Recent zbMATH articles in MSC 17Bhttps://zbmath.org/atom/cc/17B2023-09-22T14:21:46.120933ZWerkzeugFree fermions and Schur expansions of multi-Schur functionshttps://zbmath.org/1517.051772023-09-22T14:21:46.120933Z"Iwao, Shinsuke"https://zbmath.org/authors/?q=ai:iwao.shinsukeSummary: Multi-Schur functions are symmetric functions that generalize the supersymmetric Schur functions, the flagged Schur functions, and the refined dual Grothendieck functions, which have been intensively studied by \textit{A. Lascoux} [Symmetric functions and combinatorial operators on polynomials. Providence, RI: American Mathematical Society (AMS) (2003; Zbl 1039.05066)]. In this paper, we give a new free-fermionic presentation of them. The multi-Schur functions are indexed by a partition and two ``tuples of tuples'' of indeterminates. We construct a family of linear bases of the fermionic Fock space that are indexed by such data and prove that they correspond to the multi-Schur functions through the boson-fermion correspondence. By focusing on some special bases, which we call refined bases, we give a straightforward method of expanding a multi-Schur function in the refined dual Grothendieck polynomials. We also present a sufficient condition for a multi-Schur function to have its Hall-dual function in the completed ring of symmetric functions.On Ecalle's and Brown's polar solutions to the double shuffle equations modulo productshttps://zbmath.org/1517.111152023-09-22T14:21:46.120933Z"Matthes, Nils"https://zbmath.org/authors/?q=ai:matthes.nils"Tasaka, Koji"https://zbmath.org/authors/?q=ai:tasaka.kojiFor a multiple zeta value, (MZV),
\[ \zeta(m_1,\dots,m_r)=\sum_{0<n_1<\dots<n_r}\frac1{n_1^{m_1}\cdots n_r^{m_r}}
\]
the \textit{weight} is \(k=m_1+\dots+m_r\). For \(k\in\mathbb{N}\) let \(\mathcal Z_k\) denote the \(\mathbb{Q}\)-span of the finitely many MZVs of weight \(k\). \textit{D. Zagier} [Prog. Math. 120, 497--512 (1994; Zbl 0822.11001)] conjectured, that \(\dim_{\mathbb{Q}}\mathcal{Z}_k=d_k\), where \(d_k\) is defined by \(\sum_{k=0}^\infty d_k t^k=\frac1{1-t^2-t^3}\). \textit{A. B. Goncharov} [``Multiple polylogarithms and mixed Tate motives'', Preprint, \url{arXiv:Math/0103059}] and \textit{T. Terasoma} [Invent. Math. 149, No. 2, 339--369 (2002; Zbl 1042.11043)] independently proved, that \(\dim_{\mathbb{Q}}\mathcal{Z}_k\le d_k\). This means that there are many relations among the MZV. Several sets of relations are known. For instance in ascending order we have the sets of double shuffle relations, of associator relations and of motivic relations. The latter are due to \textit{F. Brown} [Ann. Math. (2) 175, No. 2, 949--976 (2012; Zbl 1278.19008)], who defined certain elements \(\sigma_{2k+1}\) of a motivic Lie algebra which define relations. It is expected, that these three classes actually coincide, which, among other things, would mean that the relators \(\sigma_{2k+1}\) should be expressible in terms of the double shuffle relators (DSR). The paper under review deals with the question for explicit expressions of (some of) the \(\sigma_{2k+1}\) relators in terms of the DSR. In the literature, there are two approaches, one by \textit{F. Brown} [``Anatomy of an associator'', Preprint, \url{arXiv:1709.02765}], and the other by \textit{J. Ecalle} [J. Théor. Nombres Bordx. 15, No. 2, 411--478 (2003; Zbl 1094.11032)]. In the paper, these two are put in a common algebraic context and similarities and differences are given. For instance, they show that the relators agree in depth \(\le 4\), but not in depth 5.
Reviewer: Anton Deitmar (Tübingen)Invariant Hilbert scheme of the Cox realization of the nilpotent cone in \(\mathfrak{sl}(n)\)https://zbmath.org/1517.140122023-09-22T14:21:46.120933Z"Kubota, Ayako"https://zbmath.org/authors/?q=ai:kubota.ayakoSummary: We pose a question asking if the main component of the invariant Hilbert scheme of the Cox realization of a given singularity is a resolution of singularities. As a concrete example, we consider the case where the singularity is the nilpotent cone in \(\mathfrak{sl}(n)\), and we see that in this case the answer to the question is positive.
For the entire collection see [Zbl 1516.14004].On generalized Steinberg theory for type AIIIhttps://zbmath.org/1517.140322023-09-22T14:21:46.120933Z"Fresse, Lucas"https://zbmath.org/authors/?q=ai:fresse.lucas"Nishiyama, Kyo"https://zbmath.org/authors/?q=ai:nishiyama.kyoSummary: The multiple flag variety \(\mathfrak{X}=\mathrm{Gr}(\mathbb{C}^{p+q},r)\times (\mathrm{Fl}(\mathbb{C}^p)\times \mathrm{Fl}(\mathbb{C}^q))\) can be considered as a double flag variety associated to the symmetric pair \((G,K)=(\mathrm{GL}_{p+q}(\mathbb{C}),\mathrm{GL}_p(\mathbb{C})\times \mathrm{GL}_q(\mathbb{C}))\) of type AIII. We consider the diagonal action of \(K\) on \(\mathfrak{X}\). There is a finite number of orbits for this action, and our first result is a description of these orbits: parametrization (by a certain set of graphs), dimensions, closure relations and cover relations.
In [Int. Math. Res. Not. 2022, No. 1, 18828--18889 (2022; Zbl 1493.14073)], we defined two generalized Steinberg maps from the \(K\)-orbits of \(\mathfrak{X}\) to the nilpotent \(K\)-orbits in \(\mathfrak{k}\) and those in the Cartan complement of \(\mathfrak{k}\), respectively. The main result in the present paper is a complete, explicit description of these two Steinberg maps by means of a combinatorial algorithm which extends the classical Robinson-Schensted correspondence.A basis of a certain module for the hyperalgebra of \((\mathrm{SL}_2)_r\) and some applicationshttps://zbmath.org/1517.160172023-09-22T14:21:46.120933Z"Yoshii, Yutaka"https://zbmath.org/authors/?q=ai:yoshii.yutakaSummary: In the hyperalgebra \(\mathcal{U}_r\) of the \(r\) th Frobenius kernel \((\mathrm{SL}_2)_r\) of the algebraic group \(\mathrm{SL}_2\), we construct a basis of the \(\mathcal{U}_r\)-module generated by a certain element which was given by the author before. As its applications, we also prove some results on the \(\mathcal{U}_r\)-modules and the algebra \(\mathcal{U}_r\).Bound for the cocharacters of the identities of irreducible representations of \(\mathfrak{sl}_2(\mathbb{C})\)https://zbmath.org/1517.160232023-09-22T14:21:46.120933Z"Domokos, Mátyás"https://zbmath.org/authors/?q=ai:domokos.matyasLet \(L=\mathfrak{sl}_2(\mathbb{C})\) be the three dimensional simple Lie algebra over the complex numbers. The author of the paper under review studies numerical invariants of the polynomial identities of the irreducible f.d. representations of \(L\). The identities of representations of \(L\) were first studied by \textit{Yu. P. Razmyslov} [Transl., Ser. 2, Am. Math. Soc. 140, 101--109 (1988; Zbl 0658.17014); translation from Algebra, Work Collect., dedic. O. Yu. Shmidt, Moskva 1982, 139--150 (1982)], see also [\textit{Yu. P. Razmyslov}, Identities of algebras and their representations. Transl. from the Russian by A. M. Shtern. Transl. ed. by Simeon Ivanov. Providence, RI: American Mathematical Society (1994; Zbl 0827.17001)]. When studying polynomial identities in characteristic 0, one can restrict the considerations to the multilinear identities only. The multilinear identities of given degree \(n\) form a module over the symmetric group \(S_n\) and one applies the theory of representations of \(S_n\) in order to study ideals of identities. Another, equivalent, approach is often much better. It is well known that the representations of \(S_n\), and the polynomial representations of the general linear group \(\mathrm{GL}_m(\mathbb{C})\), are described in the same terms, and are ``dual'' to each other. Let \(\rho\) be an irreducible representation of \(L\), and let \(I(L,\rho)\) be the ideal of identities of \(\rho\). In other words, this is the ideal of weak identities of the pair \((V,L)\) (or \((V,\rho)\)) where \(\rho\colon L\to V\). The author studies the cocharacter sequence of \(I(L,\rho)\). The main theorem of the paper under review states that the only non-zero irreducible modules that appear in the decomposition of the multilinear part of the ``non-identities'' correspond to partitions of at most three parts. If \(\dim V=d\) then these multiplicities are bounded by \(3^{d-2}\).
It should be noted that if one considers the ordinary identities of the full matrix algebra of order 2, its cocharacter has no bounded multiplicities. (Though for every PI algebra \(A\) the multiplicities of the irreducible modules that appear in its cocharacter are polynomially bounded.) The author proves that in case \(\dim V=d\) then there exists a multiplicity which is \(\ge d-1\); thus one cannot expect any uniform (that is independent of \(d\)) bound.
Reviewer: Plamen Koshlukov (Campinas)Quasi-bialgebras from set-theoretic type solutions of the Yang-Baxter equationhttps://zbmath.org/1517.160282023-09-22T14:21:46.120933Z"Doikou, Anastasia"https://zbmath.org/authors/?q=ai:doikou.anastasia"Ghionis, Alexandros"https://zbmath.org/authors/?q=ai:ghionis.alexandros"Vlaar, Bart"https://zbmath.org/authors/?q=ai:vlaar.bartThere is an interest in studying classes of quantum algebras arising from set-theoretic solutions of the Yang-Baxter equation. The motivations for deepening this investigation lie in the paper [\textit{A. Doikou} and \textit{A. Smoktunowicz}, Lett. Math. Phys. 111, No. 4, Paper No. 105, 40 p. (2021; Zbl 1486.16039)].
In the paper under review, the authors investigate algebras coming from involutive and non-degenerate solutions and their \(q\)-analogues. It is shown that they are quasi-triangular quasi-bialgebras. To get this result, they provide some universal results on quasi-bialgebras and admissible Drinfeld twists. In fact, the property of being quasi-triangular (quasi-)bialgebra is preserved by twisting (see [\textit{V. G. Drinfel'd}, Sov. Math., Dokl. 32, 256--258 (1985; Zbl 0588.17015); translation from Dokl. Akad. Nauk SSSR 283, 1060--1064 (1985)]). Moreover, they make use of some first results on the admissible Drinfeld twist for involutive set-theoretic solution already derived in [\textit{A. Doikou}, J. Phys. A, Math. Theor. 54, No. 41, Article ID 415201, 21 p. (2021; Zbl 07654521)].
Reviewer: Marzia Mazzotta (Lecce)Pre-Lie analogues of Poisson-Nijenhuis structures and Maurer-Cartan equationshttps://zbmath.org/1517.170012023-09-22T14:21:46.120933Z"Liu, Jiefeng"https://zbmath.org/authors/?q=ai:liu.jiefeng"Wang, Qi"https://zbmath.org/authors/?q=ai:wang.qi.31Summary: In this paper, we study pre-Lie analogues of Poisson-Nijenhuis structures and introduce \(\mathcal{O}\mathrm{N}\)-structures on bimodules over pre-Lie algebras. We show that an \(\mathcal{O}\mathrm{N}\)-structure gives rise to a hierarchy of pairwise compatible \(\mathcal{O}\)-operators. We study solutions of the strong Maurer-Cartan equation on the twilled pre-Lie algebra associated to an \(\mathcal{O}\)-operator, which gives rise to a pair of \(\mathcal{O}\mathrm{N}\)-structures which are naturally in duality. We show that KVN-structures and HN-structures on a pre-Lie algebra \(\mathfrak{g}\) are corresponding to \(\mathcal{O}\mathrm{N}\)-structures on the bimodule \((\mathfrak{g}^{\ast}; \mathrm{ad}^{\ast},-R^{\ast})\), and KVB-structures are corresponding to solutions of the strong Maurer-Cartan equation on a twilled pre-Lie algebra associated to an \(\mathfrak{s}\)-matrix.Relative Rota-Baxter systems on Leibniz algebrashttps://zbmath.org/1517.170022023-09-22T14:21:46.120933Z"Das, Apurba"https://zbmath.org/authors/?q=ai:das.apurba|das.apurba-narayan"Guo, Shuangjian"https://zbmath.org/authors/?q=ai:guo.shuangjian|guo.shuangjian.1Summary: In this paper, we introduce relative Rota-Baxter systems on Leibniz algebras and give some characterizations and new constructions. Then we construct a graded Lie algebra whose Maurer-Cartan elements are relative Rota-Baxter systems. This allows us to define a cohomology theory associated with a relative Rota-Baxter system. Finally, we study formal deformations and extendibility of finite order deformations of a relative Rota-Baxter system in terms of the cohomology theory.Lie \(n\)-algebras and cohomologies of relative Rota-Baxter operators on \(n\)-Lie algebrashttps://zbmath.org/1517.170042023-09-22T14:21:46.120933Z"Chen, Ming"https://zbmath.org/authors/?q=ai:chen.ming"Liu, Jiefeng"https://zbmath.org/authors/?q=ai:liu.jiefeng"Ma, Yao"https://zbmath.org/authors/?q=ai:ma.yaoSummary: Based on the differential graded Lie algebra controlling deformations of an \(n\)-Lie algebra with a representation (called an \(n\)-\(\mathsf{LieRep}\) pair), we construct a Lie \(n\)-algebra, whose Maurer-Cartan elements characterize relative Rota-Baxter operators on \(n\)-\(\mathsf{LieRep}\) pairs. The notion of an \(n\)-pre-Lie algebra is introduced, which is the underlying algebraic structure of the relative Rota-Baxter operator. We give the cohomology of relative Rota-Baxter operators and study infinitesimal deformations and extensions of order \(m\) deformations to order \(m + 1\) deformations of relative Rota-Baxter operators through the cohomology groups of relative Rota-Baxter operators. Moreover, we build the relation between the cohomology groups of relative Rota-Baxter operators on \(n\)-\(\mathsf{LieRep}\) pairs and those on \((n + 1)\)-\(\mathsf{LieRep}\) pairs by certain linear functions.3-post-Lie algebras and relative Rota-Baxter operators of nonzero weight on 3-Lie algebrashttps://zbmath.org/1517.170052023-09-22T14:21:46.120933Z"Hou, Shuai"https://zbmath.org/authors/?q=ai:hou.shuai"Sheng, Yunhe"https://zbmath.org/authors/?q=ai:sheng.yunhe"Zhou, Yanqiu"https://zbmath.org/authors/?q=ai:zhou.yanqiuSummary: In this paper, first we introduce the notions of relative Rota-Baxter operators of nonzero weight on 3-Lie algebras and 3-post-Lie algebras. A 3-post-Lie algebra consists of a 3-Lie algebra structure and a ternary operation such that some compatibility conditions are satisfied. We show that a relative Rota-Baxter operator of nonzero weight induces a 3-post-Lie algebra naturally. Conversely, a 3-post-Lie algebra gives rise to a new 3-Lie algebra, which is called the subadjacent 3-Lie algebra, and an action on the original 3-Lie algebra. Then we construct an \(L_\infty \)-algebra whose Maurer-Cartan elements are relative Rota-Baxter operators of nonzero weight. Consequently, we obtain the twisted \(L_\infty \)-algebra that controls deformations of a given relative Rota-Baxter operator of nonzero weight on 3-Lie algebras. Finally, we introduce a cohomology theory for a relative Rota-Baxter operator of nonzero weight on 3-Lie algebras and use the second cohomology group to classify infinitesimal deformations.Ghost center and representations of the diagonal reduction algebra of \(\mathfrak{osp}(1 | 2)\)https://zbmath.org/1517.170062023-09-22T14:21:46.120933Z"Hartwig, Jonas T."https://zbmath.org/authors/?q=ai:hartwig.jonas-t"Williams, Dwight Anderson II"https://zbmath.org/authors/?q=ai:williams.dwight-anderson-iiSummary: Reduction algebras are known by many names in the literature, including step algebras, Mickelsson algebras, Zhelobenko algebras, and transvector algebras, to name a few. These algebras, realized by raising and lowering operators, allow for the calculation of Clebsch-Gordan coefficients, branching rules, and intertwining operators; and have connections to extremal equations and dynamical R-matrices in integrable face models.
In this paper we continue the study of the diagonal reduction superalgebra \(A\) of the orthosymplectic Lie superalgebra \(\mathfrak{osp}(1 | 2)\). We construct a Harish-Chandra homomorphism, Verma modules, and study the Shapovalov form on each Verma module. Using these results, we prove that the ghost center (center plus anti-center) of \(A\) is generated by two central elements and one anti-central element (analogous to the Scasimir due to Leśniewski for \(\mathfrak{osp}(1 | 2))\). As another application, we classify all finite-dimensional irreducible representations of \(A\). Lastly, we calculate an infinite-dimensional tensor product decomposition explicitly.Whittaker modules and quasi-Whittaker modules for the Schrödinger algebra in \((2+1)\)-dimensional spacetimehttps://zbmath.org/1517.170092023-09-22T14:21:46.120933Z"Cai, Yan-an"https://zbmath.org/authors/?q=ai:cai.yanan"Liu, Zedong"https://zbmath.org/authors/?q=ai:liu.zedongSummary: In this paper, we classify simple (quasi)-Whittaker modules for the Schrödinger Lie algebra \(\mathfrak{s}_2\) in \((2 + 1)\)-dimensional space-time. Moreover, we show that a simple \(\mathfrak{s}_n\)-module is a Whittaker module if and only if it is a locally finite \(\mathfrak{s}_n^+\)-module, and it is a quasi-Whittaker module if and only if it is a locally finite \(\mathfrak{h}_n\)-module. Also, we determine the center of \(\mathcal{U}(\mathfrak{s}_2)\) and \(\mathcal{U}(\mathfrak{s}_2 /\mathbb{C}z)\).Representations having vectors fixed by a Levi subgroup associated to a real formhttps://zbmath.org/1517.170102023-09-22T14:21:46.120933Z"Smilga, Ilia"https://zbmath.org/authors/?q=ai:smilga.iliaFor a semisimple real Lie group \(G_{\mathbb{R}}\) consider the restricted Weyl group is the group \(W=N_{G_{\mathbb{R}}}(a)/Z_{G_{\mathbb{R}}} (a)\), where \(a\) is the Cartan subspace of \(G_{\mathbb{R}}\).
The longest element of \(W\) is the unique element that maps all positive restricted roots to negative restricted roots.
In [\textit{I. Smilga}, Math. Ann. 382, No. 1--2, 513--605 (2022; Zbl 1510.20043)] the following theorem was proved.
Let \(\rho\) be a representation of \(G_{\mathbb{R}}\) on a finite-dimensional real vector space \(V\). Assume that \(\rho\) satisfies the following algebraic condition:
(*) the longest element \(w_0\) of the restricted Weyl group \(WS\) of \(G_{\mathbb{R}}\) acts (via \(\rho\)) nontrivially on the subspace \(V_L\) of vectors of \(V\) that are fixed by all elements of \(L\) = centralizer of \(Z_{G_{\mathbb{R}}} (a)\).
Then the representation \(\rho\) has the following geometric property:
(**) The affine group \(G_{\mathbb{R}}\ltimes_{\rho} V\) contains a Zariski-dense subgroup \(\Gamma\) that is free (of rank at least \(2\)) and acts properly discontinuously on the affine space corresponding to \(V\).
In the present paper the representations that satisfy the condition (*) are classified.
Reviewer: Dmitry Artamonov (Moskva)Non-weight representations of Lie superalgebras of Block type. Ihttps://zbmath.org/1517.170112023-09-22T14:21:46.120933Z"Wang, Huidong"https://zbmath.org/authors/?q=ai:wang.huidong"Xia, Chunguang"https://zbmath.org/authors/?q=ai:xia.chunguang"Zhang, Xiufu"https://zbmath.org/authors/?q=ai:zhang.xiufuSummary: In this paper, we study non-weight representations of two classes of Lie algebras and two classes of Lie superalgebras of Block type. More precisely, we completely classify the free \(\mathcal{U}(\mathfrak{h})\)-modules of rank one or two over these Lie superalgebras. Then we determine the simplicity and isomorphism classes of these modules. As a byproduct, we also obtain the non-weight representation results for a class of non-semisimple finitely graded extensions of the Virasoro algebra.A quantum cluster algebra approach to representations of simply laced quantum affine algebrashttps://zbmath.org/1517.170122023-09-22T14:21:46.120933Z"Bittmann, Léa"https://zbmath.org/authors/?q=ai:bittmann.leaFinite-dimensional representations of quantum affine algebras have been classified by Chari and Pressley with a quantum affine analog of Cartan's highest weight classification of finite-dimensional representations of simple Lie algebras. Their classification also implies that every simple module can be obtained as a subquotient of a tensor product of certain fundamental modules. Although this classification is a major result, it gives limited information on the module structure. Frenkel and Reshetikhin thus developed a theory of \(q\)-characters, giving the decomposition of the modules into generalized eigenspaces for the action of a large commutative subalgebra of the quantum affine algebra of a finite-dimensional simple Lie algebra \(\mathfrak{g}\). When \(\mathfrak{g}\) is of simply-laced type, Nakajima used geometry, and more precisely perverse sheaves on quiver varieties, to construct \(t\)-deformations of these \(q\)-characters, called \((q, t)\)-characters, as elements of a quantum Grothendieck ring.
L. Bittmann establishes a quantum cluster algebra structure on the quantum Grothendieck ring of a certain monoidal subcategory of the category of finite-dimensional representations of a simply-laced quantum affine algebra. Moreover, the \((q,t)\)-characters of certain irreducible representations, among which fundamental representations, are obtained as quantum cluster variables. This approach gives a new algorithm to compute these \((q,t)\)-characters. As an application, she proves that the quantum Grothendieck ring of a larger category of representations of the Borel subalgebra of the quantum affine algebra, defined in a previous work as a quantum cluster algebra, contains indeed the well-known quantum Grothendieck ring of the category of finite-dimensional representations.
Reviewer: Mee Seong Im (Annapolis)Principal subspaces for the quantum affine vertex algebra in type \(A_1^{( 1 )}\)https://zbmath.org/1517.170132023-09-22T14:21:46.120933Z"Butorac, Marijana"https://zbmath.org/authors/?q=ai:butorac.marijana"Kožić, Slaven"https://zbmath.org/authors/?q=ai:kozic.slavenThe study of principal subspaces was introduced by Feigin and Stoyanovsky and later developed by several authors including the present authors in the context of representations of Kac-Moody algebras and vertex operator algebras. In the present paper, they consider a quantum affine vertex algebra \(\mathcal{V}_c(\mathfrak{sl}_2)\) and study a quantum counterpart of principal subspaces. They thus construct quantum quasi-particles and use them to obtain bases. As in the affine Lie algebra case, these bases give an interpretation of the sum side of certain combinatorial identities.
Reviewer: Stefano Capparelli (Roma)Elliptic quantum toroidal algebra \(U_{q,t,p}(\mathfrak{gl}_{1,tor})\) and affine quiver gauge theorieshttps://zbmath.org/1517.170142023-09-22T14:21:46.120933Z"Konno, Hitoshi"https://zbmath.org/authors/?q=ai:konno.hitoshi"Oshima, Kazuyuki"https://zbmath.org/authors/?q=ai:oshima.kazuyukiIn this paper, a new toroidal quantum algebra \(U_{q,t,p}(gl_{1,tor})\) is introduced. It is an elliptic analogue of the quantum toroidal algebra \(U_{q,t}(gl_{1,tor})\). The quantum algebra \(U_{q,t,p}(gl_{1,tor})\) is defined in terms of generators and relations in the same scheme as the elliptic quantum group \(U_{q,p}(\hat{g})\) associated with the affine Lie algebra \(\hat{g}\). The defined algebra has application in the \(4d\) \(N = 2^*\) \(U(M)\) theories and provides a new Alday-Gaiotto-Tachikawa correspondence.
Reviewer: Dmitry Artamonov (Moskva)Rota-Baxter operators on Clifford semigroups and the Yang-Baxter equationhttps://zbmath.org/1517.170152023-09-22T14:21:46.120933Z"Catino, Francesco"https://zbmath.org/authors/?q=ai:catino.francesco"Mazzotta, Marzia"https://zbmath.org/authors/?q=ai:mazzotta.marzia"Stefanelli, Paola"https://zbmath.org/authors/?q=ai:stefanelli.paolaThis paper aims to show how to obtain weak braces from Rota-Baxter operators defined on Clifford semigroups, for which the authors introduce the notion of Rota-Baxter operator on a Clifford semigroup \(\left(S,+\right)\), i.e., a map
\[
\mathfrak{R}:S\rightarrow S
\]
abiding by the relations
\begin{align*}
\mathfrak{R}\left(a\right)+\mathfrak{R}\left(b\right) & =\left(a+\mathfrak{R}\left(a\right)+b-\mathfrak{R}\left(a\right)\right)\\
a+\mathfrak{R}\left(a\right)-\mathfrak{R}\left(a\right) & =a
\end{align*}
for all \(a,b\in S\).
The synopsis of the paper goes as follows.
\begin{itemize}
\item[\S 1] gives essential results on the structures of weak braces introduced in [\textit{F. Catino} et al., Semigroup Forum 104, No. 2, 228--255 (2022; Zbl 07533946)] to find solutions of the Yang-Baxter equation. Some basics on Clifford semigroups are recalled [\textit{A. H. Clifford} and \textit{G. B. Preston}, The algebraic theory of semigroups. Vol. I. Providence, RI: American Mathematical Society (AMS) (1961; Zbl 0111.03403); \textit{G. Cooperman} and \textit{L. Finkelstein}, J. Symb. Comput. 17, No. 6, 513--528 (1994; Zbl 0835.20007); \textit{M. Petrich}, Inverse semigroups. Pure and Applied Mathematics. A Wiley-Interscience Publication. New York etc.: John Wiley \& Sons. (1984; Zbl 0546.20053)].
\item[\S 2] presents and investigates the Rota-Baxter operators on a Clifford semigroup, consistently with that introduced for groups in [\textit{L. Guo} et al., Adv. Math. 387, Article ID 107834, 34 p. (2021; Zbl 1468.17026)].
\item[\S 3] focuses on Rota-Baxter endomorphisms, giving a description of such maps \(f\) for which \(\mathrm{Im}\,f\) is commutative. The authors characterize Rota-Baxter endomorphisms of groups that are also idempotents.
\item[\S 4] illustrates a method for obtaining a dual weak brace \(S\) starting from a given Rota-Baxter operator on a Clifford semigroup \(\left(S,+\right) \). The construction is inspired by that of skew braces due to \textit{V. G. Bardakov} and \textit{V. Gubarev} [J. Algebra 596, 328--351 (2022; Zbl 1492.17019), Proposition 3.1]. It is shown that every commutative Rota-Baxter endormorphism determines a bi-weak brace.
\item[\S 5] makes explicit the solutions associated to dual weak braces obtained through Rota-Baxter operators.
\item[\S 6] deals with the ideals of a dual weak brace, by extending the already known theory for skew braces. In particular, the notion of the socle of a dual weak brace is introduced. The ideals of dual weak braces associated to a given Rota-Baxter operator are finally described.
\end{itemize}
Reviewer: Hirokazu Nishimura (Tsukuba)Typed angularly decorated planar rooted trees and generalized Rota-Baxter algebrashttps://zbmath.org/1517.170162023-09-22T14:21:46.120933Z"Foissy, Loïc"https://zbmath.org/authors/?q=ai:foissy.loic"Peng, Xiao-Song"https://zbmath.org/authors/?q=ai:peng.xiao-songA Rota-Baxter algebra is an associative algebra \(A\)\ with a linear endomorphism \(P\), such that for any \(a,b\in A\), we have
\[
P\left(a\right)P\left(b\right)=P\left(aP\left(b\right)\right)+P\left(P\left(a\right)b\right)+\lambda P\left(ab\right)
\]
where \(\lambda\) is a scalar called the weight of the Rota-Baxter operator \(P\). Firstly introduced by \textit{G. Baxter} [Pac. J. Math. 10, 731--742 (1960; Zbl 0095.12705)] in a context of probability theory and popularized by \textit{G.-C. Rota} and \textit{D. A. Smith} [in: Sympos. math. 9, Calcolo Probab., Teor. Turbolenza 1971, 179--201 (1972; Zbl 0255.08003); \textit{G. C. Rota}, Bull. Am. Math. Soc. 75, 325--329 (1969; Zbl 0192.33801)], they now appear in various fields of mathematics and physics. The first appearance of family Rota-Baxter algebra seems to be in [\textit{K. Ebrahimi-Fard} et al., Commun. Math. Phys. 276, No. 2, 519--549 (2007; Zbl 1136.81395)] in the context of renormalization of quantum field theories. This terminology, due to \textit{L. Guo} [J. Algebr. Comb. 29, No. 1, 35--62 (2009; Zbl 1227.05271)], refers to an associative algebra \(A\)\ with a family of linear endomorphisms \(P_{\alpha}:A\rightarrow A\) indexed by the elements of a semigroup \(\left(\Omega,\ast\right)\) such that for any \(a,b\in A\), for any \(\alpha,\beta\in\Omega\),
\[
P_{\alpha}\left(a\right)P_{\beta}\left(b\right)=P_{\alpha\ast\beta}\left(P_{\alpha}\left(a\right)b+aP_{\beta}\left(b\right)+\lambda ab\right)
\]
The notion of matching Rota-Baxter algebra was introduced in [\textit{Y. Zhang} et al., J. Algebra 552, 134--170 (2020; Zbl 1444.16058)], where the Rota-Baxter operators are indexed by the elements of a set \(\Omega\) with no structure, and the weight are given by a family of scales \(\left(\lambda_{\alpha}\right)_{\alpha\in\Omega}\), in which, for any \(a,b\in A\), we have
\[
P_{\alpha}\left(a\right)P_{\beta}\left(b\right) =P_{\beta}\left(P_{\alpha}\left(a\right)\right) b+P_{\alpha}\left(aP_{\beta}\left(b\right)\right) +\lambda_{\beta}P_{\alpha}\left(ab\right)
\]
This paper aims to generalize both family and matching Rota-Baxter algebras in the spirit of what was made in [\textit{L. Foissy}, J. Algebra 586, 1--61 (2021; Zbl 1478.16028)] for dendriform algebras. The authors study the structure needed on the set \(\Omega\) of parameters in order to obtain that free \(\Omega\)-Rota-Baxter algebras are described in terms of typed and angularly decorated planar rooted trees. They also describe free commutative \(\Omega\)-Rota-Baxter algebras generated by a commutative algebra \(A\) in terms of typed words.
Reviewer: Hirokazu Nishimura (Tsukuba)Rota-Baxter operations on \(\mathrm{Cur}(\mathrm{sl}_2 (\mathbb{C}))\)https://zbmath.org/1517.170172023-09-22T14:21:46.120933Z"Gubarev, Vsevolod"https://zbmath.org/authors/?q=ai:gubarev.vsevolod-yurevich"Kozlov, Roman"https://zbmath.org/authors/?q=ai:kozlov.roman-a|kozlov.romanIn [Vertex algebras for beginners. Providence, RI: American Mathematical Society (AMS) (1996; Zbl 0861.17017); 2nd ed. (1998; Zbl 0924.17023)], \textit{V. G. Kac} introduced the notion of a Lie conformal algebra, and then in [Sel. Math., New Ser. 4, No. 3, 377--418 (1998; Zbl 0918.17019)], \textit{A. D'Andrea} and \textit{V. G. Kac} proved that there are only two classes of simple Lie conformal algebras of finite type up to isomorphism: the Virasoro conformal algebra and the Lie conformal current algebra \(\mathrm{Cur}(\mathfrak{g})\) associated to a simple finite-dimensional Lie algebra \(\mathfrak{g}\).
In [Lett. Math. Phys. 110, No. 5, 885--909 (2020; Zbl 1472.17077)], \textit{Y. Hong} and \textit{C. Bai} introduced the notion of a ``Rota-Baxter operator on a Lie conformal algebra and showed that every skew-symmetric solution to the conformal classical Yang-Baxter equation on a Lie conformal algebra \(L\) endowed with an invariant bilinear nondegenerate form gives rise to a Rota-Baxter operator on \(L\)''. They also showed that ``there are no nontrivial Rota-Baxter operators on [the Virasoro conformal algebra]''. (Quotes are from the introduction of the paper under review.)
In the paper under review, the authors classify the Rota-Baxter operators on \(\mathrm{Cur}(\mathfrak{sl}_2(\mathbb{C}))\) and describe those which arise from solutions to the conformal classical Yang-Baxter equation via the aforementioned connection by Hong and Bai.
Reviewer: Cindy Tsang (Tokyo)Representations and modules of Rota-Baxter algebrashttps://zbmath.org/1517.170182023-09-22T14:21:46.120933Z"Guo, Li"https://zbmath.org/authors/?q=ai:guo.li|guo.li.1|guo.li.2"Lin, Zongzhu"https://zbmath.org/authors/?q=ai:lin.zongzhuThis paper aims to give a general study of representations and module theory of Rota-Baxter algebras. Based on this work, representation of the Rota-Baxter algebra of Laurent series were discussed in [\textit{Z. Lin} and \textit{L. Qiao}, ``Representations of Rota-Baxter algebras and regular singular decompositions'', Preprint, \url{arXiv:1603.05912}], where an interesting connection was found with class numbers in algebraic number theory, a similar approach to the Rota-Baxter algebra of polynomial algebras was taken in [\textit{L. Qiao} and \textit{J. Pei}, J. Pure Appl. Algebra 222, No. 7, 1738--1757 (2018; Zbl 1416.16036)], and derived functors of Rota-Baxter modules were studied in [\textit{L. Qiao} et al., Algebr. Represent. Theory 22, No. 2, 321--343 (2019; Zbl 1408.16018)].
\begin{itemize}
\item[\S 2] introduces the concept of a Rota-Baxter module over a Rota-Baxter algebra, providing the regular-singular decomposition of a quasi-idempotent Rota-Baxter module, The classical additive Atkinson factorization of a Rota-Baxter algebra is discussed in terms of quiver representations.
\item[\S 3] introduces the ring of Rota-Baxter operators on a given Rota-Baxter algebra, establishing its relation with Rota-Baxter modules by an equivalence between the category of Rota-Baxter modules and the category of modules over the ring of Rota-Baxter operators.
\item[\S 4] gives a construction of the ring of Rota-Baxter operators with more detailed description for the special cases of divided powers and Laurent series.
\item[\S 5] revisits the topic of matrix representations, giving a class of representation of convolution Rota-Baxter algebras by endomorphisms and matrix Rota-Baxter algebras. An algebraic Birkhoff factorization for Rota-Baxter modules is also established.
\item[\S 6] gives a brief discussion on Rota-Baxter algebras in the tensor category context.
\end{itemize}
Reviewer: Hirokazu Nishimura (Tsukuba)Local properties of Virasoro-like algebrahttps://zbmath.org/1517.170192023-09-22T14:21:46.120933Z"Tang, Xiaomin"https://zbmath.org/authors/?q=ai:tang.xiaomin.1|tang.xiaomin"Xiao, Mingyue"https://zbmath.org/authors/?q=ai:xiao.mingyue"Wang, Peng"https://zbmath.org/authors/?q=ai:wang.peng.2|wang.peng.25|wang.peng.4|wang.peng.7|wang.peng.5Summary: This paper studies local properties of the Virasoro-like algebra. Namely, the 2-local derivations and 2-local automorphisms of the Virasoro-like algebra are considered. It is proved that every 2-local derivation of this Lie algebra is a derivation, and every 2-local automorphism of this Lie algebra is an automorphism.Operator identities on Lie algebras, rewriting systems and Gröbner-Shirshov baseshttps://zbmath.org/1517.170202023-09-22T14:21:46.120933Z"Zhang, Huhu"https://zbmath.org/authors/?q=ai:zhang.huhu"Gao, Xing"https://zbmath.org/authors/?q=ai:gao.xing"Guo, Li"https://zbmath.org/authors/?q=ai:guo.liSummary: Motivated by the pivotal role played by linear operators, many years ago Rota proposed to determine algebraic operator identities satisfied by linear operators on associative algebras, later called Rota's Program on Algebraic Operators. Recent progresses on this program have been achieved in the contexts of operated algebra, rewriting systems and Gröbner-Shirshov bases. These developments also suggest that Rota's insight can be applied to determine operator identities on Lie algebras, and thus to put the various linear operators on Lie algebras in a uniform perspective. This paper carries out this approach, utilizing operated polynomial Lie algebras spanned by nonassociative Lyndon-Shirshov bracketed words. The Lie algebra analog of Rota's program is formulated in terms convergent rewriting systems and equivalently in terms of Gröbner-Shirshov bases. The relation of this Lie algebra analog of Rota's program with Rota's program for associative algebras is established. Applications are given to modified Rota-Baxter operators, differential type operators and Rota-Baxter type operators.Invariant bilinear operators and the second \(\mathfrak{sl}(2)\)-relative cohomology of the Lie algebra of vector fields on \(\mathbb{R}\)https://zbmath.org/1517.170212023-09-22T14:21:46.120933Z"Abdaoui, Meher"https://zbmath.org/authors/?q=ai:abdaoui.meherLet \(\mathrm{Vect}(\mathbb{R})\) be the Lie algebra of smooth vector fields on \(\mathbb{R}\). By direct computations, the author classifies \(\mathfrak{sl}(2)\)-invariant skew-symmetric binary differential operators from \(\mathrm{Vect}(\mathbb{R})\wedge \mathrm{Vect}(\mathbb{R})\) to the space of bilinear differential operators acting on weighted densities \(\mathcal{D}_{\lambda,\mu;\nu}\) vanishing on \(\mathfrak{sl}(2)\). As an application, the second differential \(\mathfrak{sl}(2)\)-relative cohomology of \(\mathrm{Vect}(\mathbb{R})\) with coefficients in \(\mathcal{D}_{\lambda,\mu;\nu}\) is computed.
Reviewer: Andrea Galasso (Taipei)Bilinear differential operators and \(\mathfrak{osp}(1|2)\)-relative cohomology on \(\mathbb{R}^{1|1}\)https://zbmath.org/1517.170222023-09-22T14:21:46.120933Z"Ghallabi, Abderraouf"https://zbmath.org/authors/?q=ai:ghallabi.abderraouf"Abdaoui, Meher"https://zbmath.org/authors/?q=ai:abdaoui.meherThe authors study the simplest super analog of the problem solved by \textit{S. Bouarroudj} [J. Nonlinear Math. Phys. 14, No. 1--4, 112--127 (2007; Zbl 1170.17006)]. Namely, they consider the superspace \(\mathbb R^{1|1}\) equipped with the contact structure determined by a 1-form \(\alpha\), and the Lie superalgebra \(\mathcal K(1)\) of contact vector fields on \(\mathbb R^{1|1}\). The conformal Lie superalgebra \(\mathcal K(1)\) acts on \(\mathbb R^{1|1}\) as the Lie superalgebra of contact vector fields, it contains the Möbius superalgebra \(\mathfrak{osp}(1|2)\). The authors classify \(\mathfrak{osp}(1|2)\)-invariant superskew-symmetric binary differential operators from \(\mathcal K(1)\wedge\mathcal K(1)\) to \(\mathcal D_{\lambda, \mu;\nu}\) vanishing on \(\mathfrak{osp}(1|2)\), where \(\mathcal D_{\lambda, \mu;\nu}\) is the superspace of bilinear differential operators between the superspaces of weighted densities. As a consequence they compute the second differential \(\mathfrak{osp}(1|2)\)-relative cohomology of \(\mathcal K(1)\) with coefficients in \(\mathcal D_{\lambda, \mu;\nu}\). Especially, they show that nonzero cohomology \(H^2_{\mathrm{diff}}(\mathcal K(1), \mathfrak{osp}(1|2);\mathcal D_{\lambda, \mu;\nu})\) only appears for some values of weights that satisfy \(\nu-\mu-\lambda\in\frac{1}{2}\mathbb N+3\).
Reviewer: Béchir Dali (Bizerte)Genus of vertex algebras and mass formulahttps://zbmath.org/1517.170232023-09-22T14:21:46.120933Z"Moriwaki, Yuto"https://zbmath.org/authors/?q=ai:moriwaki.yutoLattice vertex algebras present an important class of vertex algebras which are associated with positive definite even integral lattices. In the lattice theory, one can naturally introduce the notions of equivalent lattices, i.e. lattices in the same genus, and mass of the genus of a lattice. Moreover, these concepts play an important role in the classification of lattices. The goal of this paper is to introduce and investigate the analogous notions for vertex algebras in order to obtain a new method of their construction and classification, which employs techniques coming from the theory of lattices.
As the construction of lattice vertex algebras is not a functor from the category of lattices to the category of vertex algebras, the author considers the (monoidal) categories of AH pairs and VH pairs. Roughly speaking, the former consists of pairs \((A,H)\), where \(A\) is an associative algebra and \(H\) a finite dimensional vector space equipped with a bilinear form satisfying some properties, while the later consists of pairs \((V,H)\), where \(V\) is a vertex algebra and \(H\) the so-called Heisenberg subspace of \(V\) (which are again subject to certain conditions). The author constructs a functor from the category of AH pairs to the category of VH pairs, along with its right adjoint functor, such that, in particular, the pair \((\mathbb{C}\left\{L\right\},\mathbb{C} \otimes_{\mathbb{Z}} L)\), where \(\mathbb{C}\left\{L\right\}\) is the group algebra of an even lattice \(L\), is mapped to the lattice vertex algebra \(V_L\) associated with \(L\). Next, the notions of genus of VH pair and mass of the VH pair in the genus are introduced and studied. In particular, for a certain wide class of VH pairs, the so-called good pairs, a formula which relates the mass of VH pair to the mass of lattice is derived. Finally, as an application, a new characterization of some holomorphic vertex operator algebras is obtained.
Reviewer: Slaven Kozic (Zagreb)Graded torsion-free \(\mathfrak{sl}_2(\mathbb{C})\)-modules of rank 2https://zbmath.org/1517.170242023-09-22T14:21:46.120933Z"Bahturin, Yuri"https://zbmath.org/authors/?q=ai:bahturin.yuri-a"Shihadeh, Abdallah"https://zbmath.org/authors/?q=ai:shihadeh.abdallahThis paper looks for simple examples of infinite-dimensional graded \(\mathfrak{sl}_2(\mathbb C)\)-modules, compatible with the Pauli \(\mathbb{Z}_2^2\)-grading on the Lie algebra \( \mathfrak{sl}_2(\mathbb C)\) given by zero \( (\bar0,\bar0) \)-component and the other three homogeneous components one dimensional and spanned, respectively, by
\[
h=\begin{pmatrix}1&0\\
0&-1\end{pmatrix},\quad B=\begin{pmatrix}0&1\\
1&0\end{pmatrix},\quad C=\begin{pmatrix}0&1\\
-1&0\end{pmatrix}.
\]
It is well known that every simple \(\mathfrak{sl}_2(\mathbb C)\)-module is either a weight module or torsion-free, that is, free as a module over the polynomial algebra \(\mathbb C[h]\). Besides, there are torsion-free \(\mathfrak{sl}_2(\mathbb C)\)-modules of arbitrary finite rank. But this is not the case for graded \(\mathfrak{sl}_2(\mathbb C)\)-modules, for instance, there does not exist a torsion-free graded \(\mathfrak{sl}_2(\mathbb C)\)-module of rank 1. The main purpose of this paper is to construct simple torsion-free \(\mathbb{Z}_2^2\)-graded \(\mathfrak{sl}_2(\mathbb C)\)-modules (of rank 2).
Thus, extend the \(\mathbb{Z}_2^2\)-grading on \(\mathfrak{sl}_2(\mathbb C)\) to the universal enveloping algebra \(U(\mathfrak{sl}_2(\mathbb C) )\). The Casimir element for \(\mathfrak{sl}_2(\mathbb C)\) can be written as \(c=h^2+B^2-C^2+1\), so \(c\in U(\mathfrak{sl}_2(\mathbb C) )\) is a homogeneous element. For each scalar \(\lambda\in\mathbb C\), consider \(I_\lambda\) the ideal of \(U(\mathfrak{sl}_2(\mathbb C) )\) generated by the central element \(c-(\lambda+1)^2\). Take \(U(I_\lambda):= U(\mathfrak{sl}_2(\mathbb C) )/I_\lambda\) and the \(U(I_\lambda)\)-module \(M_\lambda^C:=U(I_\lambda)/U(I_\lambda)C\). This is a torsion-free graded \(\mathfrak{sl}_2(\mathbb C)\)-module of rank 2, with basis \(\{1,B\}\) as \(\mathbb C[h]\)-module. For any complex number \(\lambda\) which is not an even integer, it is proved that the graded \(\mathfrak{sl}_2(\mathbb C)\)-module \(M_\lambda^C\) is (graded) simple. The condition on the scalar is necessary, because it is also proved that whenever \(\lambda\in 2\mathbb Z\), then \(M_\lambda^C\) is not simple and its unique maximal (graded) submodule is direct sum of two simple torsion-free \(\mathfrak{sl}_2(\mathbb C)\)-modules of rank 1.
Finally, \(\mathbb Z\)-gradings are considered too, due to the fact that the Lie algebra \(\mathfrak{sl}_2(\mathbb C)\) only admits two fine gradings, the above mentioned \(\mathbb{Z}_2^2\)-grading and the usual root decomposition which is a \(\mathbb{Z}\)-grading. In this regard, simple torsion-free \(\mathfrak{sl}_2(\mathbb C)\)-modules of finite rank are never compatible with the \(\mathbb Z\)-grading. In particular, \(M_\lambda^C\) is not a \(\mathbb Z\)-graded \(\mathfrak{sl}_2(\mathbb C)\)-module if \(\lambda\notin 2\mathbb Z\).
Reviewer: Cristina Draper Fontanals (Malaga)Saturated Majorana representations of \(A_{12}\)https://zbmath.org/1517.200192023-09-22T14:21:46.120933Z"Franchi, Clara"https://zbmath.org/authors/?q=ai:franchi.clara"Ivanov, Alexander A."https://zbmath.org/authors/?q=ai:ivanov.alexander-a"Mainardis, Mario"https://zbmath.org/authors/?q=ai:mainardis.marioThe main ingredients of a Majorana representation, as defined by Ivanov, are a finite group \(G\), a \(G\)-stable set \(T\) of involutions in \(G\) and an action of \(G\) on a commutative non-associative algebra \(V\) called Majorana algebra. The motivating example of a Majorana representation is the Monster simple group \(M\), together with its set of Fischer involutions and its action on the Griess algebra.
In the paper under review, the authors investigate the case where \(G\) is the alternating group \(A_{12}\) and \(T\) is the set of involutions of cycle type \(2^2\) or \(2^6\). The authors prove that there is a unique (up to equivalence) corresponding Majorana representation. The relevant Majorana algebra has dimension 3960, and the decomposition of \(V\), viewed as an \(\mathbb{R}A_{12}\)-module, into simple submodules is determined. Thus this Majorana representation comes from an embedding of \(A_{12}\) into the Monster \(M\).
As a consequence of these results, the authors show that the Harada-Norton simple group \(HN\) also has a unique (up to equivalence) Majorana representation. Thus, this similarly comes from an embedding of \(HN\) into \(M\). Some results on Majorana representations of smaller alternating groups are also derived.
Reviewer: Burkhard Külshammer (Jena)Functions whose orbital integrals and those of their Fourier transforms are at topologically nilpotent supporthttps://zbmath.org/1517.200752023-09-22T14:21:46.120933Z"Waldspurger, J.-L."https://zbmath.org/authors/?q=ai:waldspurger.jean-loupSummary: Let \(F\) be a \(p\)-adic field and let \(G\) be a connected reductive group defined over \(F\). We assume \(p\) is large. Denote \(\mathfrak{g}\) the Lie algebra of \(G\). To each vertex \(s\) of the reduced Bruhat-Tits' building of \(G\), we associate as usual a reductive Lie algebra \({\mathfrak{g}_s}\) defined over the residual field \({\mathbb{F}_q} \). We normalize suitably a Fourier-transform \(f\mapsto \hat{f}\) on \({C_c^{\infty }}(\mathfrak{g}(F))\). We study the subspace of functions \(f\in{C_c^{\infty }}(\mathfrak{g}(F))\) such that the orbital integrals of \(f\) and of \(\hat{f}\) are 0 for each element of \(\mathfrak{g}(F)\) which is not topologically nilpotent. This space is related to the characteristic functions of the character-sheaves on the spaces \({\mathfrak{g}_s}({\mathbb{F}_q})\), for each vertex \(s\), which are cuspidal and with nilpotent support. We prove that our subspace behave well under endoscopy.On the algebra generated by \(\overline{\mu}\), \(\overline{\partial}\), \(\partial\), \(\mu\)https://zbmath.org/1517.320802023-09-22T14:21:46.120933Z"Auyeung, Shamuel"https://zbmath.org/authors/?q=ai:auyeung.shamuel"Guu, Jin-Cheng"https://zbmath.org/authors/?q=ai:guu.jin-cheng"Hu, Jiahao"https://zbmath.org/authors/?q=ai:hu.jiahaoSummary: In this note, we determine the structure of the associative algebra generated by the differential operators \(\overline{\mu}\), \(\overline{\partial}\), \(\partial\), and \(\mu\) that act on complex-valued differential forms of almost complex manifolds. This is done by showing that it is the universal enveloping algebra of the graded Lie algebra generated by these operators and determining the structure of the corresponding graded Lie algebra. We then determine the cohomology of this graded Lie algebra with respect to its canonical inner differential \([d, -]\), as well as its cohomology with respect to all its inner differentials.On finitely Levi non degenerate homogeneous CR manifoldshttps://zbmath.org/1517.321142023-09-22T14:21:46.120933Z"Marini, Stefano"https://zbmath.org/authors/?q=ai:marini.stefanoSummary: A \(CR\) manifold \(M\) is a differentiable manifold together with a complex subbundle of the complexification of its tangent bundle, which is formally integrable and has zero intersection with its conjugate bundle. A fundamental invariant of a \(CR\) manifold \(M\) is its vector-valued Levi form. A Levi non degenerate \(CR\) manifold of order \(k\ge 1\) has non degenerate Levi form in a higher order sense. For a (locally) homogeneous manifold Levi non degeneracy of order \(k\) can be described in terms of its \(CR\) algebra, i.e. a pair of Lie algebras encoding the structure of (locally) homogeneous \(CR\) manifolds. I will give an introduction to these topics presenting some recent results.The classificatory function of diagrams: two examples from mathematicshttps://zbmath.org/1517.570052023-09-22T14:21:46.120933Z"Eckes, Christophe"https://zbmath.org/authors/?q=ai:eckes.christophe"Giardino, Valeria"https://zbmath.org/authors/?q=ai:giardino.valeriaSummary: In a recent paper, \textit{S. De Toffoli} and the second author analyzed the practice of knot theory, by focusing in particular on the use of diagrams to represent and study knots
[Erkenntnis 79, No. 4, 829--842 (2014; Zbl 1304.57013)].
To this aim, they distinguished between \textit{illustrations} and \textit{diagrams}. An illustration is \textit{static}; by contrast, a diagram is \textit{dynamic}, that is, it is closely related to some specific inferential procedures. In the case of knot diagrams, a diagram is also a well-defined mathematical object in itself, which can be used to classify knots. The objective of the present paper is to reply to the following questions: Can the classificatory function characterizing knot diagrams be generalized to other fields of mathematics? Our hypothesis is that dynamic diagrams that are mathematical objects in themselves are used to solve classification problems. To argue in favor of our hypothesis, we will present and compare two examples of classifications involving them: (i) the classification of compact connected surfaces (orientable or not, with or without boundary) in combinatorial topology; (ii) the classification of complex semisimple Lie algebras.
For the entire collection see [Zbl 1387.68019].Relative (pre)-modular categories from special linear Lie superalgebrashttps://zbmath.org/1517.570192023-09-22T14:21:46.120933Z"Anghel, Cristina Ana-Maria"https://zbmath.org/authors/?q=ai:anghel.cristina-ana-maria"Geer, Nathan"https://zbmath.org/authors/?q=ai:geer.nathan"Patureau-Mirand, Bertrand"https://zbmath.org/authors/?q=ai:patureau-mirand.bertrandSummary: We examine two different m-traces in the category of representations over the quantum Lie superalgebra associated to \(\mathfrak{sl}(m|n)\) at root of unity. The first m-trace is on the ideal of projective modules and leads to new Extended Topological Quantum Field Theories. The second m-trace is on the ideal of perturbative typical modules. We consider the quotient with respect to negligible morphisms coming from this m-trace and show that in the case of \(\mathfrak{sl}(2|1)\) this quotient leads to 3-manifolds invariants. We conjecture that the quotient category of perturbatives over quantum \(\mathfrak{sl}(m|n)\) leads to 3-manifold invariants and more generally ETQFTs.On integrations and cross ratios on supermanifoldshttps://zbmath.org/1517.580022023-09-22T14:21:46.120933Z"Leites, Dimitry"https://zbmath.org/authors/?q=ai:leites.dimitry-aSummary: (A) The conventional integration theory on supermanifolds had been constructed in order to have (an analog of) the Stokes formula in which a sub-supermanifold is of codimension \(1 = (1|0)\). I review other integrations and formulate related open problems:
\begin{itemize}
\item[(1)] On the \(1|1\)-dimensional superstring associated with the trivial bundle, in presence of a contact structure there is a special integration useful in describing super versions of elliptic functions. It is needed to construct a particular spinor representation of the Neveu-Schwarz superalgebra.
\item[(2)] Versions of the Stokes formula with ``over-supermanifold'' of codimension \((0|-1)\) due to Shander and Palamodov should be developed further.
\item[(3)] Apply Shander's integration with odd parameters over chains to inverse problems.
\item[(4)] Establish existence of conjectural integrations (apparently, not leading to any analog of the Stokes formula) related to various (super)traces on various Lie superalgebras and the corresponding (super)determinants.
\end{itemize}
(B) I offer analogs of the cross ratio for ``classical superspaces'', including infinite-dimensional versions. Open problem: apply these invariants to the matrix-valued Riccati equations.Riesz potential for \((k,1)\)-generalized Fourier transformhttps://zbmath.org/1517.810522023-09-22T14:21:46.120933Z"Ivanov, Valeriĭ Ivanovich"https://zbmath.org/authors/?q=ai:ivanov.valerii-ivanovichSummary: In spaces with weight \(|x|^{-1}v_k(x)\), where \(v_k(x)\) is the Dunkl weight, there is the \((k,1)\)-generalized Fourier transform. Harmonic analysis in these spaces is important, in particular, in problems of quantum mechanics. We define the Riesz potential for the \((k,1)\)-generalized Fourier transform and prove for it, a \((L^q,L^p)\)-inequality with radial power weights, which is an analogue of the well-known Stein-Weiss inequality for the classical Riesz potential. For the Riesz potential we calculate the sharp value of the \(L^p\)-norm with radial power weights. The sharp value of the \(L^p\)-norm with radial power weights for the classical Riesz potential was obtained independently by W. Beckner and S. Samko.Semiclassical asymptotics of oscillating tunneling for a quadratic Hamiltonian on the algebra \(\operatorname{su}(1,1)\)https://zbmath.org/1517.810562023-09-22T14:21:46.120933Z"Vybornyi, E. V."https://zbmath.org/authors/?q=ai:vybornyi.e-v"Rumyantseva, S. V."https://zbmath.org/authors/?q=ai:rumyantseva.s-vSummary: In this paper, we consider the problem of constructing semiclassical asymptotics for the tunnel splitting of the spectrum of an operator defined on an irreducible representation of the Lie algebra \(\operatorname{su}(1,1)\). It is assumed that the operator is a quadratic function of the generators of the algebra. We present coherent states and a unitary coherent transform that allow us to reduce the problem to the analysis of a second-order differential operator in the space of holomorphic functions. Semiclassical asymptotic spectral series and the corresponding wave functions are constructed as decompositions in coherent states. For some values of the system parameters, the minimal energy corresponds to a pair of nondegenerate equilibria, and the discrete spectrum of the operator has an exponentially small tunnel splitting of the levels. We apply the complex WKB method to prove asymptotic formulas for the tunnel splitting of the energies. We also show that, in contrast to the one-dimensional Schrödinger operator, the tunnel splitting in this problem not only decays exponentially but also contains an oscillating factor, which can be interpreted as tunneling interference between distinct instantons. We also show that, for some parameter values, the tunneling is completely suppressed and some of the spectral levels are doubly degenerate, which is not typical of one-dimensional systems.