known that the Virasoro algebra acts on highest-weight modules for affine Kac-Moody Lie algebras of level \(1\) (for example) through the Sugawara construction. The affine-Virasoro algebra is the Lie algebra spanned by both the affine and Virasoro operators acting on such highest-weight modules. In the paper under review, the affine-Virasoro algebra associated to \(\mathfrak{sl}_2\) is considered. This Lie algebra has a central basis element \(C\), basis elements \(e_i\), \(f_i\), \(h_i\), \(i\in\mathbb{Z}\), whose commutation relations are those of the affine Lie algebra \(\widehat{\mathfrak{sl}}_2\), and basis elements \(d_i\), \(i\in\mathbb{Z}\), whose commutation relations are those of the Virasoro algebra. The remaining commutation relations are
\[
[d_i, a_j]=j a_{i+j}
\]
for \(a\in\lbrace e,f,h\rbrace\) and \(i,j\in\mathbb{Z}\).
In [\textit{Q. Chen} and \textit{J. Han}, J. Math. Phys. 60, No. 7, Article ID 071707, 9 p. (2019; Zbl 1416.81077)], four families of irreducible non-weight modules for the affine-Virasoro algebra of \(\mathfrak{sl}_2\) were constructed. The Cartan subalgebra acts non-semisimply on these modules: they are free of rank \(1\) as \(\mathbb{C}[h_0,d_0]\)-modules. In the paper under review, the authors consider tensor products of finitely many of these irreducible non-weight modules with one irreducible highest-weight module. They find simple necessary and sufficient conditions for such tensor products to remain irreducible as modules for the affine-Virasoro algebra, and they show that two such tensor product modules are isomorphic to each other if and only if their tensor factors are pairwise isomorphic (possibly after rearrangement). They also show that these tensor product modules are not isomorphic to previously-constructed irreducible non-weight modules for the affine-Virasoro algebra of \(\mathfrak{sl}_2\), such as Whittaker modules and \(\mathbb{C}[h_0,d_0]\)-free modules of rank \(1\).
Reviewer: Robert McRae (Beijing)Gluing vertex algebrashttps://zbmath.org/1500.170232023-01-20T17:58:23.823708Z"Creutzig, Thomas"https://zbmath.org/authors/?q=ai:creutzig.thomas"Kanade, Shashank"https://zbmath.org/authors/?q=ai:kanade.shashank"McRae, Robert"https://zbmath.org/authors/?q=ai:mcrae.robertSummary: We relate commutative algebras in braided tensor categories to braid-reversed tensor equivalences, motivated by vertex algebra representation theory. First, for \(\mathcal{C}\) a braided tensor category, we give a detailed account of the canonical algebra construction in the Deligne product \(\mathcal{C} \boxtimes \mathcal{C}^{\operatorname{rev}} \). Especially, we show that if \(\mathcal{C}\) is semisimple but not necessarily finite or rigid, then \(\bigoplus_{\mathsf{X} \in \operatorname{Irr} ( \mathcal{C} )} \mathsf{X}^\prime \boxtimes \mathsf{X}\) is a commutative algebra, where \(\mathsf{X}^\prime\) is a representing object for the functor \(\operatorname{Hom}_{\mathcal{C}}(\bullet \otimes_{\mathcal{C}} \mathsf{X}, 1_{\mathcal{C}})\) (assuming \(\mathsf{X}^\prime\) exists) and the sum runs over all inequivalent simple objects of \(\mathcal{U} \). Conversely, let \(\mathsf{A} = \bigoplus_{i \in I} \mathsf{U}_i \boxtimes \mathsf{V}_i\) be a simple commutative algebra in a Deligne product \(\mathcal{U} \boxtimes \mathcal{V}\) with \(\mathcal{U}\) semisimple and rigid but not necessarily finite, and \(\mathcal{V}\) rigid but not necessarily semisimple. We show that if the unit objects \(1_{\mathcal{U}}\) and \(1_{\mathcal{V}}\) form a commuting pair in \(\mathsf{A}\) in a suitable sense, then there is a braid-reversed equivalence between (sub)categories of \(\mathcal{U}\) and \(\mathcal{V}\) that sends \(\mathsf{U}_i\) to \(\mathsf{V}_i^\ast \). These results apply when \(\mathcal{U}\) and \(\mathcal{V}\) are braided (vertex) tensor categories of modules for simple vertex operator algebras \(\mathsf{U}\) and \(\mathsf{V} \), respectively: Given \(\tau : \operatorname{Irr}(\mathcal{U}) \to \operatorname{Obj}(\mathcal{V})\) such that \(\tau(\mathsf{U}) = \mathsf{V} \), we glue \(\mathsf{U}\) and \(\mathsf{V}\) along \(\mathcal{U} \boxtimes \mathcal{V}\) via \(\tau\) to create \(\mathsf{A} = \bigoplus_{\mathsf{X} \in \operatorname{Irr} ( \mathcal{U} )} \mathsf{X}^\prime \otimes \tau(\mathsf{X})\). Then under certain conditions, \( \tau\) extends to a braid-reversed equivalence between \(\mathcal{U}\) and \(\mathcal{V}\) if and only if \(\mathsf{A}\) has a simple conformal vertex algebra structure that (conformally) extends \(\mathsf{U} \otimes \mathsf{V} \). As examples, we glue suitable Kazhdan-Lusztig categories at generic levels to construct new vertex algebras extending the tensor product of two affine vertex subalgebras, and we prove braid-reversed equivalences between certain module subcategories for affine vertex algebras and \(W\)-algebras at admissible levels.On a family of vertex operator superalgebrashttps://zbmath.org/1500.170242023-01-20T17:58:23.823708Z"Li, Haisheng"https://zbmath.org/authors/?q=ai:li.haisheng"Yu, Nina"https://zbmath.org/authors/?q=ai:yu.ninaSummary: This paper is to study vertex operator superalgebras which are strongly generated by their weight-2 and weight-\( \frac{ 3}{ 2}\) homogeneous subspaces. Among the main results, it is proved that if such a vertex operator superalgebra \(V\) is simple, then \(V_{(2)}\) has a canonical commutative associative algebra structure equipped with a non-degenerate symmetric associative bilinear form and \(V_{(\frac{3}{2})}\) is naturally a \(V_{(2)}\)-module equipped with a \(V_{(2)}\)-valued symmetric bilinear form and a non-degenerate \(( \mathbb{C} \)-valued) symmetric bilinear form, satisfying a set of conditions. On the other hand, assume that \(A\) is any commutative associative algebra equipped with a non-degenerate symmetric associative bilinear form and assume that \(U\) is an \(A\)-module equipped with a symmetric \(A\)-valued bilinear form and a non-degenerate \((\mathbb{C} \)-valued) symmetric bilinear form, satisfying the corresponding conditions. Then we construct a Lie superalgebra \(\mathcal{L}(A, U)\) and a simple vertex operator superalgebra \(L_{\mathcal{L} (A, U)}(\ell, 0)\) for every nonzero number \(\ell\) such that \(L_{\mathcal{L} (A, U)} (\ell, 0)_{(2)} = A\) and \(L_{\mathcal{L} (A, U)} (\ell, 0)_{(\frac{3}{2})} = U\).On a problem by Nathan Jacobsonhttps://zbmath.org/1500.170252023-01-20T17:58:23.823708Z"López Solís, Victor Hugo"https://zbmath.org/authors/?q=ai:lopez-solis.victor-hugo"Shestakov, Ivan Pavlovich"https://zbmath.org/authors/?q=ai:shestakov.ivan-pThe classical Wedderburn coordinatization theorem says that if a unital associative algebra \(A\) contains a matrix algebra \(M_n(F)\) with the same identity element, then it is itself a matrix algebra, \( A \cong M_n(D)\), ``coordinated'' by \(D\). Generalizations and analogs of this theorem were proved for various classes of algebras and superalgebras. The common content of all these results is that if an algebra (or superalgebra) contains a certain subalgebra (matrix algebra, octonions, Albert algebra) with the same identity element, then the algebra itself has the same structure, but not over the basic field rather over a certain algebra that ``coordinatizes'' it. In the present paper, the authors proved a coordinatization theorem for unital alternative algebras containing \(2 \times 2\) matrix algebra with the same identity element.
Let \(B\) be an associative unital algebra and let \(V\) be a left \(B\)-module such that \([B,B]\) annihilates \(V.\) Clearly, in this case \(V\) has a structure of a commutative \(B\)-bimodule with \(v \cdot b = b \cdot v,\) \(v \in V, b \in B.\) Assume that there exists a \(B\)-bilinear skew-symmetric mapping \(\langle, \rangle: V \times V \to B\) such that \(\langle V,V \rangle \subseteq Z(B)\) and formula
\[
\langle u,v \rangle w+ \langle v,w \rangle u +\langle w, u\rangle v =0
\]
holds for any \(u, v, w \in V.\) Let \(A= M_2(B) \oplus V^2,\) where \(V^2 = \{(u, v) | u, v \in V \} \cong V \oplus V.\) Let \(X=X_a +(x, y),\) \(Y=Y_a +(z, t),\) where \(X_a, Y_a \in M_2(B)\) and \((x, y), (z, t) \in V^2.\) Define a product in \(A\) by the formula: \[
XY = X_aY_a + \left( \begin{array}{cc} -\langle x,t \rangle & - \langle y,t \rangle \\
\langle x,z \rangle & \langle y,z \rangle \end{array}\right) +(z, t) X_a + (x, y) Y_a^*, \]
where \(\left( \begin{array}{cc} a & b \\
c & d \end{array}\right)^*=\left( \begin{array}{cc} d & -b \\
-c & a \end{array}\right).\) The main result of the paper is given in the following theorem.
Theorem 5.1. The algebra \(A\) with the product defined above is an alternative unital algebra containing \(M_2(F)\) with the same unit. Conversely, every unital alternative algebra that contains the matrix algebra \(M_2(F)\) with the same unit has this form.
Reviewer: Ivan Kaygorodov (Covilhã)Cocycle deformations for Hom-Hopf algebrashttps://zbmath.org/1500.170262023-01-20T17:58:23.823708Z"Alonso Álvarez, J. N."https://zbmath.org/authors/?q=ai:alonso-alvarez.jose-nicanor"Fernández Vilaboa, J. M."https://zbmath.org/authors/?q=ai:fernandez-vilaboa.jose-manuel"González Rodríguez, R."https://zbmath.org/authors/?q=ai:gonzalez-rodriguez.ramonThis article is quite extensive, so we only give a review of the main result therein. In this work the authors introduce the analogue of 2-cocycle for Hopf algebra in the setting of Hom-algebras, that they called it \textit{Hom-2-cocycle}. With this notion of cocycle, the authors give a theory of multiplication alteration for Hom-Hopf-algebras. The authors prove that if the endomorphism \(\alpha\) associated to a Hom-Hopf algebra \(H\) satisfies \(\alpha^4=\alpha\) and \(H\) has a convolution invertible Hom-2-cocycle, then it is possible to define a new product in \(H\) to get a new Hom-Hopf algebra. The authors assume that the endomorphism \(\alpha\) is not an isomorphism. Due to in a previous work, is shown that if \(\alpha\) is an isomorphism then the Hom-algebras arising are classical algebras in a suitable category implying that the classical results of Hopf-algebras can be translated to the Hom setting.
\smallskip
There are different approaches to state a definition of Hom-algebra, but all known coincide with the classical definition of algebra when \(\alpha\) is the identity. To define a Hom-algebra, the authors begin by considering a strict symmetric monoidal category \(\mathcal{C}\) with unit object \(K\), tensor product \(\otimes\), and a natural isomorphism of symmetry \(c\). A \textit{magma} in \(\mathcal{C}\) is a pair \((A,\mu_A)\) such that \(\mu_A:A \otimes A \to A\) (product) is a morphism in \(\mathcal{C}\). A \textit{unital magma} in \(\mathcal{C}\) is a triple \(\mathbf{A}=(A,\eta_A,\mu_A)\) where \((A,\mu_A)\) is a magma in \(\mathcal{C}\) and \(\eta_A:K \to A\) (unit) is a morphism in \(\mathcal{C}\) such that \(\mu_A \circ (A \otimes \eta_A)=\operatorname{Id}_A=\mu_A \circ (\eta_A \otimes A)\). By reversing arrows and duality, it can be defined a \textit{comagma} in \(\mathcal{C}\). Thus, a \textit{Hom-algebra} in \(\mathcal{C}\) is a magma \((A,\mu_A)\) with morphism \(\alpha:A \to A\) such that \(\mu_A \circ (\alpha \otimes \mu_A)=\mu_A \circ (\mu_A \otimes \alpha)\). The Hom-algebra \((A,\mu_A)\) is \textit{unital} if there exists \(\eta_A:K \to A\) such that \(\alpha=\mu_A \circ (A \otimes \eta_A)\) and \(\alpha \circ \eta_A=\eta_A\). If \(\mathbf{A}\) and \(\mathbf{B}\) are unital Hom-algebras then \(\mathbf{A} \otimes \mathbf{B}\) is a unital Hom-algebra. By reversing arrows, the authors define the notion of \textit{monoidal Hom-coalgebra}.
Let \(\mathbf{A}=(A,\mu_A, \alpha)\) be a Hom-algebra and let \(\mathbf{C}=(C,\delta_C,\gamma)\) be a Hom-coalgebra. If \(f,g:C \to A\) are two morphisms, the author define what they called the \textit{convolution operation} \(f \ast g=\mu_A \circ (f \otimes g) \circ \delta_C\). When \(\mathbf{A}\) is unital and \(\mathbf{C}\) is counital, a morphism \(h:C \to A\) is \textit{invertible} if there exists \(h^{-1}:C \to A\) such that \(h \ast h^{-1}=h^{-1} \ast h=\epsilon_C \otimes \eta_A\), \(h=\alpha \circ h \circ \gamma\) and \(h^{-1}=\alpha \circ h^{-1} \circ \gamma\). The set consisting of convolution invertible morphisms is denoted by \(\mathrm{Reg}(C,A)\), that is a group under the convolution product.
A \textit{Hom-bialgebra} in \(\mathcal{C}\) is a sextuple \(\mathbf{H}=(H,\eta_H,\mu_H,\epsilon_H,\delta_H,\alpha)\) where \((H,\eta_H,\mu_H,\alpha)\) is a unital Hom-algebra and \((H,\epsilon_H,\delta_H,\alpha)\) is a counital Hom-coalgebra satisfying:
\begin{itemize}
\item[(c1)] \(\delta_H \circ \mu_H=(\mu_H \otimes \mu_H) \circ \delta_{H \otimes H},\)
\item[(c2)] \(\delta_H \circ \eta_H=\eta_H \otimes \eta_H, \)
\item[(c3)] \(\epsilon_H \circ \mu_H=\epsilon_H \otimes \epsilon_H,\)
\item[(c4)] \(\epsilon_H \circ \eta_H=\operatorname{Id}_K.\)
\end{itemize}
If there exists \(\lambda_H:H \to H\), called the antipode, antimultiplicative (\(\lambda_H \circ \mu_H=\mu_H \circ c_{H,H} \circ (\lambda_H \otimes \lambda_H)\)) and anticomultiplicative (\(\delta_H \circ \lambda_H=(\lambda_H \otimes \lambda_H) \circ c_{H,H} \circ \delta_H\)) such that:
\begin{itemize}
\item[(c5)] \(\alpha \circ \lambda_H=\lambda_H \circ \alpha,\)
\item[(c6)] \(\operatorname{Id}_H \ast \lambda_H=\eta_H \otimes \epsilon_H=\lambda_H \ast \operatorname{Id}_H,\)
\end{itemize}
then \((\mathbf{H},\lambda_H)\) is a \textit{Hom-Hopf algebra} in \(\mathcal{C}\). The authors introduce the notion of Hom-2-cocycle in order to provide a way of altering the product of a Hom-Hopf algebra. A \textit{Hom-2-cocycle} is a morphism \(\sigma:H \otimes H \to K\) such that \(\sigma=\sigma \circ (\alpha \otimes \alpha)\) and \(\partial_1(\sigma) \ast \partial_3{\sigma}=\partial_4(\sigma) \ast \partial_2(\sigma)\), where:
\[
\partial_1(\sigma)=\epsilon_H \otimes \sigma,\,\partial_2(\sigma)=\sigma \circ (\mu_H \otimes H),\,\partial_3(\sigma)=\sigma \circ (H \otimes \mu_H),\,\partial_4(\sigma)=\sigma \otimes \epsilon_H.
\]
The Hom-2-cocycle \(\sigma\) is \textit{normal} if \(\sigma \circ (\eta_H \otimes H)=\epsilon_H=\sigma \circ (H \otimes \eta_H)\). In addition, \(\sigma\) is invertible if \(\sigma \in \operatorname{Reg}(H \otimes H,K)\). A property that the authors prove is that if \(\sigma\) is invertible, then \(\sigma\) is normal if and only if so is its inverse \(\sigma^{-1}\). If \(\sigma\) is an invertible Hom-2-cocycle then \(\partial_{\ell}(\sigma)\) belong to \(\operatorname{Reg}(H \otimes H \otimes H,K)\) with inverse \(\partial_{\ell}(\sigma^{-1})\), for all \(1 \leq \ell \leq 4\). If \(\sigma\) is a normal invertible Hom-2-cocycle and define \(t_{\sigma}=\sigma \circ (\eta_H \otimes \eta_H)\) and \(t_{\sigma}^{\prime}=\sigma^{-1} \circ (\eta_H \otimes \eta_H)\) then \(\xi_{\sigma}=t^{\prime}_{\sigma} \otimes \sigma\) is a normal invertible 2-cocycle with inverse \(\xi_{\sigma}^{-1}=t_{\sigma} \otimes \sigma^{-1}\). Then, the authors define the following products:
\begin{align*}
& \mu_1^{\sigma}=\alpha \circ (\sigma \otimes \sigma^{-1})\circ \delta_{H \otimes H},\\
& \mu_2^{\sigma}=\alpha \circ (\mu_H \otimes \sigma^{-1}) \circ (\sigma \otimes \delta_{H,H})\circ \delta_{H\otimes H},\\
& \mu_3^{\sigma}=\alpha \circ (\sigma \otimes \mu_H \otimes \sigma^{-1})\circ (H \otimes c_{H,H} \otimes c_{H,H} \otimes H) \circ (\delta_H \otimes H \otimes H \otimes \delta_H) \circ \delta_{H \otimes H},\\
& \mu_4^{\sigma}=\alpha \circ (\sigma \otimes \mu_H \otimes \sigma^{-1})\circ (H \otimes c_{H,H} \otimes c_{H,H} \otimes H) \circ (H \otimes \delta_{H \otimes H} \otimes H) \circ (\delta_H \otimes \delta_H).
\end{align*}
Similarly construct \(\mu_{\ell}^{\xi_{\sigma}}\) for all \(\ell \in \{1,2,3,4\}\). Then \(\mu_{\ell}^{\sigma}=\mu_{\ell}^{\xi_{\sigma}}\) for all \(\ell \in \{1,2,3,4\}\). In addition, if \(\omega\) is an invertible Hom-2-cocycle then \(\mu_1^{\omega}=\mu_2^{\omega}=\mu_3^{\omega}=\mu_4^{\omega}\) and if \(\sigma\) is invertible then \(\mu_1^{\sigma}=\mu_2^{\sigma}=\mu_3^{\sigma}=\mu_4^{\sigma}\). When this is the case, is written \(\mu_{\ell}^{\sigma}=\mu_{H^{\sigma}}\).
On the other hand, the following conditions are equivalents:
\[
\alpha^4=\alpha \Leftrightarrow \alpha=\mu_{H^{\sigma}} \circ (\eta_H \otimes H) \Leftrightarrow \alpha=\mu_{H^{\sigma}}\circ (H \otimes \eta_H).
\]
Then the authors sate the main result of the work, which is: if \(\mathbf{H}=(H,\eta_H,\mu_H,\epsilon_H,\delta_H,\alpha,\lambda_H)\) is a Hom-Hopf algebra such that \(\alpha^4=\alpha\) and \(\sigma\) is a convolution invertible Hom-2-cocycle, then\\ \(H^{\sigma}=(H,\eta_H,\mu_{H^{\sigma}},\epsilon_H,\delta_H,\alpha,\lambda_{H^{\sigma}})\) is a Hom-Hopf algebra, where \(\lambda_{H^{\sigma}}=(f \otimes (\alpha \circ \lambda_H) \otimes f^{-1})\circ (H \otimes \delta_H) \circ \delta_H\) and \(f=\sigma \circ (H \otimes \lambda_H) \circ \delta_H \in \operatorname{Reg}(H,K)\) with \(f^{-1}=\sigma^{-1} \circ (\lambda_H \otimes H) \circ \delta_H\).
Reviewer: García Rosendo (Guanajuato)The \(b\)-radical of generalized alternative \(b\)-algebras IIhttps://zbmath.org/1500.170272023-01-20T17:58:23.823708Z"Ferreira, B. L. M."https://zbmath.org/authors/?q=ai:ferreira.bruno-leonardo-macedoA \(b\)-algebra over a field \(F\) is a pair \((U,\omega)\), where \(U\) is an algebra over \(F\) and \(\omega:U\longrightarrow F\) is a homomorphism called the \textit{weight function}.
Al algebra over a field \(F\) of characteritic different from 2 and 3 is called a \textit{generalized alternative algebra II} provided the following conditions hold:
\begin{itemize}
\item \((wx,y,z)+(w,x,[y,z])=w(x,y,z)+(w,y,z)x\)
\item \((x,y,x)=0\)
\end{itemize}
where \([x,y]\) and \((x,y,z)\) denote, as usual, the commutator and the associator of two and three elements, respectively.
In this paper the author proves that, for every finite dimensional generalized alternative \(b\)-algebra II over a field of characteristic different from 2 and 3, it holds that \(\operatorname{rad}(U)=R(U)\cap (bar(U))^3\).
Reviewer: Antonio M. Oller Marcén (Zaragoza)Volterra evolution algebras and their graphshttps://zbmath.org/1500.170282023-01-20T17:58:23.823708Z"Qaralleh, Izzat"https://zbmath.org/authors/?q=ai:qaralleh.izzat"Mukhamedov, Farrukh"https://zbmath.org/authors/?q=ai:mukhamedov.farruh-mAn \textit{evolution algebra} is a finite dimensional vector space \(\mathcal{E}\) endowed with a multiplication such that, for some basis \(\{e_1,\dots,e_n\}\) the multiplication table is given by
\[
e_ie_j=0\ \forall i\neq j,
\]
\[
e_i^2=\sum_k a_{ik}e_k.
\]
Such a basis is called a \textit{natural basis}, and \(A=(a_{ij})_{1\leq i,j\leq n}\) is called the \textit{matrix of the algebra \(\mathcal{E}\)}. In this setting, if \(A\) turns out to be skew-symmetric, the algebra \(\mathcal{E}\) is called a \textit{volterra evolution algebra}.
In this paper, the authors investigate different aspects of this class of algebras, such as their nilpotency, their isomorphism classes, and their derivations. To do so, the so-called \textit{weighted graphs} of a Volterra evolution algebra is introduced.
It is proved, for example, that if the weighted graph of a given Volterra evolution algebra is complete, then all its derivations are trivial.
Reviewer: Antonio M. Oller Marcén (Zaragoza)Derivation of some nilpotent evolution algebrahttps://zbmath.org/1500.170292023-01-20T17:58:23.823708Z"Zannon, Mohammad"https://zbmath.org/authors/?q=ai:zannon.mohammadThe paper is devoted to investigation of the derivations of n-dimensional nilpotent evolution algebras, depending on the choice of structure matrix. By choosing the matrix of structural constants in particular cases authors have found the spaces of derivations for nilpotent evolution algebras and described the space of local and 2-local derivation of such choices.
For the \(n\)-dimensional nilpotent evolution algebra with rank \((n-2)\) showed the forms of the derivation for the given conditions. Also proven that every local derivation of the \(n\)-dimensional nilpotent evolution algebra is a derivation and showed that every 2-local derivation of nilpotent evolution is a derivation.
Reviewer: Sherzod N. Murodov (Bukhara City)Nonlinear evolution equations related to Kac-Moody algebras \(A_r^{(1)}\): spectral aspectshttps://zbmath.org/1500.352582023-01-20T17:58:23.823708Z"Gerdjikov, Vladimir S."https://zbmath.org/authors/?q=ai:gerdzhikov.vladimir-stefanovThe paper contains constructions of Lax representations and an investigation of the inverse scattering method for three types of nonlinear evolution equations related to the Kac-Moody algebras: The derivative non-linear Schrödinger equation, multicomponent mKdV equations and 2-dimensional Toda field theories. The analysis involves spectral properties of the Lax operators and their connections to Riemann-Hilbert problems.
Reviewer: Ayman Kachmar (Nabaṭiyya)Quantisation ideals of nonabelian integrable systemshttps://zbmath.org/1500.370442023-01-20T17:58:23.823708Z"Mikhailov, A. V."https://zbmath.org/authors/?q=ai:mikhailov.a-v|mikhailov.alexander-vSummary: We consider dynamical systems on the space of functions taking values in a free associative algebra. The system is said to be integrable if it possesses an infinite dimensional Lie algebra of commuting symmetries. In this paper we propose a new approach to the problem of quantisation of dynamical systems, introduce the concept of quantisation ideals and provide meaningful examples.On the von Neumann algebra of class functions on a compact quantum grouphttps://zbmath.org/1500.460542023-01-20T17:58:23.823708Z"Krajczok, Jacek"https://zbmath.org/authors/?q=ai:krajczok.jacek"Wasilewski, Mateusz"https://zbmath.org/authors/?q=ai:wasilewski.mateuszSummary: We study analogues of the radial subalgebras in free group factors (called the algebras of class functions) in the setting of compact quantum groups. For the free orthogonal quantum groups we show that they are not MASAs, as soon as we are in a non-Kac situation. The most important notion to our present work is that of a (quasi-)split inclusion. We prove that the inclusion of the algebra of class functions is quasi-split for some unitary quantum groups -- in this case the subalgebra is non-abelian and we also obtain a result concerning its relative commutant. In the positive direction, we construct certain bicrossed products from the quantum group \(\operatorname{SU}_q(2)\) for which the algebra of class functions is a MASA.Generalizations of the drift Laplace equation in the Heisenberg group and Grushin-type spaceshttps://zbmath.org/1500.530392023-01-20T17:58:23.823708Z"Bieske, Thomas"https://zbmath.org/authors/?q=ai:bieske.thomas-j"Blackwell, Keller"https://zbmath.org/authors/?q=ai:blackwell.kellerSummary: We find fundamental solutions to \(p\)-Laplace equations with drift terms in the Heisenberg group and Grushin-type planes. These solutions are natural generalizations of the fundamental solutions discovered by Beals, Gaveau, and Greiner for the Laplace equation with drift term. Our results are independent of the results of Bieske and Childers, in that Bieske and Childers consider a generalization that focuses on the \(p\)-Laplace-type equation while we primarily concentrate on a generalization of the drift term.Linearization of Poisson-Lie structures on the \(2D\) Euclidean and \((1+1)\) Poincaré groupshttps://zbmath.org/1500.530882023-01-20T17:58:23.823708Z"Ganbouri, Bousselham"https://zbmath.org/authors/?q=ai:ganbouri.bousselham"Mansouri, Mohamed Wadia"https://zbmath.org/authors/?q=ai:mansouri.mohamed-wadiaSummary: The paper deals with linearization problem of Poisson-Lie structures on the \((1+1)\) Poincaré and \(2D\) Euclidean groups. We construct the explicit form of linearizing coordinates of all these Poisson-Lie structures. For this, we calculate all Poisson-Lie structures on these two groups mentioned above, through the correspondence with Lie Bialgebra structures on their Lie algebras which we first determine.On semisimple standard compact Clifford-Klein formshttps://zbmath.org/1500.570242023-01-20T17:58:23.823708Z"Bocheński, Maciej"https://zbmath.org/authors/?q=ai:bochenski.maciej-franciszekThe homogeneous space \(G/H\) admits a compact Clifford-Klein form if there exists a discrete subgroup \(\Gamma \subset G\) such that \(\Gamma\) acts freely, properly and co-compactly on \(G/H\). If \(G,H\) are reductive Lie groups and there exists a reductive connected subgroup \(L \subset G\) such that \(L\) acts properly and co-compactly on \(G/H\), then it is always possible to choose a co-compact lattice \(\Gamma\) in \(L\) so that \(\Gamma\) acts properly, co-compactly and freely on \(G/H\). In such case it is said that \(G/H\) admits a standard compact Clifford-Klein form.
In this paper a classification of standard compact Clifford-Klein forms, corresponding to triples \((\mathfrak{g}, \mathfrak{h}, \mathfrak{l})\) such that \(\mathfrak{g}= \mathfrak{h}+ \mathfrak{l}\), \(\mathfrak{h}, \mathfrak{l}\) have no nontrivial ideals of compact type and the Lie algebra \(\mathfrak{g}\) of \(G\) is a sum of two absolutely simple ideals (for such ideals their complexifications are simple) of noncompact type, is given. The classification uses Onishchik's results about semisimple decompositions of semisimple Lie algebras [\textit{A. L. Onishchik}, Math. USSR, Sb. 9, 515--554 (1970; Zbl 0227.22013) and Tr. Mosk. Mat. O.-va 11, 199--242 (1962; Zbl 0192.12601)]. Some new examples of reductive homogeneous spaces admitting non-standard compact Clifford-Klein forms are presented. They are the spaces \(SO(4, 4) \times SO(2, 4)/\Delta_1\) and \(SO(3, 4) \times SO(2, 4)/\Delta_2\), where \(\Delta_1,\Delta_2\) are some ``diagonal'' subgroups, isomorphic to \(SO(2,4)\), in \(SO(4,4) \times SO(2,4)\) and \(SO(3,4) \times SO(2,4)\) correspondingly.
Reviewer: V. V. Gorbatsevich (Moskva)Probability distributions characterisations on a homogeneous conehttps://zbmath.org/1500.600072023-01-20T17:58:23.823708Z"Boutouria, Imen"https://zbmath.org/authors/?q=ai:boutouria.imen"Bouzida, Imed"https://zbmath.org/authors/?q=ai:bouzida.imedIn this paper, the authors study probability distributions on a homogeneous cone and generalize Gindikin results to Vinberg algebras.
Reviewer: Alessandro Selvitella (Fort Wayne)Classification of four-rebit stateshttps://zbmath.org/1500.810172023-01-20T17:58:23.823708Z"Dietrich, Heiko"https://zbmath.org/authors/?q=ai:dietrich.heiko"de Graaf, Willem A."https://zbmath.org/authors/?q=ai:de-graaf.willem-adriaan"Marrani, Alessio"https://zbmath.org/authors/?q=ai:marrani.alessio"Origlia, Marcos"https://zbmath.org/authors/?q=ai:origlia.marcosSummary: We classify states of four rebits, that is, we classify the orbits of the group \(\hat{G}(\mathbb{R}) = \operatorname{SL} ( 2 , \mathbb{R} )^4\) in the space \(( \mathbb{R}^2 )^{\otimes 4} \). This is the real analogon of the well-known SLOCC operations in quantum information theory. By constructing the \(\hat{G}(\mathbb{R})\)-module \(( \mathbb{R}^2 )^{\otimes 4}\) via a \(\mathbb{Z} / 2 \mathbb{Z} \)-grading of the simple split real Lie algebra of type \(D_4\), the orbits are divided into three groups: semisimple, nilpotent and mixed. The nilpotent orbits have been classified in [\textit{H. Dietrich} et al., Fortschr. Phys. 65, No. 2, 1600118, 25 p. (2017; Zbl 1371.83177)], yielding applications in theoretical physics (extremal black holes in the STU model of \(\mathcal{N} = 2, D = 4\) supergravity, see [\textit{D. Ruggeri} and \textit{M. Trigiante}, Fortschr. Phys. 65, No. 5, 1700007, 66 p. (2017; Zbl 1371.83183)]. Here we focus on the semisimple and mixed orbits which we classify with recently developed methods based on Galois cohomology, see [\textit{M. Borovoi}, \textit{W. A. de Graaf} and \textit{H. V. Lê}, ``Real graded Lie algebras, Galois cohomology, and classification of trivectors in \(\mathbb{R}^9\)'', Preprint, \url{arXiv:2106.00246}; ``Classification of real trivectors in dimension nine'', Preprint, \url{arXiv:2108.00790}, see also Zbl 1489.15020]. These orbits are relevant to the classification of non-extremal (or extremal over-rotating) and two-center extremal black hole solutions in the STU model.BRST ghost-vertex operator in Witten's cubic open string field theory on multiple \(Dp\)-braneshttps://zbmath.org/1500.810592023-01-20T17:58:23.823708Z"Lee, Taejin"https://zbmath.org/authors/?q=ai:lee.taejinSummary: The Becchi-Rouet-Stora-Tyutin (BRST) ghost field is a key element in constructing Witten's cubic open string field theory. However, to date, the ghost sector of the string field theory has not received a great deal of attention. In this study, we address the BRST ghost on multiple \(Dp\)-branes, which carries non-Abelian indices and couples to a non-Abelian gauge field. We found that the massless components of the BRST ghost field can play the role of the Faddeev-Popov ghost in the non-Abelian gauge field, such that the string field theory maintains the local non-Abelian gauge invariance.Derived brackets in bosonic string sigma-modelhttps://zbmath.org/1500.810932023-01-20T17:58:23.823708Z"Bernardes, Vinícius"https://zbmath.org/authors/?q=ai:bernardes.vinicius"Mikhailov, Andrei"https://zbmath.org/authors/?q=ai:mikhailov.andrei-igorevich|mikhailov.andrei-yu"Viana, Eggon"https://zbmath.org/authors/?q=ai:viana.eggonSummary: We study the worldsheet theory of bosonic string from the point of view of the BV formalism. We explicitly describe the derived Poisson structure which arises when we expand the Master Action near a Lagrangian submanifold. The resulting higher Poisson brackets are all degenerate and essentially constant along their symplectic leaves. Deformations of the worldsheet complex structures define a family of Lagrangian submanifolds, parametrized by Beltrami differential. The worldsheet action depends nonlinearly on the Beltrami differential, but the structure of nonlinearity is governed by the BV Master Equation. This helps to clarify the mechanism of holomorphic factorization of string amplitudes.