Recent zbMATH articles in MSC 17Bhttps://www.zbmath.org/atom/cc/17B2022-05-16T20:40:13.078697ZWerkzeugA composite order generalization of modular moonshinehttps://www.zbmath.org/1483.110802022-05-16T20:40:13.078697Z"Urano, Satoru"https://www.zbmath.org/authors/?q=ai:urano.satoruSummary: We introduce a generalization of Brauer character to allow arbitrary finite length modules over discrete valuation rings. We show that the generalized super Brauer character of Tate cohomology is a linear combination of trace functions. Using this result, we find a counterexample to a conjecture of Borcherds about vanishing of Tate cohomology for Fricke elements of the Monster.\(\widehat{sl(2)}\) decomposition of denominator formulae of some BKM Lie superalgebrashttps://www.zbmath.org/1483.110912022-05-16T20:40:13.078697Z"Govindarajan, Suresh"https://www.zbmath.org/authors/?q=ai:govindarajan.suresh"Shabbir, Mohammad"https://www.zbmath.org/authors/?q=ai:shabbir.mohammad"Viswanath, Sankaran"https://www.zbmath.org/authors/?q=ai:viswanath.sankaranSummary: We study a family of Siegel modular forms that are constructed using Jacobi forms that arise in Umbral moonshine. All but one of them arise as the Weyl-Kac-Borcherds denominator formula of some Borcherds-Kac-Moody (BKM) Lie superalgebras. These Lie superalgebras have a \(\widehat{sl(2)}\) subalgebra which we use to study the Siegel modular forms. We show that the expansion of the Umbral Jacobi forms in terms of \(\widehat{sl(2)}\) characters leads to vector-valued modular forms. We obtain closed formulae for these vector-valued modular forms. In the Lie algebraic context, the Fourier coefficients of these vector-valued modular forms are related to multiplicities of roots appearing on the sum side of the Weyl-Kac-Borcherds denominator formulae.Hodge theory of the Turaev cobracket and the Kashiwara-Vergne problemhttps://www.zbmath.org/1483.140152022-05-16T20:40:13.078697Z"Hain, Richard"https://www.zbmath.org/authors/?q=ai:hain.richard-mDenote the set of free homotopy classes of maps \(S^1 \to X\) in a topological space \(X\) by \(\lambda(X)\) and the free \(\mathbb{Q}\)-module it generates by \(\mathbb{Q}\lambda(X)\). When \(X\) is an oriented surface with a nowhere vanishing vector field \(\xi\), there is a map
\[
\delta_\xi : \mathbb{Q}\lambda(X) \to \mathbb{Q}\lambda(X)\otimes \mathbb{Q}\lambda(X),
\]
called the \textit{Turaev cobracket}, that gives \(\mathbb{Q}\lambda(X)\) the structure of a Lie coalgebra. The cobracket was first defined by [\textit{V. G. Turaev}, Mat. Sb., Nov. Ser. 106(148), 566--588 (1978; Zbl 0384.57004)] on \(\mathbb{Q}\lambda(M)/\mathbb{Q}\) (with no framing) and lifted to \(\mathbb{Q}\lambda(M)\) for framed surfaces in [\textit{V. G. Turaev}, Ann. Sci. Éc. Norm. Supér. (4) 24, No. 6, 635--704 (1991; Zbl 0758.57011)] and [\textit{A. Alekseev} et al., ``The Goldman-Turaev Lie bialgebra and the Kashiwara-Vergne problem in higher genera'', Preprint, \url{arXiv:1804.09566}]. The cobracket \(\delta_\xi\) and the Goldman bracket [\textit{R. Hain}, ``Hodge theory of the Goldman bracket'', Preprint, \url{arXiv:1710.06053}]
\[\{\ ,\ \} : \mathbb{Q}\lambda(X)\otimes \mathbb{Q}\lambda(X) \to \mathbb{Q}\lambda(X)
\] endow \(\mathbb{Q}\lambda(X)\) with the structure of an involutive Lie bialgebra [\textit{V. G. Turaev}, Ann. Sci. Éc. Norm. Supér. (4) 24, No. 6, 635--704 (1991; Zbl 0758.57011); \textit{M. Chas}, Topology 43, No. 3, 543--568 (2004; Zbl 1050.57014); \textit{N. Kawazumi} and \textit{Y. Kuno}, Ann. Inst. Fourier 65, No. 6, 2711--2762 (2015; Zbl 1370.57009)].
The value of the cobracket on a loop \(a \in \lambda(X)\) is obtained by representing it by an immersed circle \(\alpha : S^1 \to X\) with transverse self intersections and trivial winding number relative to \(\xi\). Each double point \(P\) of \(\alpha\) divides it into two loops based at \(P\), which we denote by \(\alpha'_P\) and \(\alpha_P''\). Let \(\epsilon_P = \pm 1\) be the intersection number of the initial arcs of \(\alpha_P'\) and \(\alpha_P''\). The cobracket of \(a\) is then defined by
\[
\delta_\xi(a) = \sum_P \epsilon_P(a'_P \otimes a''_P - a''_P \otimes a'_P),\tag{1}
\]
where \(a_P'\) and \(a_P''\) are the classes of \(\alpha_P'\) and \(\alpha_P''\), respectively.
The powers of the augmentation ideal \(I\) of \(\mathbb{Q}\pi_1(X,x)\) define the \(I\)-adic topology on it and induce a topology on \(\mathbb{Q}\lambda(X)\). \textit{N. Kawazumi} and \textit{Y. Kuno} [Ann. Inst. Fourier 65, No. 6, 2711--2762 (2015; Zbl 1370.57009)] showed that \(\delta_\xi\) is continuous in the \(I\)-adic topology and thus induces a map
\[
\delta_\xi : \mathbb{Q}\lambda(X)^\wedge \to \mathbb{Q}\lambda(X)^\wedge\widehat{\otimes}\mathbb{Q}\lambda(X)^\wedge
\]
on \(I\)-adic completions. This and the completed Goldman bracket give \(\mathbb{Q}\lambda(X)^\wedge\) the structure of an involutive completed Lie bialgebra [loc. cit.].
Now suppose that \(X\) is a smooth affine curve over \(\mathbb C\) or, equivalently, the complement of a non-empty finite set \(D\) in a compact Riemann surface \(\overline{X}\). In this case \(\mathbb Q\lambda(X)^\wedge\) has a canonical pro-mixed Hodge structure [\textit{R. M. Hain}, \(K\)-Theory 1, No. 3, 271--324 (1987; Zbl 0637.55006)]. In particular, it has a \textit{weight filtration}
\[
\cdots \subseteq W_{-2} \mathbb Q\lambda(X)^\wedge \subseteq W_{-1} \mathbb Q\lambda(X)^\wedge \subseteq W_0 \mathbb Q\lambda(X)^\wedge = \mathbb Q\lambda(X)^\wedge
\]
and its complexification \(\mathbb C\lambda(X)^\wedge\) has a \textit{Hodge filtration}
\[
\cdots \supset F^{-2} \mathbb C\lambda(X)^\wedge \supset F^{-1} \mathbb C\lambda(X)^\wedge \supset F^{0} \mathbb C\lambda(X)^\wedge \supset F^1\mathbb C\lambda(X)^\wedge = 0.
\]
The Hodge filtration depends on the algebraic structure on \(X\), while the weight filtration is topologically determined and so does not depend on the complex structure. The weight filtration on \(\mathbb Q\lambda(X)^\wedge\) is the image of the weight filtration of \(\mathbb Q\pi_1(X,x)^\wedge\), which is determined uniquely by the conditions that \(W_{-1} \mathbb Q\pi_1(X,x)^\wedge = I\), \(W_{-2} \mathbb Q\pi_1(X,x)^\wedge=\ker\{I\to H_1(\overline{X})\}\), and by the condition that \(W_{-m-2}\mathbb Q\pi_1(X,x)^\wedge\) is the ideal generated by \(W_{-1}W_{-m-1}\) and \(W_{-2}W_{-m}\). This pro-mixed Hodge structure contains subtle geometric and arithmetic information about \(X\). The first main result of the paper is that the Turaev cobracket is compatible with this structure.
Theorem 1.
If \(\xi\) is a nowhere vanishing holomorphic vector field on \(X\) that is meromorphic on \(\overline{X}\), then
\[
\delta_\xi : \mathbb Q\lambda(X)^\wedge\otimes\mathbb Q(-1) \to\mathbb Q\lambda(X)^\wedge\widehat{\otimes}\mathbb Q\lambda(X)^\wedge
\]
is a morphism of pro-mixed Hodge structures, so that \(\mathbb Q\lambda(X)^\wedge\otimes\mathbb Q(1)\) is a complete Lie coalgebra in the category of pro-mixed Hodge structures.
We call such a framing \(\xi\) an \textit{algebraic framing}. The main result of [\textit{R. Hain}, Geom. Topol. 24, No. 4, 1841--1906 (2020; Zbl 1470.14017)] asserts that
\[
\{\ ,\ \} : \mathbb Q\lambda(X)^\wedge\otimes\mathbb Q\lambda(X)^\wedge \to \mathbb Q\lambda(X)^\wedge \otimes \mathbb Q(1)
\]
is a morphism of mixed Hodge structure (MHS), so that \(\mathbb Q\lambda(X)^\wedge\otimes\mathbb Q(-1)\) is a complete Lie algebra in the category of pro-mixed Hodge structures.
Corollary 1. If \(\xi\) is a quasi-algebraic framing of \(X\), then \(\big(\mathbb Q\lambda(X)^\wedge,\{\ ,\ \},\delta_\xi\big)\) is a ``twisted'' completed Lie bialgebra in the category of pro-mixed Hodge structures.
By ``twisted'' one means that one has to twist both the bracket and cobracket by \(\mathbb Q(\pm 1)\) to make them morphisms of MHS. There is no one twist of \(\mathbb Q\lambda(X)\) that makes them simultaneously morphisms of MHS. Let \(\vec{\mathsf v}\) be a non-zero tangent vector of \(\overline{X}\) at a point of \(D\). Standard results in Hodge theory (see [\textit{R. Hain}, Geom. Topol. 24, No. 4, 1841--1906 (2020; Zbl 1470.14017)]) imply:
Corollary 2. Hodge theory determines torsors of compatible isomorphisms
\[
\big(\mathbb Q\lambda(X)^\wedge,\{\ ,\ \},\delta_\xi\big) \overset{\simeq}\longrightarrow \Big(\prod_{m\ge 0}\operatorname{Gr}^W_{-m}\mathbb Q\lambda(X)^\wedge,\operatorname{Gr}^W_\bullet\{\ ,\ \},\operatorname{Gr}^W_\bullet\delta_\xi\Big)\tag{2}
\]
of the Goldman-Turaev Lie bialgebra with the associated weight graded Lie bialgebra and of the completed Hopf algebras
\[
\mathbb Q\pi_1(X,\vec{\mathsf v})^\wedge \overset{\simeq}\longrightarrow\prod_{m\ge 0}\operatorname{Gr}^W_{-m}\mathbb Q\pi_1(X,\vec{\mathsf v})^\wedge\tag{3}
\]
under which the logarithm of the boundary circle lies in \(\operatorname{Gr}^W_{-2} \mathbb Q\pi_1(X,\vec{\mathsf v})^\wedge\). These isomorphisms are torsors under the prounipotent radical \(U^{\mathrm{MT}}_{X,\vec{\mathsf v}}\) of the Mumford-Tate group of the MHS on \(\mathbb Q\pi_1(X,\vec{\mathsf v})^\wedge\).
Let \(\overline{S}\) be a closed oriented surface of genus \(g\) and \(P=\{x_0,\dots,x_n\}\) a finite subset. Set \(S=\overline{S}-P\).
Assume that \(S\) is hyperbolic; that is, \(2g-1+n>0\). Suppose that \(\xi_o\) is a framing of \(S\). Denote the index (or local degree) of \(\xi_o\) at \(x_j\) by \(d_j\). Let \({\mathbf{d}} = (d_0,\dots,d_n) \in \mathbb Z^{n+1}\) be the vector of local degrees of \(\xi_o\). The Poincaré-Hopf Theorem implies that \(\sum d_j = 2-2g\). Also denote the category of mixed Tate motives unramified over \(\mathbb Z\) by \({\mathsf{MTM}}(\mathbb Z)\). Denote the prounipotent radical of its tannakian fundamental group \(\pi_1({\mathsf{MTM}},\omega^B)\) (with respect to the Betti realization \(\omega^B\)) by \(\mathcal K\). Denote the relative completion of the mapping class group of \((\overline{S},P,\vec{\mathsf v}_o)\) by \(\mathcal G_{g,n+u}\) and its prounipotent radical by \(\mathcal U_{g,n+u}\). (See [\textit{R. Hain}, J. Am. Math. Soc. 10, No. 3, 597--651 (1997; Zbl 0915.57001)] for definitions.) These act on \(\mathbb Q\pi_1(S,\vec{\mathsf v}_o)^\wedge\). Denote the image of \(\mathcal U_{g,n+u}\) in \(\operatorname{Aut}\mathbb Q\pi_1(S,\vec{\mathsf v}_o)^\wedge\) by \(\overline{\mathbb U}_{g,n+u}\). The vector field \(\xi_o\) determines a homomorphism \(\overline{\mathbb U}_{g,n+u} \to H_1(\overline{S})\) that depends only on the vector \({\mathbf{d}}\) of local degrees of \(\xi\). Denote its kernel by \(\overline{\mathbb U}_{g,n+u}^{\mathbf{d}}\). \textit{Y. Ihara} and \textit{H. Nakamura} [J. Reine Angew. Math. 487, 125--151 (1997; Zbl 0910.14010)] construct canonical smoothings of each maximally degenerate stable curve \(X_0\) of type \((g,n+1)\) over \(\mathbb Z[[q_1,\dots,q_N]]\) for all \(n \ge 0\), where \(N= \dim \mathbb M_{g,n+1}\). Associated to each tangent vector \(\vec{\mathsf v} = \pm \partial/\partial q_j\) of \(\overline{\mathbb M}_{g,n+1}\) at the point corresponding to \(X_0\), there is a limit pro-MHS on \(\mathbb Q\lambda(X)^\wedge\), that we denote by \(\mathbb Q\lambda(X_{\vec{\mathsf v}})^\wedge\).
Hypothesis. The limit MHS on \(\mathbb Q\lambda(X_{\vec{\mathsf v}})^\wedge\) is the Hodge realization of a pro-object of \({\mathsf{MTM}}(\mathbb Z)\). Equivalently, the Mumford-Tate group of the MHS on \(\mathbb Q\lambda(X_{\vec{\mathsf v}})^\wedge\) is isomorphic to \(\pi_1({\mathsf{MTM}},\omega^B)\).
Theorem 4. If \(2g+n>1\) (i.e., \(S\) is hyperbolic), then the group \(\widehat{\mathbb U}_{g,n+u}^{\mathbf{d}}\) does not depend on the choice of a quasi-algebraic structure \(\phi : (\overline{S},P,\vec{\mathsf v}_o,\xi_o) \to (\overline{X},D,\vec{\mathsf v},\xi)\). The group \(\overline{\mathbb U}_{g,n+u}^{\mathbf{d}}\) is normal in \(\widehat{\mathbb U}_{g,n+u}^{\mathbf{d}}\). If we assume Hypothesis 1, there is a canonical surjective group homomorphism \(\mathcal K \to \widehat{\mathbb U}_{g,n+u}^{\mathbf{d}}/\overline{\mathbb U}_{g,n+u}^{\mathbf{d}}\), where \(\mathcal K\) denotes the prounipotent radical of \(\pi_1({\mathsf{MTM}})\).
From [\textit{R. Hain}, J. Am. Math. Soc. 10, No. 3, 597--651 (1997; Zbl 0915.57001)] that the completion of \(\Gamma_{g,m+\vec{r}}\) relative to \(\rho : \Gamma_{g,m+\vec{r}} \to {\mathrm{Sp}}(H_{\mathbb Q})\) is an affine \(\mathbb Q\)-group \(\mathcal G_{g,m+\vec{r}}\) that is an extension
\[
1 \to \mathcal U_{g,m+\vec{r}} \to \mathcal G_{g,m+\vec{r}} \to {\mathrm{Sp}}(H) \to 1
\]
of affine \(\mathbb Q\)-groups, where \(\mathcal U_{g,m+\vec{r}}\) is prounipotent. There is a Zariski dense homomorphism \(\tilde{\rho} : \Gamma_{g,m+\vec{r}} \to\mathcal G_{g,m+\vec{r}}(\mathbb Q)\) whose composition with the homomorphism \(\mathcal G_{g,m+\vec{r}}(\mathbb Q) \to {\mathrm{Sp}}(H_{\mathbb Q})\) is \(\rho\). When \(g=0\), \({\mathrm{Sp}}(H)\) is trivial and \(\mathcal G_{0,m+\vec{r}}\) is the unipotent completion \(\Gamma_{0,m+\vec{r}}^{\mathrm{un}}\). The action of the mapping class group \(\Gamma_{g,n+u}\) on \(\mathbb Q\pi_1(S,\vec{\mathsf v}_o)\) induces an action on \(\mathbb Q\lambda(S)\) which preserves the Goldman bracket. The stabilizer of \(\xi_o\) preserves the Turaev cobracket. The universal mapping property of relative completion implies that \(\mathcal G_{g,n+u}\) acts on \(\mathbb Q\pi_1(S,\vec{\mathsf v}_o)^\wedge\) and \(\mathbb Q\lambda(S)^\wedge\). Since the image of the mapping class group in \(\mathcal G_{g,n+u}\) is Zariski dense, this action preserves the Goldman bracket.
A quasi-complex structure
\[
\phi : (\overline{S},P,\vec{\mathsf v}_o,\xi_o)\to (\overline{X},D,\vec{\mathsf v},\xi)
\]
on \((\overline{S},P,\vec{\mathsf v}_o,\xi_o)\) determines an isomorphism \(\Gamma_{g,n+u} \cong \pi_1(\mathbb M_{g,n+u},\phi_o)\). The corresponding MHS on the relative completion \(\mathcal G_{g,n+u}\) corresponds to an action of \(\pi_1(\mathsf{MHS})\) on \(\mathcal G_{g,n+u}\). The quasi-complex structure \(\phi\) determines a semi-direct product
\[
\pi_1(\mathsf{MHS}) \ltimes \mathcal G_{g,n+u}.
\]
Since the natural homomorphism \(\mathcal G_{g,n+u} \to \operatorname{Aut} \mathbb Q\pi_1(X,\vec{\mathsf v}_o)^\wedge\) is a morphism of MHS [\textit{R. Hain}, J. Am. Math. Soc. 10, No. 3, 597--651 (1997; Zbl 0915.57001)], the monodromy homomorphism extends to a homomorphism
\[
\pi_1(\mathsf{MHS}) \ltimes\mathcal G_{g,n+u} \to \operatorname{Aut}\mathbb Q\pi_1(X,\vec{\mathsf v}_o)^\wedge.
\]
Denote its image by \(\widehat{\mathcal G}_{g,n+u}\) and the image of \(\mathcal G_{g,n+u}\) by \(\overline{\mathcal G}_{g,n+u}\). It is normal in \(\widehat{\mathcal G}_{g,n+u}\). The group \(\widehat{\mathcal G}_{g,n+u}\) is an extension
\[
1 \to \widehat{\mathbb U}_{g,n+u} \to \widehat{\mathcal G}_{g,n+u} \to {\mathrm{GSp}}(H) \to 1,
\]
where \({\mathrm{GSp}}\) denotes the general symplectic group and \(\widehat{\mathbb U}_{g,n+u}\) is prounipotent. One can argue as in [\textit{R. Hain} and \textit{M. Matsumoto}, J. Inst. Math. Jussieu 4, No. 3, 363--403 (2005; Zbl 1094.14013)] that, if \(g\ge 3\), then then \(\mathcal U^{\mathrm{MT}}_{X,\vec{\mathsf v}} \to \widehat{\mathbb U}_{g,n+u}\) is an isomorphism if and only if \(\pi_1(\mathsf{MHS}) \to{\mathrm{GSp}}(H)\) is surjective; the Griffiths invariant \(\nu(\overline{X}) \in \operatorname{Ext}^1_{\mathsf{MHS}}(\mathbb Q,PH^3(\operatorname{Jac}\overline{X}(2)))\) of the Ceresa cycle in \(\operatorname{Jac}\overline{X}\) is non-zero; and if the points \(\kappa_j := (2g-2)x_j - K_{\overline{X}} \in (\operatorname{Jac} \overline{X})\otimes \mathbb Q\), \(0\le j \le n\), are linearly independent over \(\mathbb Q\). This holds for general \((\overline{X},D,\vec{\mathsf v})\).
Proposition. For each complex structure \(\phi : (\overline{S},P,\vec{\mathsf v}_o) \to (\overline{X},D,\vec{\mathsf v})\), the coordinate ring \(\mathcal O(\widehat{\mathcal G}_{g,n+u}/\overline{\mathcal G}_{g,n+u})\) has a canonical MHS. These form an admissible variation of MHS over \(\mathbb M_{g,n+u}\) with trivial monodromy. Consequently, the MHS on \(\mathcal O(\widehat{\mathbb U}_{g,n+u}/\overline{\mathbb U}_{g,n+u})\) does not depend on the complex structure \(\phi\).
Reviewer: Mohammad Reza Rahmati (León)Gauss-Manin Lie algebra of mirror elliptic \(K3\) surfaceshttps://www.zbmath.org/1483.140682022-05-16T20:40:13.078697Z"Alim, Murad"https://www.zbmath.org/authors/?q=ai:alim.murad"Vogrin, Martin"https://www.zbmath.org/authors/?q=ai:vogrin.martinThis paper is about mirror symmetry for elliptic \(K3\) surfaces, which are \(K3\) surfaces together with a map to the projective line such that the general fiber is an elliptic curve.
The authors first describe a moduli space \(T\) of the mirror \(K3\) surfaces, from the data of the holomorphic Gauss-Manin connection of the middle dimensional cohomology of the mirrors. They show that away from a discriminant locus \(T\) is a locally ringed space with the local ring \(\mathcal{O}_T\), which admits an isomorphism to a graded ring of quasi-modular forms for a congruence subgroup of \(SL(2,\mathbb{Z})\). The authors furthermore construct the Gauss-Manin Lie algebra attached to \(T\) and prove that it is isomorphic to \(\mathrm{sl}_2(\mathbb{C}) \oplus \mathrm{sl}_2(\mathbb{C})\).
Reviewer: Hulya Arguz (London)Polynomial tau-functions for the multicomponent KP hierarchyhttps://www.zbmath.org/1483.140852022-05-16T20:40:13.078697Z"Kac, Victor G."https://www.zbmath.org/authors/?q=ai:kac.victor-g"De Leur, Johan W. van"https://www.zbmath.org/authors/?q=ai:van-de-leur.johan-wIn a previous paper [Jpn. J. Math. (3) 13, No. 2, 235--271 (2018; Zbl 1401.14209)], the authors constructed all polynomial tau-functions of the \(1\)-component KP hierarchy, namely, they showed that any such tau-function is obtained from a Schur polynomial \(s_\lambda(t)\) by certain shifts of arguments. In the present paper they give a simpler proof of this result, using the (\(1\)-component) boson-fermion correspondence. Moreover, they show that this approach can be applied to the \(s\)-component KP hierarchy, using the \(s\)-component boson-fermion correspondence, finding thereby all its polynomial tau-functions. The authors also find all polynomial tau-functions for the reduction of the \(s\)-component KP hierarchy, associated to any partition consisting of \(s\) positive parts.
The paper is organized as follows. The first section is an introduction to the subject. Section 2 is devoted to the fermionic formulation of the KP hierarchy and Section 3 to the bosonic formulation of KP. Section 4 deals with polynomial solutions of KP. In Section 5 the authors introduce the \(s\)-component KP, where \(s\) is a positive integer. Section 6 is devoted to the \(n\)-KdV where \(n\) is an integer, \(n\geq2\). In Section 7 the authors consider a reduction of the \(s\)-component KP hierarchy, which describes the loop group orbit of \(SL_n\), where \(n=n_1+n_2+\cdots+n_s\), with \(n_1\geq n_2\geq\cdots\geq n_s\geq1\). The case \(s=1\) is the nth Gelfand-Dickey hierarchy. The case \(n=s=2\), i.e., \(n_1=n_2=1\), is the AKNS (or nonlinear Schrödinger) hierarchy. Section 8 is devoted to the AKNS hierarchy.
Reviewer: Ahmed Lesfari (El Jadida)Differential operators on quantized flag manifolds at roots of unity. IIIhttps://www.zbmath.org/1483.140862022-05-16T20:40:13.078697Z"Tanisaki, Toshiyuki"https://www.zbmath.org/authors/?q=ai:tanisaki.toshiyukiLet \(\mathfrak{g}_k\) be the Lie algebra of a connected semisimple algebraic group over an algebraically closed field \(k\) of positive characteristic. Two important results concerning the sheaf \(\mathcal{D}\) of twisted differential operators on the corresponding flag manifold, which are Beilinson-Bernstein type derived equivalence between the category of certain representations of \(\mathfrak{g}_k\) and that of \(\mathcal{D}\)-modules, and the split Azumaya property of \(\mathcal{D}\) over a certain central subalgebra.
The author gives an analogue using quantized flag manifolds and quantized enveloping algebras at roots of unity instead of ordinary flag manifolds and ordinary enveloping algebras in positive characteristics. More specifically, the author describes the cohomology of the sheaf of twisted differential operators on the quantized flag manifold at a root of unity whose order is a prime power. For the De Concini-Kac type quantized enveloping algebra, where the parameter \(q\) is specialized to a root of unity whose order is a prime power, it follows that the number of irreducible modules with a certain specified central character coincides with the dimension of the total cohomology group of the corresponding Springer fiber, giving a weak version of a conjecture of Lusztig concerning non-restricted representations of the quantized enveloping algebra.
Reviewer: Mee Seong Im (Annapolis)Schofield sequences in the Euclidean casehttps://www.zbmath.org/1483.160212022-05-16T20:40:13.078697Z"Szántó, Csaba"https://www.zbmath.org/authors/?q=ai:szanto.csaba"Szöllősi, István"https://www.zbmath.org/authors/?q=ai:szollosi.istvanSummary: Let \(k\) be a field and consider the path algebra \textit{kQ} of the quiver \(Q\). A pair of indecomposable \textit{kQ}-modules \((Y, X)\) is called an orthogonal exceptional pair if the modules are exceptional and \(\mathrm{Hom}(X,Y)=\mathrm{Hom}(Y, X)=\mathrm{Ext}^1(X,Y)=0\). Denote by \(\mathcal{F}(X, Y)\) the full subcategory of objects having a filtration with factors \(X\) and \(Y\). By a theorem of Schofield if \(Z\) is exceptional but not simple, then \(Z \in \mathcal{F}(X, Y)\) for some orthogonal exceptional pair \((Y, X)\), and \(Z\) is not a simple object in \(\mathcal{F}(X, Y)\). In fact, there are precisely \(s(Z) - 1\) such pairs, where \(s(Z)\) is the support of \(Z\) (i.e. the number of nonzero components in \(\dim_{\_} Z)\). Whereas it is easy to construct \(Z\) given \(X\) and \(Y\), there is no convenient procedure yet to determine the possible modules \(X\) (called Schofield submodules of \(Z)\) and then \(Y\) (called Schofield factors of \(Z)\), when \(Z\) is given. We present such an explicit procedure in the tame case, i.e. when \(Q\) is Euclidean.Galois and cleft monoidal cowreaths. Applicationshttps://www.zbmath.org/1483.160312022-05-16T20:40:13.078697Z"Bulacu, D."https://www.zbmath.org/authors/?q=ai:bulacu.daniel"Torrecillas, B."https://www.zbmath.org/authors/?q=ai:torrecillas.blasThe theory of Hopf-Galois extensions and weak cleft extensions has been developed in the last decades for various Hopf algebraic structures such as Hopf algebras, weak Hopf algebras, Hopf algebroids, Frobenius Hopf algebroids, Hopf quasigroups, weak Hopf quasigroups, etc. Also, the study of this theory was performed in a general monoidal setting and in the literature we can find extensions of the Hopf-Galois theory to entwining structures and weak entwining structures. In all the previous cases the main objective has been to characterize cleft extensions as Galois extensions with the normal basis property (this property is a generalization of the normal basis property of an extension of fields) or as some kind of cross product defined by an action and a cocycle. Taking all this into account, the main motivation of this book is to show how cowreaths in monoidal categories are the tool that permits to unify a considerable part of the above theories and results and on the other hand to introduce a theory of Hopf-Galois and cleft extensions for quasi-Hopf algebras since in this context such theory did not exist.
Let \({\mathcal C}\) be a monoidal category. A cowreath is a comonad in the Eilenberg-Moore category associated to \({\mathcal C}\). If we denote it by \(EM({\mathcal C})\), a cowreath is a couple \((A,X)\) where \(A\) is an algebra in \({\mathcal C}\) and \(X\) is a coalgebra in the monoidal category \({\mathcal T}^{\sharp}_{A}:=EM({\mathcal C})(A)\). As was pointed by the authors, any cowreath \((A,X)\) admits a category of entwined modules, denoted by \({\mathcal C}(\psi)_{A}^{X}\), and \((A,X)\) is called pre-Galois if \(A\in {\mathcal C}(\psi)_{A}^{X}\). Therefore, \((A,X)\) is pre-Galois if the coalgebra part in \({\mathcal T}^{\sharp}_{A}\) admits an almost group-like element. The existence of this element permits to define the subalgebra of coinvariants of \(A\), \(B=A^{co(X)}\), as well as the canonical morphisms \(\mathrm{can}\;:\;A\otimes_{B}A\rightarrow A\otimes X\) under suitable conditions, for example if the category \({\mathcal C}\) admits equalizers and any object is coflat and robust. The canonical morphisms can be defined for any object \({\mathfrak M}\) in the category \({\mathcal T}^{\sharp}_{A}\) and, as was proved by the authors, can is an isomorphism if, and only if, it is an isomorphism for any object in \({\mathcal T}^{\sharp}_{A}\). When the latter happens it will be said, by analogy with the Hopf algebra case, that \((A,X)\) is a Galois cowreath. Under these conditions it is possible to guarantee that the functor \(L:=-\otimes_{B}A:{\mathcal C}_{B}\rightarrow {\mathcal C}(\psi)_{A}^{X}\) admits as a right adjoint the functor of coinvariants \(R=(-)^{co(X)}\) (see Theorem 3.10). Then, as a consequence, the authors prove in Theorem 4.9 that the adjuntion \(L\dashv R\) is an equivalence of categories if, and only if, \((A,X)\) is a Galois cowreath and the functor \(-\otimes_{B}A:{\mathcal C}_{B}\rightarrow {\mathcal C}\) preserves and reflects equalizers, i.e., the object \(_{A}B\) is faithfully flat. Next, in Chapter 5, the authors find sufficient conditions to obtain that \(L\dashv R\) is an equivalence. More concretely, in Theorem 5.10 they prove that any Galois cowreath \((A,X)\), satisfying that the coalgebra part is coseparable in the category \({\mathcal T}^{\sharp}_{A}\), gives rise to the desired categorical equivalence. Also, they obtain that the coseparability is linked with the existence of a total integral \(\lambda:X\rightarrow A\) and, as a consequence, any object of \({\mathcal C}(\psi)_{A}^{X}\) is \(A\)-relative injective and, moreover, \(A\in\) \(_{B}{\mathcal C}(\psi)_{A}^{X}\) is \(_{B}{\mathcal C}_{A}\)-relative injective. Therefore, under these new conditions, the existence of a total integral or the conditions of relative injectivity, \(L\dashv R\) is an equivalence of categories.
As was pointed in the first paragraph of this review, it is a well-known fact (proved by \textit{Y. Doi} and \textit{M. Takeuchi} [Commun. Algebra 14, 801--817 (1986; Zbl 0589.16011)]) that in the classical Hopf-Galois theory Galois extensions with the normal basis property are nothing but cleft extensions. The main target of the sixth chapter of the book is to prove a similar result for cowreaths. To achieve it the authors introduce the notion of cleft cowreath requiring the existence of two morphisms \(\Phi\), \(\phi\) \(:X\rightarrow A\) satisfying the conditions contained in Definition 6.1. This is an interesting definition because if \(X\) is a coalgebra in \({\mathcal T}_{A}\) the conditions of the previous definition reduces to the clasical conditions for the so called cleaving morphism. On the other hand, it is applicable to the quasi-Hopf algebra setting since quasi-Hopf algebras give rise to examples of cowreaths in \({\mathcal T}^{\sharp}_{A}\) and not in \({\mathcal T}_{A}\). Finally, in Theorem 6.8 the authors prove the following: Let \((A,X)\) be a pre-Galois cowreath satisfying the condition 6.3 (see page 57) and such that \(X\) is a left flat object in the category \({\mathcal C}\). Then, \((A,X)\) is cleft iff it is Galois and satisfies the normal basis property.
In Chapter 7, the authors present a one to one correspondence between cleft cowreaths and cleft wreaths in a monoidal category. In the Hopf algebra context cleft extensions can be identified to a sort of crossed products defined by an action and a convolution invertible cocycle. In this chapter they show that for cowreaths in place of the classical crossed products, it is more convenient to consider wreath algebras in the sense of \textit{S. Lack} and \textit{R. Street} [J. Pure Appl. Algebra 175, No. 1--3, 243--265 (2002; Zbl 1019.18002)]), equipped with two additional morphisms satisfying the conditions contained in (7.6). The authors call these wreaths cleft and in Theorem 7.11 and Corollary 7.12 they prove that these kind of wreaths describe completely the cleft cowreaths, up to a unitally cowreath isomorphism.
The final part of the book is dedicated to the applications of the results contained in the previous chapters to Galois-cleft theory for quasi-Hopf algebras. For example in Chapter 8 for a quasibialgebra \(H\), a right \(H\)-comodule algebra \({\mathfrak A}\) and a right \(H\)-module coalgebra \(C\) the authors explain the structure of the cowreath \(({\mathfrak A}, C)\) and when it is Galois. Also, they find the conditions under which we can obtain the categorical equivalence presented in Theorem 4.9. Moreover, in Chapter 12, the authors prove that right comodule algebras over \(H\) induces examples of Galois cowreaths satisfying the normal basis property and therefore they can guarantee that in this context there exist examples of cleft cowreaths.
In Chapter 9, the authors investigate when a pre-Galois cowreath defined by an entwining structure \((A,C)_{\psi}\) is cleft. They prove in Theorem 9.6 that this happens iff the algebra \(A\) is isomorphic as algebra and as right \(C\)-comodule to a crossed product by a coalgebra \({\mathfrak B}\sharp C\) for which the canonical embedding morphism from \(C\) to \({\mathfrak B}\sharp C\) is convolution invertible. Chapter 10 and 11 are dedicated to the study of cowreaths associated to \(\nu\)-Doi-Hopf modules and to the study of those who appear when we work with generalized crossed products.
Finally, as the authors state, the results presented in this book can recover somehow of the Hopf-Galois and cleft theory developed by \textit{G. Böhm} and \textit{T. Brzeziński} [Appl. Categ. Struct. 14, No. 5--6, 431--469 (2006; Zbl 1133.16024)] for Hopf algebroids, the Schneider type theorem (see [\textit{A. Ardizzoni} et al., J. Algebra 321, No. 6, 1786--1796 (2009; Zbl 1165.16305)]), and consequently the Hopf-Galois theory for weak Hopf algebras. Also, it can be applied to braided Hopf algebras, Hom-Hopf algebras [\textit{S. Caenepeel} and \textit{I. Goyvaerts}, Commun. Algebra 39, No. 6, 2216--2240 (2011; Zbl 1255.16032)] and to Hopf group (co)algebras [\textit{S. Caenepeel} and \textit{M. De Lombaerde}, Commun. Algebra 34, No. 7, 2631--2657 (2006; Zbl 1103.16024)].
Reviewer: Ramón González Rodríguez (Vigo)Generalised quantum determinantal rings are maximal ordershttps://www.zbmath.org/1483.160352022-05-16T20:40:13.078697Z"Lenagan, T. H."https://www.zbmath.org/authors/?q=ai:lenagan.thomas-h"Rigal, L."https://www.zbmath.org/authors/?q=ai:rigal.laurentSummary: Generalised quantum determinantal rings are the analogue in quantum matrices of Schubert varieties. Maximal orders are the noncommutative version of integrally closed rings. In this paper, we show that generalised quantum determinantal rings are maximal orders. The cornerstone of the proof is a description of generalised quantum determinantal rings, up to a localisation, as skew polynomial extensions.Skew braces as remnants of co-quasitriangular Hopf algebras in suplathttps://www.zbmath.org/1483.160362022-05-16T20:40:13.078697Z"Ghobadi, Aryan"https://www.zbmath.org/authors/?q=ai:ghobadi.aryanSkew braces are sets with two compatible group structures and they have been introduced by \textit{L. Guarnieri} and \textit{L. Vendramin} [Math. Comput. 86, No. 307, 2519--2534 (2017; Zbl 1371.16037)] to study set-theoretical solutions of the Yang-Baxter equation.
Considering linear solutions on vector spaces, there exists a relation between these kind of solutions and (co-)quasitriangular Hopf algebras. Into the specific, (co-)quasitriangular Hopf algebras and bialgebras provide solutions via their (co-)representation theory. Conversely, the Fadeev-Reshetikhin-Takhtajan construction produces such a bialgebra from any linear solution [\textit{N. Yu. Reshetikhin} et al., Leningr. Math. J. 1, No. 1, 193--225 (1990; Zbl 0715.17015); translation from Algebra Anal. 1, No. 1, 178--206 (1989)]. A question is whether skew braces can be viewed as Hopf algebras in a suitable category related to sets.
In the paper under review, the author finds the correct category to consider, that is the category SupLat of complete lattices and join preserving morphisms. Specifically, it is shown that any Hopf algebra \(H\) in SupLat, has a corresponding group \(R(H)\), and a co-quasitriangular structure on \(H\) induces a solution on \(R(H)\), which is compatible with its group structure. Conversely, any group with a compatible solution can be realised in this way.
Reviewer: Marzia Mazzotta (Lecce)Principal Galois orders and Gelfand-Zeitlin moduleshttps://www.zbmath.org/1483.160442022-05-16T20:40:13.078697Z"Hartwig, Jonas T."https://www.zbmath.org/authors/?q=ai:hartwig.jonas-tSummary: We show that the ring of invariants in a skew monoid ring contains a so called standard Galois order. Any Galois ring contained in the standard Galois order is automatically itself a Galois order and we call such rings principal Galois orders. We give two applications. First, we obtain a simple sufficient criterion for a Galois ring to be a Galois order and hence for its Gelfand-Zeitlin subalgebra to be maximal commutative. Second, generalizing a recent result by Early-Mazorchuk-Vishnyakova, we construct canonical simple Gelfand-Zeitlin modules over any principal Galois order.
As an example, we introduce the notion of a rational Galois order, attached an arbitrary finite reflection group and a set of rational difference operators, and show that they are principal Galois orders. Building on results by Futorny-Molev-Ovsienko, we show that parabolic subalgebras of finite W-algebras are rational Galois orders. Similarly we show that Mazorchuk's orthogonal Gelfand-Zeitlin algebras of type \(A\), and their parabolic subalgebras, are rational Galois orders. Consequently we produce canonical simple Gelfand-Zeitlin modules for these algebras and prove that their Gelfand-Zeitlin subalgebras are maximal commutative.
Lastly, we show that quantum OGZ algebras, previously defined by the author, and their parabolic subalgebras, are principal Galois orders. This in particular proves the long-standing Mazorchuk-Turowska conjecture that, if \(q\) is not a root of unity, the Gelfand-Zeitlin subalgebra of \(U_q(\mathfrak{gl}_n)\) is maximal commutative and that its Gelfand-Zeitlin fibers are non-empty and (by Futorny-Ovsienko theory) finite.From Schouten to Mackenzie: notes on bracketshttps://www.zbmath.org/1483.170012022-05-16T20:40:13.078697Z"Kosmann-Schwarzbach, Yvette"https://www.zbmath.org/authors/?q=ai:kosmann-schwarzbach.yvetteSummary: In this paper, dedicated to the memory of Kirill Mackenzie, I relate the origins and early development of the theory of graded Lie brackets, first in the publications on differential geometry of Schouten, Nijenhuis, and Frölicher-Nijenhuis, then in the work of Gerstenhaber and Nijenhuis-Richardson in cohomology theory.The second relative homology of Leibniz algebrashttps://www.zbmath.org/1483.170062022-05-16T20:40:13.078697Z"Hosseini, Seyedeh Narges"https://www.zbmath.org/authors/?q=ai:hosseini.seyedeh-narges"Edalatzadeh, Behrouz"https://www.zbmath.org/authors/?q=ai:edalatzadeh.behrouz"Salemkar, Ali Reza"https://www.zbmath.org/authors/?q=ai:salemkar.ali-rezaStudied in the paper under review are the relative Leibniz homology groups in dimension two of a pair \(( \mathfrak{g}, \, \mathfrak{n})\), where \(\mathfrak{g}\) is a right Leibniz algebra over a field \(\mathbf{F}\) and \(\mathfrak{n}\) is a two-sided ideal of \(\mathfrak{g}\). Recall that a right Leibniz algebra in this case is a vector space over \(\mathbf{F}\) equipped with a bilinear map \(\circ :\mathfrak{g} \times\mathfrak{g} \to \mathfrak{g}\) that is a derivation from the right, meaning that
\[ ( x \circ y) \circ z = x \circ ( y \circ z) + ( x \circ z) \circ y, \quad x, y, z \in\mathfrak{g}. \]
The paper adopts the older notation with \(x \circ y\) denoted by \([x, y]\), although the bracket is not necessarily skew-symmetric. The Leibniz homology of \(\mathfrak{g}\) with coefficients in \(\mathbf{F}\) is denoted \(HL_* (\mathfrak{g} )\), and is the homology of the chain complex
\[ CL_k (\mathfrak{g} ) := ( \mathfrak{g}^{\otimes k}, \, d), \quad k \geq 0, \]
where the boundary map \(d\) is defined in [\textit{J.-L. Loday} and \textit{T. Pirashvili}, Math. Ann. 296, No. 1, 139--158 (1993; Zbl 0821.17022)]. Let \(\mathfrak{n}\) be a two-sided ideal of \(\mathfrak{g}\) and let \(\pi :\mathfrak{g} \to\mathfrak{g}/\mathfrak{n}\) be the quotient map of Leibniz algebras. There is chain map
\[ \pi : (\mathfrak{g}^{\otimes k}, \, d) \to ( (\mathfrak{g}/ \mathfrak{n})^{\otimes k}, \, d) \]
with mapping cone
\[ M_k( \pi ) = (\mathfrak{g}^{\otimes (k-1)} \oplus (\mathfrak{g}/\mathfrak{n})^{\otimes k}, \, \delta ). \]
Then \(( M_* ( \pi ), \, \delta )\) becomes a chain complex with the authors using the sign convention
\[ \delta (a, \, b) = ( -d(a), \, \pi(a) + d(b)), \ \ \ a \in\mathfrak{g}^{\otimes (k-1)}, \ b \in (\mathfrak{g}/\mathfrak{n})^{\otimes k}. \]
By standard results in homological algebra, there is a long exact sequence in homology:
\[ \longrightarrow HL_k (\mathfrak{g}/ \mathfrak{n}) \longrightarrow H_k (M_*( \pi )) \longrightarrow HL_{k-1} (\mathfrak{g}) \overset{\pi_*}{\longrightarrow} HL_{k-1}(\mathfrak{g}/\mathfrak{n}) \longrightarrow \]
By definition the relative homology groups are \(HL_k (\mathfrak{g}, \, \mathfrak{n} ) := H_{k+1} ( M_* (\pi) )\).
A basic result about \(HL_2 (\mathfrak{g}, \, \mathfrak{n} )\) is given in [\textit{G. Donadze} et al., Rev. Mat. Complut. 31, No. 1, 217--236 (2018; Zbl 1403.18015)], where the non-abelian exterior product \(\mathfrak{m} \curlywedge \mathfrak{n}\) of two ideals \(\mathfrak{m}\), \(\mathfrak{n}\) of \(\mathfrak{g}\) is given, and the kernel of the commutator map \([ \ , \ ] : \mathfrak{g} \curlywedge \mathfrak{n} \to\mathfrak{n}\) is proven to be isomorphic to \(HL_2 (\mathfrak{g}, \, \mathfrak{n} )\).
Let \(\mathfrak{g}^2 = [\mathfrak{g}, \, \mathfrak{g}]\) be the derived subalgebra of \(\mathfrak{g}\) and let \(Z( \mathfrak{g} )\) denote the center of \(\mathfrak{g}\). Proven in the paper under review is that if \(\mathfrak{n} \subseteq Z(\mathfrak{g} ) \cap \mathfrak{g}^2\), then
\[ HL_2 (\mathfrak{g}, \, \mathfrak{n} ) \simeq ( \mathfrak{g}/\mathfrak{g}^2 \otimes \mathfrak{n}) \oplus (\mathfrak{n} \otimes \mathfrak{g}/\mathfrak{g}^2 ) . \]
Suppose that \(\mathfrak{r} \to\mathfrak{f} \overset{\pi}{\to} \mathfrak{g}\) is a free presentation of \(\mathfrak{g}\), where \(\mathfrak{f}\) is the free Leibniz algebra on the set \(\mathfrak{g}\). Suppose further that \(\eta\) is an ideal of \(\mathfrak{f}\) generated by \(\mathfrak{n}\). Proven is the ``Hopf formula''
\[ HL_2 (\mathfrak{g}, \, \mathfrak{n} ) \simeq \frac{ \mathfrak{r} \cap [ \eta , \, \mathfrak{f} ]} { [\mathfrak{r}, \, \eta] + [\mathfrak{f}, \,\mathfrak{r} \cap \eta]}. \]
Furthermore, the notions of a stem extension and a stem cover from [\textit{J. M. Casas} and \textit{M. Ladra}, Georgian Math. J. 9, No. 4, 659--669 (2002; Zbl 1051.17001)] are generalized to the relative case, and it is proven that the pair \((\mathfrak{g}, \,\mathfrak{n})\) admits a relative stem cover.
Let \(\mathfrak{g}\) be a Leibniz algeba with \(\dim_{\mathbf{F}} (\mathfrak{g}/\mathfrak{n}) = m\), \(\dim_{\mathbf{F}} ( \mathfrak{n} / Z(\mathfrak{g}, \,\mathfrak{n}) ) = n\). Proven is that (i) \(\dim_{\mathbf{F}} ( [\mathfrak{g}, \,\mathfrak{n}] ) \leq n(n + 2m)\), and (ii) if \(\mathfrak{g}\) is nilpotent and \(\dim_{\mathbf{F}} ( [\mathfrak{g}, \,\mathfrak{n} ]) = n(n + 2m)\) or \(n(n + 2m) - 1\), then \( \mathfrak{n}/ Z(\mathfrak{g}, \, \mathfrak{n})\) is a central ideal of \(\mathfrak{g} / Z( \mathfrak{g}, \,\mathfrak{n})\). Let \(d(\mathfrak{g})\) denote the minimal number of generators of the Leibniz algebra \(\mathfrak{g}\). If \(\mathfrak{g}\) is finite dimensional and nilpotent, then \(d(\mathfrak{g} ) = \dim_{\mathbf{F}} (\mathfrak{g} / \mathfrak{g}^2 )\). For such a Leibniz algebra \(\mathfrak{g}\) with an ideal \(\mathfrak{n}\) of codimension 1, proven is that
\[ \dim_{\mathbf{F}} ( HL_2 ( \mathfrak{g}, \, \mathfrak{n} ) ) \leq\dim_{\mathbf{F}} (HL_2 (\mathfrak{n} )) + 2 d(n). \] Equality of the bound is attained if \(Z( \mathfrak{g} ) \nsubseteq \mathfrak{n}\).
Reviewer: Jerry M. Lodder (Las Cruces)On the elliptic Kashiwara-Vergne Lie algebrahttps://www.zbmath.org/1483.170072022-05-16T20:40:13.078697Z"Raphael, Élise"https://www.zbmath.org/authors/?q=ai:raphael.elise"Schneps, Leila"https://www.zbmath.org/authors/?q=ai:schneps.leilaThe present paper is concerned with a comparison of two definitions of an elliptic Kashiwara-Vergne Lie algebra. Recall that the original Kashiwara-Vergne Lie algebra was introduced by Alekseev-Torossian in the context of the Kashiwara-Vergne conjecture [\textit{M. Kashiwara} and \textit{M. Vergne}, Invent. Math. 47, 249--272 (1978; Zbl 0404.22012)]. Subsequently, two different constructions in genus one were given by \textit{A. Alekseev} et al. [C. R., Math., Acad. Sci. Paris 355, No. 2, 123--127 (2017; Zbl 1420.57052)] and by \textit{E. Raphael} ans \textit{L. Schneps} [``On linearised and elliptic versions of the Kashiwara-Vergne Lie algebra'', Preprint, \url{arXiv:1706.08299}].
In this paper, the authors prove that the two Lie algebras thus obtained are canonically isomorphic to one another.
For the entire collection see [Zbl 1459.11005].
Reviewer: Nils Matthes (Oxford)A construction by deformation of unitary irreducible representations of \(\mathrm{SU}(1, n)\) and \(S\mathrm{SU}(n + 1)\)https://www.zbmath.org/1483.170082022-05-16T20:40:13.078697Z"Cahen, Benjamin"https://www.zbmath.org/authors/?q=ai:cahen.benjaminIn the paper under review, the author constructs holomorphic discrete series representations of \(\mathrm{SU}(1, n)\) and some unitary irreducible representations of \(\mathrm{SU}(n)\) by deforming a minimal realization of \(\mathfrak{g}=\mathfrak{sl}(n+ 1,\mathbb C)\). The minimal realization refers to a representation \(\rho_0\) of \(\mathfrak{g}\) in the space of complex polynomials with \(n\) variables, which is given by the classical Weyl correspondence. The term ``minimal'' indicates that the construction of \(\rho_0\) is closely related to the minimal nilpotent coadjoint orbit of \(\mathfrak{g}\). The deformation of \(\rho_0\) is given over the space \(M\) of the complex polynomials with \(2n\) variables and is controlled by the first Chevalley-Eilenberg cohomology space \(H^1(\mathfrak{g},M)\).
Reviewer: Husileng Xiao (Harbin)The classification of blocks in BGG category \(\mathcal{O}\)https://www.zbmath.org/1483.170092022-05-16T20:40:13.078697Z"Coulembier, Kevin"https://www.zbmath.org/authors/?q=ai:coulembier.kevinA block in the BGG category \(\mathcal{O}\) of a complex semisimple Lie algebra is determined up to equivalence by its integral Weyl group \(W\) and a parabolic subgroup \(W^\prime \leq W\) by results of [\textit{W. Soergel}, J. Am. Math. Soc. 3, No. 2, 421--445 (1990; Zbl 0747.17008)]. Therefore, a block can be denoted unambigously by \(\mathcal{O}(W,W^\prime)\).
The main result of this article is that, for any two finite Weyl groups \(W\) and \(U\) with parabolic subgroups \(W^\prime \leq W\) and \(U^\prime \leq U\), the blocks \(\mathcal{O}(W,W^\prime)\) and \(\mathcal{O}(U,U^\prime)\) are equivalent if and only if the Bruhat orders on \(W / W^\prime\) and \(U / U^\prime\) are isomorphic. As part of the proof, it is observed that any finite-dimensional algebra with simple preserving duality admits at most one quasi-hereditary structure. The author further determines all pairs \((W,W^\prime)\) and \((U,U^\prime)\) such that there exists an isomorphisms between the Bruhat orders on \(W / W^\prime\) and \(U / U^\prime\) and thus obtains a complete classification of the blocks in category \(\mathcal{O}\).
Reviewer: Jonathan Gruber (Lausanne)Free 2-step nilpotent Lie algebras and indecomposable representationshttps://www.zbmath.org/1483.170102022-05-16T20:40:13.078697Z"Cagliero, Leandro"https://www.zbmath.org/authors/?q=ai:cagliero.leandro"Gutiérrez Frez, Luis "https://www.zbmath.org/authors/?q=ai:gutierrez-frez.luis"Szechtman, Fernando"https://www.zbmath.org/authors/?q=ai:szechtman.fernandoSummary: Given an algebraically closed field \(F\) of characteristic 0 and an \(F\)-vector space \(V\), let \(L(V)=V\oplus\Lambda^2(V)\) denote the free 2-step nilpotent Lie algebra associated to \(V\). In this paper, we classify all uniserial representations of the solvable Lie algebra \(\mathfrak{g}=\langle x\rangle\ltimes L(V)\), where \(x\) acts on \(V\) via an arbitrary invertible Jordan block.Advances relating to \(R\)-matrices and their applications [after Maulik-Okounkov, Kang-Kashiwara-Kim-Oh,\ldots]https://www.zbmath.org/1483.170112022-05-16T20:40:13.078697Z"Hernandez, David"https://www.zbmath.org/authors/?q=ai:hernandez.david-romero|hernandez.davidSummary: \(R\)-matrices are the solutions of the Yang-Baxter equation. At the origin of the quantum group theory, they may be interpreted as intertwining operators. Recent advances have been made independently in different directions. \textit{D. Maulik} and \textit{A. Okounkov} [Quantum groups and quantum cohomology. Paris: Société Mathématique de France (SMF) (2019; Zbl 1422.14002)] have given a geometric approach to \(R\)-matrices with new tools in symplectic geometry, the stable envelopes. Kang-Kashiwara-Kim-Oh [\textit{S.-J. Kang} et al., J. Am. Math. Soc. 31, No. 2, 349--426 (2018; Zbl 1460.13039)] proved a conjecture on the categorification of cluster algebras by using \(R\)-matrices in a crucial way. Eventually, a better understanding of the action of transfer-matrices obtained from R-matrices led to the proof of several conjectures about the corresponding quantum integrable systems.
For the entire collection see [Zbl 1416.00029].Positive representations of split real simply-laced quantum groupshttps://www.zbmath.org/1483.170122022-05-16T20:40:13.078697Z"Ip, Ivan C. H."https://www.zbmath.org/authors/?q=ai:ip.ivan-chi-hoSummary: We construct the positive principal series representations for \(\mathcal U_q(\mathfrak g_\mathbb R)\) where \(\mathfrak g\) is of simply-laced type, parametrized by \(\mathbb R_{\geq 0}^r\) where \(r\) is the rank of \(\mathfrak g\). We describe explicitly the actions of the generators in the positive representations as positive essentially self-adjoint operators on a Hilbert space, and prove the transcendental relations between the generators of the modular double. We define the modified quantum group \(\mathbf U_{\mathfrak q \tilde{\mathfrak q}}(\mathfrak g_\mathbb R)\) of the modular double and show that the representations of both parts of the modular double commute weakly with each other, there is an embedding into a quantum torus algebra, and the commutant contains its Langlands dual.Positive representations: recent developmentshttps://www.zbmath.org/1483.170132022-05-16T20:40:13.078697Z"Ip, Ivan C. H."https://www.zbmath.org/authors/?q=ai:ip.ivan-chi-hoSummary: We give a historical motivation of the theory of positive representations of split real quantum groups, first initiated in at new research program by I. Frenkel and the author, and we survey its recent developments utilizing the cluster realization of quantum groups discovered recently. This allows us to demonstrate, for the moment in type \(A_n\), a continuous braided tensor category structure of positive representations, as well as proving an analogue of the Peter-Weyl theorem for split real quantum groups.
For the entire collection see [Zbl 1454.00057].Derived Grothendieck-Teichmüller group and graph complexes [after T. Willwacher]https://www.zbmath.org/1483.170142022-05-16T20:40:13.078697Z"Kontsevich, Maxim"https://www.zbmath.org/authors/?q=ai:kontsevich.maximSummary: Graph complex is spanned by equivalence classes of finite connected graphs with the dual differential given by the sum of all contractions of edges, with appropriate signs. This complex forms a differential graded Lie algebra, and acts as a universal derived infinitesimal symmetry of all graded Lie algebras of polyvector fields on finite-dimensional manifolds. Grothendieck-Teichmüller group, as defined by V. Drinfeld, is the group of symmetries of the tower of rationally completed braid groups. Recent breakthrough by T. Willwacher identifies the graph complex with the derived version of GT group. This result settles essentially all open questions in the subject of deformation quantization and little disk operads.
For the entire collection see [Zbl 1416.00029].Cohomology of the vector fields Lie algebras on \(\mathbb{R}\) acting on trilinear differential operators, vanishing on \(\mathfrak{sl}(2)\)https://www.zbmath.org/1483.170152022-05-16T20:40:13.078697Z"Boujelben, Maha"https://www.zbmath.org/authors/?q=ai:boujelben.maha"Amina, Jabeur"https://www.zbmath.org/authors/?q=ai:amina.jabeur"Lerbet, Jean"https://www.zbmath.org/authors/?q=ai:lerbet.jeanHere, the name of polynomial vector fields Lie algebra on \(\mathbb{R}\) with \((x)\) as a coordinate system, is \(\mathrm{Vect}\left(\mathbb{R}\right)\) and \(\mathcal{F}_\mu=\{f\,dx^\mu,f\in\mathbb{R}[x]\}\) where \(\mu\in\mathbb{R}\). Naturally, \(\mathrm{Vect}\left(\mathbb{R}\right)\) acts on \(\mathcal{F}_\mu\) by
\[
X_h.\left(fdx^{\mu}\right)=\left(h\frac{df}{dx}+\mu\frac{dh}{dx}f\right)dx^\mu
\]
with \(X_h=h\frac{d}{dx}\in \mathrm{Vect}\left(\mathbb{R}\right)\). In the following, \(n\in\mathbb{N}^\ast\).
The authors of the article under review state the space of \(n\)-ary differential operators in which \(\mathrm{Vect}\left(\mathbb{R}\right)\) acts again: \(\lambda=(\lambda_1,\dots,\lambda_n)\in\mathbb{R}^n\), \(\mu\in\mathbb{R}\), this space is \(D_{\bar{\lambda},\mu}\) of these operators on \(\mathcal{F}_{\lambda_1}\otimes\dots\otimes\mathcal{F}_{\lambda_n}\) towards \(\mathcal{F}_\mu\). The fact on the sub-Lie algebra \(\mathfrak{sl}(2)=\left\langle X_1,X_x,X_{x^2}\right\rangle\) of \(\mathrm{Vect}\left(\mathbb{R}\right)\) is to be noted in order that \(\mathcal{F}_\mu\) and \(D_{\bar{\lambda},\mu}\) are both \(\mathfrak{sl}(2)\)-modules. Then, in term of relative cohomology, recalled by the authors in the beginning of the paper, \(H^1_{\text{diff}}\left(\mathrm{Vect}\left(\mathbb{R}\right),\mathfrak{sl}(2), D_{\bar{\lambda},\mu}\right)\) can be computed.
For \(n=1\) or \(2\), the results have been found respectively by \textit{S. Bouarroudj} and \textit{V. Yu. Ovsienko} [Int. Math. Res. Not. 1998, No. 1, 25--39 (1998; Zbl 0919.57026)] and \textit{S. Bouarroudj} [Int. J. Geom. Methods Mod. Phys. 2, No. 1, 23--40 (2005; Zbl 1062.17014)]. By hard calculations around cocycles and coboundaries relative to the above cohomology, they find the corresponding result when \(n=3\). The case where \(n\ge 4\) is still an open question.
Reviewer: Princy Randriambololondrantomalala (Antananarivo)Surjectivity of certain adjoint operators and applicationshttps://www.zbmath.org/1483.170162022-05-16T20:40:13.078697Z"Cherifi Hadjiat, Amina"https://www.zbmath.org/authors/?q=ai:cherifi-hadjiat.amina"Lansari, Azzeddine"https://www.zbmath.org/authors/?q=ai:lansari.azzeddineLet \(E\) be the Lie-Fréchet space that consists of all vector fields \(X\) of class \(C^{\infty}\) on \(\mathbb R^{n}\) with a graduation of seminorms \(\| X\| _{r}=\sup _{x\in \mathbb R^n}\max _{k+|\alpha | \leq r}\| D^{\alpha} X(x)\|(1+\| x\|^{2})^{k/2}\) (such Lie algebras were considered by [\textit{R. S. Hamilton}, Bull. Am. Math. Soc., New Ser. 7, 65--222 (1982; Zbl 0499.58003)]. The authors study a subalgebra \(U\) of the Lie algebra \(E\) consisting of all vector fields of the form \(Y_0=X_{0}^{+}+X_{0}^{-}+Z_0\) such that \(X_0(x, y)=A(x, y)=(A^{-}(x), A^{+}(y)),\) with \(A^{-}\) (respectively, \(A^{+})\) a symmetric matrix with eigenvalues \(\lambda<0\) (respectively, \(\lambda >0\)) and \(Z_0\) are germs infinitely flat at the origin. The main result of the paper: the subalgebra \(U\) of the Lie algebra \(E\) admits a hyperbolic structure for the diffeomorphism \(\psi _{t^{\star}}=(\exp{tY_0})_{\star}.\) As an application of this result it was proved that for any admissible subalgebra \(U\) of finite codimension of the Lie algebra \(E\) that has a hyperbolic structure for the flow and satisfies some conditions it holds \(U=E.\)
Reviewer: Anatoliy Petravchuk (Kyiv)Classification of integrable representations for toroidal extended affine Lie algebrashttps://www.zbmath.org/1483.170172022-05-16T20:40:13.078697Z"Chen, Fulin"https://www.zbmath.org/authors/?q=ai:chen.fulin"Li, Zhiqiang"https://www.zbmath.org/authors/?q=ai:li.zhiqiang.1"Tan, Shaobin"https://www.zbmath.org/authors/?q=ai:tan.shao-binAn extended affine Lie algebra \(E\) is by definition a complex Lie algebra, together with a non-zero finite dimensional ad-diagonalizable subalgebra \(H\) and a non-degenerate invariant symmetric bilinear form \((\cdot|\cdot)\), that satisfies a list of natural axioms. A crucial consequence of these axioms is that the form \((\cdot|\cdot)\) on \(E\) induces a semi-positive bilinear form on the \({\mathbb R}\)-span of the root system of \(E\) relative to \(H\). Then the root system of \(E\) divides into a disjoint union of the sets of isotropic and non-isotropic roots. The subalgebra \(E_c\) of \(E\) generated by non-isotropic root vectors is called the core of \(E\), which is in fact an ideal of \(E\). An \(E\)-module is said to be integrable if it admits a weight space decomposition relative to \(H\) and all non-isotropic root vectors in \(E\) act locally nilpotent on it. From the axioms of \(E\), it follows that the rank of the group generated by the isotropic roots is finite; this rank is called the nullity of \(E\).
Toroidal extended affine Lie algebras are a class of important extended affine Lie algebras whose construction parallels with the untwisted affine Lie algebras. Let \({\dot{\mathfrak g}}\) be a finite dimensional simple Lie algebras, let \(\mathcal R={\mathbb C}[t_0^{\pm 1},\ldots,t_{N-1}^{\pm 1}]\) be the ring of Laurent polynomials in \(N\) variables and let \(\mathcal S\) be the subspace of divergence zero derivations on \(\mathcal R\) (which is also called the set of skew-derivations). The nullity \(N\) toroidal extended affine Lie algebra \(\widetilde{\mathfrak g}=\widetilde{\mathfrak g}_c \oplus {\mathcal{S}}\) is defined by adding the space \(\mathcal S\) (possibly twisted with a \(2\)-cocycle) to the universal central extension \(\widetilde{\mathfrak g}_c=(\mathcal R\otimes {\dot{\mathfrak g}})\oplus \mathcal K\) of the multi-loop Lie algebra \(\mathcal R\otimes {\dot{\mathfrak g}}\).
Paper gives a complete classification of irreducible integrable representations for the nullity \(2\) toroidal extended affine Lie algebras \(\widetilde{\mathfrak g}\) with finite dimensional weight spaces and non-trivial \(\widetilde{\mathfrak g}_c\)-action.
Reviewer: Saeid Azam (Isfahan)Integrable modules for loop affine-Virasoro algebrashttps://www.zbmath.org/1483.170182022-05-16T20:40:13.078697Z"Eswara Rao, S."https://www.zbmath.org/authors/?q=ai:eswara-rao.senapathi"Sharma, Sachin S."https://www.zbmath.org/authors/?q=ai:sharma.sachin-s"Mukherjee, Sudipta"https://www.zbmath.org/authors/?q=ai:mukherjee.sudiptaIn this paper, a class of infinite-dimensional Lie algebras called the loop affine-Virasoro algebra \(\mathfrak{L}(A)\) appearing in [\textit{S. Eswara Rao}, J. Algebra Appl. 20, No. 4, Article ID 2150055, 13 p. (2021; Zbl 1469.17023)] but with independent centers is considered. The classification of the irreducible integrable modules with finite-dimensional weight spaces is given.
Reviewer: Haibo Chen (Shanghai)Chiral versus classical operadhttps://www.zbmath.org/1483.170192022-05-16T20:40:13.078697Z"Bakalov, Bojko"https://www.zbmath.org/authors/?q=ai:bakalov.bojko"De Sole, Alberto"https://www.zbmath.org/authors/?q=ai:de-sole.alberto"Heluani, Reimundo"https://www.zbmath.org/authors/?q=ai:heluani.reimundo"Kac, Victor G."https://www.zbmath.org/authors/?q=ai:kac.victor-gSummary: We establish an explicit isomorphism between the associated graded of the filtered chiral operad and the classical operad, which is important for computing the cohomology of vertex algebras.The Virasoro vertex algebra and factorization algebras on Riemann surfaceshttps://www.zbmath.org/1483.170202022-05-16T20:40:13.078697Z"Williams, Brian"https://www.zbmath.org/authors/?q=ai:williams.brian-r|williams.brian-j|williams.brian-g|williams.brian-wesley|williams.brian-charlesSummary: This paper focuses on the connection of holomorphic two-dimensional factorization algebras and vertex algebras which has been made precise in the forthcoming book of Costello-Gwilliam. We provide a construction of the Virasoro vertex algebra starting from a local Lie algebra on the complex plane. Moreover, we discuss an extension of this factorization algebra to a factorization algebra on the category of Riemann surfaces. The factorization homology of this factorization algebra is computed as the correlation functions. We provide an example of how the Virasoro factorization algebra implements conformal symmetry of the beta-gamma system using the method of effective BV quantization.Towards automorphic to differential correspondence for vertex algebrashttps://www.zbmath.org/1483.170212022-05-16T20:40:13.078697Z"Zuevsky, Alexander"https://www.zbmath.org/authors/?q=ai:zuevsky.alexanderSummary: In these notes we propose a version of geometric correspondence between parameter spaces for projectively flat connections in vector bundles and automorphic representations of modular groups over Riemann surfaces. Principal role of vertex algebras is discussed. We then formulate a conjecture concerning an extended correspondence between categories of twisted \(D\)-modules and Hecke eigensheaves both defined on the moduli stack of modular group bundles, and obtained as sheaves of conformal blocks for vertex operator algebras with formal parameters on complex algebraic curves.
For the entire collection see [Zbl 1470.00021].Quasi Jordan algebras with involutionhttps://www.zbmath.org/1483.170222022-05-16T20:40:13.078697Z"Alhefthi, Reem K."https://www.zbmath.org/authors/?q=ai:alhefthi.reem-k"Siddiqui, Akhlaq A."https://www.zbmath.org/authors/?q=ai:siddiqui.akhlaq-ahmad"Jamjoom, Fatmah B."https://www.zbmath.org/authors/?q=ai:jamjoom.fatmah-backerA self-map \(\ast : x \mapsto x^\ast\) on a complex quasi Jordan algebra \(\mathscr{A}\) is called an involution on \(\mathscr{A}\) if it satisfies \((i)~(x^\ast)^\ast=x\) \((ii)~(x+y)^\ast=x^\ast+y^\ast\) \((iii)~(\alpha x)^\ast=\overline{\alpha}x^\ast\) \((iv)~(x\triangleleft y)^\ast=x^\ast \triangleleft y^\ast\) for all \(x,y\in \mathscr{A}\) and \(\alpha\in \mathbb{C}\). In such a case, \(\mathscr{A}\) is called a quasi Jordan \(\ast-\)algebra. If, in addition, the algebra \(\mathscr{A}\) is normed satisfying the condition \(||x^\ast||=||x||\) for all \(x\in \mathscr{A}\), then \(\mathscr{A}\) is called an involutive quasi Jordan normed algebra. An element \(x\) in a quasi Jordan \(\ast\)algebra \(\mathscr{A}\) is said to be self-adjoint if \(x^\ast=x\). An element \(u\) in a unital quasi Jordan algebra with involution \(\ast\) is said to be \(e-\)unitary if it is invertible with respect to \(e\) with inverse \(u^\ast\). An element \(u\) is said to be unitary if it is \(e-\)unitary for all units \(e\in \mathscr{A}\).
In this paper, the authors have initiated the study of involutions in the setting of complex quasi Jordan algebras \(\mathscr{A}\) and discussed the notions of self-adjoint and unitary elements. They obtained some interesting results and also proved that a quasi Jordan \(\ast-\)algebra \(\mathscr{A}\) may have nonself-adjoint zero elements. Additionally, they explored various properties of \(e-\)unitary element and obtain some appropriate analogue of the Russo-Dye theorem for unital involutive split quasi Jordan Banach algebras.
Reviewer: Mohd Arif Raza (Rabigh)A note on skew characters of symmetric groupshttps://www.zbmath.org/1483.200282022-05-16T20:40:13.078697Z"Taylor, Jay"https://www.zbmath.org/authors/?q=ai:taylor.jaySummary: In previous work [Isr. J. Math. 195, Part A, 31--35 (2013; Zbl 1279.20017)] \textit{A. Regev} used part of the representation theory of Lie superalgebras to compute the values of a character of the symmetric group whose decomposition into irreducible constituents is described by semistandard \((k,\ell)\)-tableaux. In this short note we give a new proof of Regev's result using skew characters.On conjugacy of abstract root bases of root systems of Coxeter groupshttps://www.zbmath.org/1483.200732022-05-16T20:40:13.078697Z"Dyer, Matthew"https://www.zbmath.org/authors/?q=ai:dyer.matthew-jSummary: We introduce and study a combinatorially defined notion of the root basis of a (real) root system of a possibly infinite Coxeter group. Known results on conjugacy up to sign of root bases of certain irreducible finite rank real root systems are extended to abstract root bases, to a larger class of real root systems and, with a short list of (genuine) exceptions, to infinite rank irreducible Coxeter systems.Imaginary cone and reflection subgroups of Coxeter groupshttps://www.zbmath.org/1483.200742022-05-16T20:40:13.078697Z"Dyer, Matthew J."https://www.zbmath.org/authors/?q=ai:dyer.matthew-jLet \(V\) be an \(\mathbb{R}\)-vector space equipped with a symmetric bilinear form \(\langle -,- \rangle\). Suppose \((\Phi, \Pi)\) is a based root system in \(V\) with associated Coxeter system \((W,S)\). Denote
\[
\mathscr{C} = \{v \in V \mid \langle v, \alpha \rangle \ge 0, \forall \alpha \in \Pi\}, \text{ and } \mathscr{K} = (\mathbb{R}_{\ge 0} \Pi) \cap (- \mathscr{C}).
\]
Define the imaginary cone \(\mathscr{Z}\) to be
\[
\mathscr{Z} = \bigcup_{w \in W} w \mathscr{K}.
\]
This extends the notion for Kac-Moody Lie algebras studied in [\textit{V. G. Kac}, Infinite dimensional Lie algebras. Cambridge etc.: Cambridge University Press (1990; Zbl 0716.17022)]. The paper under review provides a survey on the imaginary cone, emphasizing its relationship with reflection subgroups. There are four main results in this paper, which were unknown previously, listed as follows.
Theorem 6.3./Theorem 12.2. Let \(W^\prime\) be a reflection subgroup of \(W\), then \(\mathscr{Z}_{W^\prime} \subseteq \mathscr{Z}\), where \(\mathscr{Z}_{W^\prime}\) is the imaginary cone of \(W^\prime\).
Theorem 7.6. Suppose \(W\) is irreducible, infinite, and of finite rank. Then \(\overline{\mathscr{Z}}\) is the unique non-zero \(W\)-invariant closed pointed cone contained in \(\mathbb{R}_{\ge 0} \Pi\).
Theorem 10.3. (sketched) Suppose \(W\) is of finite rank. (a) The imaginary cone and the Tits cone is a dual pair. (b) The lattice (i.e. poset) formed by faces of \(\mathscr{Z}\) is isomorphic to the lattice formed by facial subgroups without finite components. (c) The face lattice of \(\mathscr{Z}\) is dual to that of the Tits cone. (d) If \(W^\prime\) is a facial subgroup without finite components, then its imaginary cone and its Tits cone can be described explicitly by each other.
Theorem 12.3. One has \(\mathscr{Z} = \mathbb{R}_{\ge 0} (\bigcup_{W^\prime \in \daleth} \mathscr{Z}_{W^\prime})\), where \(\daleth\) is the set of dihedral reflection subgroups of \(W\).
Besides, the hyperbolic and universal cases are discussed in Section 9. In Section 13, some motivations and applications are presented, including the dominance order, limit roots, etc. In particular, the author mentioned that a weakened version of the boundedness conjecture on Lusztig's \(a\)-function can be proved, but no more details are given. There is a list of notations at the end, which is helpful in reading.
Reviewer: Hongsheng Hu (Beijing)Some conditions under which left derivations are zerohttps://www.zbmath.org/1483.470682022-05-16T20:40:13.078697Z"Hosseini, Amin"https://www.zbmath.org/authors/?q=ai:hosseini.aminSummary: In this study, we show that every continuous Jordan left derivation on a (commutative or noncommutative) prime UMV-Banach algebra with the identity element~1 is identically zero. Moreover, we prove that every continuous left derivation on a unital finite dimensional Banach algebra, under certain conditions, is identically zero. As another result in this regard, it is proved that if \(\mathfrak{R}\) is a 2-torsion free semiprime ring such that [the annihilator] \(\operatorname{ann}\{[y,z]\mid y,z \in \mathfrak{R}\}=\{0\}\), then every Jordan left derivation \(\mathfrak{L}:\mathfrak{R}\rightarrow \mathfrak{R}\) is identically zero. In addition, we provide several other results in this regard.Multiplicative Nambu structures on Lie groupoidshttps://www.zbmath.org/1483.530982022-05-16T20:40:13.078697Z"Das, Apurba"https://www.zbmath.org/authors/?q=ai:das.apurbaWeinstein introduced the notion of coisotropic submanifold of a Poisson manifold as a natural generalisation of a Lagrangian submanifold of a symplectic manifold. The condition that the bi-vector field that satisfies \([\Pi, \Pi]= 0\) plays no role in defining a coisotropic submanifold. Thus, the notion of coisotropic submanifolds is well-defined for any bivector field and, more generally, any multivector field. The reader should consult the paper under review for more details.
Nambu-Poisson manifolds are one particular higher-order generalisation of Poisson manifolds. Recall that A Nambu-Poisson manifold of order \(n\) is a manifold equipped with an n-vector field such that the associated \(n\)-array bracket on functions satisfies Filippov's Fundamental Identity. Coisotropic submanifolds of a Nambu-Poisson manifold are submanifolds that are coisotropic for the Nambu tensor.
In the paper under review, the author presents some basic properties of coisotropic submanifolds for a given multivector field and generalises the results of \textit{A. Weinstein} [J. Math. Soc. Japan 40, No. 4, 705--727 (1988; Zbl 0642.58025)]. The notion of a Nambu-Lie groupoid, understood as a Lie groupoid equipped with a multiplicative Nambu tensor is introduced and studied.
Reviewer: Andrew Bruce (Swansea)Kowalewski top and complex Lie algebrashttps://www.zbmath.org/1483.700372022-05-16T20:40:13.078697Z"Jurdjevic, V."https://www.zbmath.org/authors/?q=ai:jurdjevic.velimirSummary: This paper identifies a natural Hamiltonian on a ten dimensional complex Lie algebra that unravels the mysteries encountered in Kowalewski's famous paper on the motions of a rigid body around its fixed point under the influence of gravity. This system reveals that the enigmatic conditions of Kowalewski, namely, two principal moments of inertia equal to each other and twice the value of the remaining moment of inertia, and the centre of gravity in the plane spanned by the directions corresponding to the equal moments of inertia, are both necessary and sufficient for the existence of an isospectral representation \(\frac{dL(\lambda)}{dt}=[M(\lambda), L(\lambda)]\) with a spectral parameter \(\lambda \). This representation then yields a crucial spectral invariant that naturally accounts for all the integrals of motion, known as Kowalewski type integrals in the literature of the top. This result is fundamentally dependent on a preliminary discovery that the equality of two principal moments of inertia and the placement of the centre of mass in the plane spanned by the corresponding directions is intimately tied to the existence of another integral of motion on whose zero level surface the above spectral representation resides. The link between mechanical tops and Hamiltonian systems on Lie algebras is provided by an earlier result in which it is shown that the equations of mechanical tops with a linear potential, (heavy tops, in particular) can be represented on certain coadjoint orbits in the semi-direct product \(\mathfrak{g}=\mathfrak{p}\rtimes\mathfrak{k}\) induced by a closed subgroup \(K\) of the underlying group \(G\). The passage to complex Lie algebras is motivated by Kowalewski's mysterious use of complex variables. It is shown that the complex variables in her paper are naturally identified with complex quaternions and the representation of \(\mathfrak{so}(4,\mathbb{C})\) as the product \(\mathfrak{sl}(2,\mathbb{C})\times \mathfrak{sl}(2,\mathbb{C})\). The paper also shows that all the equations of Kowalewski type can be solved by a uniform integration procedure over the Jacobian of a hyperelliptic curve, as in the original paper of Kowalewski.Multi-component supersymmetric D type Drinfeld-Sokolov hierarchy and its Virasoro symmetryhttps://www.zbmath.org/1483.810842022-05-16T20:40:13.078697Z"Li, Chuanzhong"https://www.zbmath.org/authors/?q=ai:li.chuanzhong.1|li.chuanzhongSummary: In this paper, we define a multi-component supersymmetric B type 2KP(MS2BKP) hierarchy. Under a reduction, we derive a multi-component supersymmetric D type Drinfeld-Sokolov hierarchy which has a multi super Virasoro algebraic structure.Boson-fermion correspondence, QQ-relations and Wronskian solutions of the T-systemhttps://www.zbmath.org/1483.810862022-05-16T20:40:13.078697Z"Tsuboi, Zengo"https://www.zbmath.org/authors/?q=ai:tsuboi.zengoSummary: It is known that there is a correspondence between representations of superalgebras and ordinary (non-graded) algebras. Keeping in mind this type of correspondence between the twisted quantum affine superalgebra \(U_q(gl(2r | 1)^{(2)})\) and the non-twisted quantum affine algebra \(U_q(so(2r + 1)^{(1)})\), we proposed, in the previous paper [the author, ibid. 870, No. 1, 92--137 (2013; Zbl 1262.17017)], a Wronskian solution of the T-system for \(U_q(so(2r + 1)^{(1)})\) as a reduction (folding) of the Wronskian solution for the non-twisted quantum affine superalgebra \(U_q(gl(2r | 1)^{(1)})\). In this paper, we elaborate on this solution, and give a proof missing in [loc. cit.]. In particular, we explain its connection to the Cherednik-Bazhanov-Reshetikhin (quantum Jacobi-Trudi) type determinant solution known in [\textit{A. Kuniba} et al., J. Phys. A, Math. Gen. 28, No. 21, 6211--6226 (1995; Zbl 0871.17015)]. We also propose Wronskian-type expressions of T-functions (eigenvalues of transfer matrices) labeled by non-rectangular Young diagrams, which are quantum affine algebra analogues of the Weyl character formula for \(so(2r + 1)\). We show that T-functions for spinorial representationsof \(U_q(so(2r + 1)^{(1)})\) are related to reductions of T-functions for asymptotic typical representations of \(U_q(gl( 2r | 1)^{(1)})\).Spinorial Snyder and Yang models from superalgebras and noncommutative quantum superspaceshttps://www.zbmath.org/1483.810912022-05-16T20:40:13.078697Z"Lukierski, Jerzy"https://www.zbmath.org/authors/?q=ai:lukierski.jerzy"Woronowicz, Mariusz"https://www.zbmath.org/authors/?q=ai:woronowicz.mariuszSummary: The relativistic Lorentz-covariant quantum space-times obtained by Snyder can be described by the coset generators of (anti) de-Sitter algebras. Similarly, the Lorentz-covariant quantum phase spaces introduced by Yang, which contain additionally quantum curved fourmomenta and quantum-deformed relativistic Heisenberg algebra, can be defined by suitably chosen coset generators of conformal algebras. We extend such algebraic construction to the respective superalgebras, which provide quantum Lorentz-covariant superspaces (SUSY Snyder model) and indicate also how to obtain the quantum relativistic phase superspaces (SUSY Yang model). In last Section we recall briefly other ways of deriving quantum phase (super)spaces and we compare the spinorial Snyder type models defining bosonic or fermionic quantum-deformed spinors.Type IIB superstring vertex operator from the -8 picturehttps://www.zbmath.org/1483.811202022-05-16T20:40:13.078697Z"Martins, Lucas N. S."https://www.zbmath.org/authors/?q=ai:martins.lucas-n-sSummary: A new procedure was recently proposed for constructing massless Type IIB vertex operators in the pure spinor formalism. Instead of expressing these closed string vertex operators as left-right products of open string vertex operators, they were instead constructed from the complex N=2 d=10 superfield whose lowest real and imaginary components are the dilaton and Ramond-Ramond axion. These Type IIB vertex operators take a simple form in the -8 picture and are related to the usual vertex operators in the zero picture by acting with picture-raising operators. In this paper, we compute explicitly this picture-raising procedure and confirm this proposal in a flat background. Work is in progress on confirming this proposal in an \(AdS_5 \times S^5\) background.Non-abelian W-representation for GKMhttps://www.zbmath.org/1483.811212022-05-16T20:40:13.078697Z"Mironov, A."https://www.zbmath.org/authors/?q=ai:mironov.andrei-d"Mishnyakov, V."https://www.zbmath.org/authors/?q=ai:mishnyakov.v"Morozov, A."https://www.zbmath.org/authors/?q=ai:morozov.alexei-yurievichSummary: \(W\)-representation is a miraculous possibility to define a non-perturbative (exact) partition function as an exponential action of somehow integrated Ward identities on unity. It is well known for numerous eigenvalue matrix models, when the relevant operators are of a kind of \(W\)-operators: for the Hermitian matrix model with the Virasoro constraints, it is a \(W_3\)-like operator, and so on. We extend this statement to the monomial generalized Kontsevich models (GKM), where the new feature is appearance of an ordered P-exponential for the set of non-commuting operators of different gradings.Physics of the inverted harmonic oscillator: From the lowest Landau level to event horizonshttps://www.zbmath.org/1483.811392022-05-16T20:40:13.078697Z"Subramanyan, Varsha"https://www.zbmath.org/authors/?q=ai:subramanyan.varsha"Hegde, Suraj S."https://www.zbmath.org/authors/?q=ai:hegde.suraj-s"Vishveshwara, Smitha"https://www.zbmath.org/authors/?q=ai:vishveshwara.smitha"Bradlyn, Barry"https://www.zbmath.org/authors/?q=ai:bradlyn.barrySummary: In this work, we present the inverted harmonic oscillator (IHO) Hamiltonian as a paradigm to understand the quantum mechanics of scattering and time-decay in a diverse set of physical systems. As one of the generators of area preserving transformations, the IHO Hamiltonian can be studied as a dilatation generator, squeeze generator, a Lorentz boost generator, or a scattering potential. In establishing these different forms, we demonstrate the physics of the IHO that underlies phenomena as disparate as the Hawking-Unruh effect and scattering in the lowest Landau level (LLL) in quantum Hall systems. We derive the emergence of the IHO Hamiltonian in the LLL in a gauge invariant way and show its exact parallels with the Rindler Hamiltonian that describes quantum mechanics near event horizons. This approach of studying distinct physical systems with symmetries described by isomorphic Lie algebras through the emergent IHO Hamiltonian enables us to reinterpret geometric response in the lowest Landau level in terms of relativistic effects such as Wigner rotation. Further, the analytic scattering matrix of the IHO points to the existence of quasinormal modes (QNMs) in the spectrum, which have quantized time-decay rates. We present a way to access these QNMs through wave packet scattering, thus proposing a novel effect in quantum Hall point contact geometries that parallels those found in black holes.An algebraic classification of solution generating techniqueshttps://www.zbmath.org/1483.830792022-05-16T20:40:13.078697Z"Borsato, Riccardo"https://www.zbmath.org/authors/?q=ai:borsato.riccardo"Driezen, Sibylle"https://www.zbmath.org/authors/?q=ai:driezen.sibylle"Hassler, Falk"https://www.zbmath.org/authors/?q=ai:hassler.falkSummary: We consider a two-fold problem: on the one hand, the classification of a family of solution-generating techniques in (modified) supergravity and, on the other hand, the classification of a family of canonical transformations of 2-dimensional \(\sigma \)-models giving rise to integrable-preserving transformations. Assuming a generalised Scherk-Schwarz ansatz, in fact, the two problems admit essentially the same algebraic formulation, emerging from an underlying double Lie algebra \(\mathfrak{d}\). After presenting our derivation of the classification, we discuss in detail the relation to modified supergravity and the additional conditions to recover the standard (unmodified) supergravity. Starting from our master equation -- that encodes all the possible continuous deformations allowed in the family of solution-generating techniques -- we show that these are classified by the Lie algebra cohomologies \(H^2(\mathfrak{h}, \mathbb{R})\) and \(H^3(\mathfrak{h}, \mathbb{R})\) of the maximally isotropic subalgebra \(\mathfrak{h}\) of the double Lie algebra \(\mathfrak{d} \). We illustrate our results with a non-trivial example, the bi-Yang-Baxter-Wess-Zumino model.