Recent zbMATH articles in MSC 17https://zbmath.org/atom/cc/172022-11-17T18:59:28.764376ZUnknown authorWerkzeugBirational geometry for the covering of a nilpotent orbit closurehttps://zbmath.org/1496.140182022-11-17T18:59:28.764376Z"Namikawa, Yoshinori"https://zbmath.org/authors/?q=ai:namikawa.yoshinoriFor a complex classical simple Lie algebra \(\mathfrak{g}\) and a nilpotent orbit \(O\) of \(\mathfrak{g}\), the fundamental group \(\pi_1(O)\) is finite. Taking the universal covering \(\pi^0: X^0 \to O\), \(\pi^0\) extends to a finite cover \(\pi: X \to \bar{O}\), where \(\bar{O}\) is the closure of \(O\) in \(\mathfrak{g}\). By using the Kirillov-Kostant form \(\omega_{KK}\) on \(O\), the normal affine variety \(X\) becomes a conical symplectic variety. Let \(\omega := (\pi^0)^*\omega_{KK}\), a \(G\)-invariant symplectic \(2\)-form on \(X^0\).
The author studies the birational geometry for the resolutions of \((X, \omega)\). A crepant projective resolution \(f: Y \to X\) of \(X\) is a projective birational morphism \(f\) from a nonsingular variety \(Y\) to \(X\) such that \(K_Y = f^*K_X\). In general, \(X\) does not have a crepant projective resolution. However, \(X\) always has a nice crepant projective partial resolution \(f: Y \to X\) known as \(\mathbb{Q}\)-factorial terminalization. A \(\mathbb{Q}\)-factorial terminalization \(f\) is a projective birational morphism from a normal variety \(Y\) to \(X\) such that \(Y\) has only \(\mathbb{Q}\)-factorial terminal singularities and \(K_Y = f^*K_X\).
The author provides an explicit construction of a \(\mathbb{Q}\)-factorial terminalization of \(X\) when \(O\) is a nilpotent orbit of a classical simple Lie algebra \(\mathfrak{g}\) and \(X^0\) is the universal covering of \(O\). That is, let \(Q \subset G\) be a parabolic subgroup of \(G\) and let \(Q = U L\) be a Levi decomposition of \(Q\) by the unipotent radical \(U\) and a Levi subgroup \(L\). Its Lie algebra \(\mathfrak{q} = \text{Lie}(Q) = \mathfrak{n} \oplus \mathfrak{l}\) decomposes as a direct sum of \(\mathfrak{n} := \text{Lie}(U)\) and \(\mathfrak{l} := \text{Lie}(L)\). Let \(O'\) be a nilpotent orbit of \(\mathfrak{l}\). Then there is a unique nilpotent orbit \(O\) of \(\mathfrak{g}\) such that \(O\) meets \(\mathfrak{n} + O'\) in a Zariski open subset of \(\mathfrak{n} + O'\). In such a case, \(O\) is induced from \(O'\) and write \(O = \mathrm{Ind}^{\mathfrak g}_{\mathfrak l}(O')\). There is a generically finite map \(\mu: G\times^Q (\mathfrak{n}+ \bar{O}') \rightarrow \bar{O}\), where \([g,z] \mapsto \text{Ad}_g(z)\), which the author calls a generalized Springer map. Let \((X')^0 \to O'\) be an etale covering and let \(X' \to \bar{O}'\) be the associated finite cover. Then the author considers the space \(\mathfrak{n} + X'\) which is a product of an affine space \(\mathfrak{n}\) and the affine variety \(X'\). There is a finite cover \(\mathfrak{n} + X' \to \mathfrak{n} + \bar{O}'\). If the \(Q\)-action on \(\mathfrak{n} + \bar{O}'\) lifts to a Q-action on \(\mathfrak{n} + X'\), then one can make \(G \times^Q(\mathfrak{n} + X')\) and get a certain commutative diagram involving the Stein factorization. For an arbitrary nilpotent orbit \(O\) of a classical Lie algebra \(\mathfrak{g}\), the author gives an explicit algorithm for finding \(Q\), \(O'\) and \(X'\) such that \(O = \mathrm{Ind}^{\mathfrak g}_{\mathfrak l}(O')\), \(X'\) has only \(\mathbb{Q}\)-factorial terminal singularities, and the Q-action on \(\mathfrak{n} + \bar{O}'\) lifts to a Q-action on \(\mathfrak{n} + X'\) and the finite covering \(Z \to \bar{O}\) in the commutative diagram coincides with the finite covering \(\pi: X \to \bar{O}\) associated with the universal covering \(X^0\) of \(O\).
Reviewer: Mee Seong Im (Annapolis)Bidiagonal triads and the tetrahedron algebrahttps://zbmath.org/1496.150012022-11-17T18:59:28.764376Z"Funk-Neubauer, Darren"https://zbmath.org/authors/?q=ai:funk-neubauer.darrenA bidiagonal triad on a finite-dimensional vector space \(V\) over a field \(\mathbb{F}\) is a triple of diagonalizable linear endomorphisms \(A_i\) of \(V\), \(i=1,2,3\), such that there exists an ordering \(\{V_i^j\}_{j=0}^{d}\) of the eigenspaces of \(A_i\), with corresponding eigenvalues \(\{\theta_i^j\}_{j=0}^d\), such that the following two conditions hold:
\begin{itemize}
\item \(A_i(V_{i'}^j)\subseteq V_{i'}^j+V_{i'}^{j+1}\) for any \(i\neq i'\) and any \(j=0,\ldots,d\), where \(V_i^{d+1}=0\) for any \(i=1,2,3\);
\item The restriction of \([A_i,A_{i'}]^{d-2j}\) to \(V_{i'}^j\) gives a bijection \(V_{i'}^j\mapsto V_{i'}^{d-j}\) for any \(i\neq i'\) and any \(j\).
\end{itemize}
The first main result of the paper asserts that the eigenvalues of a bidiagonal triad satisfy the linear recurrence relations \[\frac{\theta_i^{j+1}-\theta_i^j}{\theta_i^j-\theta_i^{j-1}}=1, \qquad i=1,2,3.\] \par Then a very natural connection between bidiagonal triads and representations of the tetrahedron Lie algebra \(\boxtimes\) (or the three-point \(\mathfrak{sl}_2\) loop algebra), defined by \textit{B. Hartwig} and \textit{P. Terwilliger} [J. Algebra 308, No. 2, 840--863 (2007; Zbl 1163.17026)] is showcased: any finite-dimensional irreducible module \(V\) for \(\boxtimes\) gives rise to several bidiagonal triads on \(V\), obtained by the action of the three generators of \(\boxtimes\) meeting at a corner of the tetrahedron. Moreover, any `reduced thin' bidiagonal triad on a vector space \(V\) may be extended to a representation of the tetrahedron algebra.
Reviewer: Alberto Elduque (Zaragoza)Cohomology and deformations of twisted Rota-Baxter operators and NS-algebrashttps://zbmath.org/1496.160132022-11-17T18:59:28.764376Z"Das, Apurba"https://zbmath.org/authors/?q=ai:das.apurba-narayan|das.apurbaLet us recall that a Rota-Baxter is an algebraic abstraction of the integral operator. In this paper, the author considers twisted Rota-Baxter operators on associative algebras. First, the author recalls H-twisted Rota-Baxter operators, Reynolds operators and NS-algebras. Then, he constructs \(L_\infty\)-algebras where Maurer-Cartan elements are given by twisted Rota-Naxter operators.
Reviewer: Angela Gammella-Mathieu (Metz)Universal enveloping of (modified) \(\lambda\)-differential Lie algebrashttps://zbmath.org/1496.160252022-11-17T18:59:28.764376Z"Peng, Xiao-Song"https://zbmath.org/authors/?q=ai:peng.xiao-song"Zhang, Yi"https://zbmath.org/authors/?q=ai:zhang.yi.3|zhang.yi.2|zhang.yi.1|zhang.yi.6|zhang.yi.12|zhang.yi.14|zhang.yi.8|zhang.yi.5|zhang.yi.10|zhang.yi.4|zhang.yi"Gao, Xing"https://zbmath.org/authors/?q=ai:gao.xing"Luo, Yan-Feng"https://zbmath.org/authors/?q=ai:luo.yan-fengThis paper deals with somewhat natural generalizations of differential (associative or Lie) algebras, namely \(\lambda\)-differential (associative or Lie) algebras and modified \(\lambda\)-differential (associative or Lie) algebras. A \(\lambda\)-differential (associative or Lie) algebra, for a constant \(\lambda\), is roughly an associative or Lie algebra \(A\) with a linear endomorphism \(A\xrightarrow{d}A\) which satisfies a relation similar to the Leibniz rule, namely \(d(xy)=xd(y)+d(x)y+\lambda d(x)d(y)\), \(x,y\in A\) (where, when \(A\) is a Lie algebra, a concatenation such as \(xy\) should be read as a Lie bracket \([x,y]\)). In the ``modified'' version the term \(\lambda d(x)d(y)\) is replaced by the term \(\lambda xy\).
The existence and uniqueness (up to a unique isomorphism) results about free objects are obtained easily by the authors because \(\lambda\)-differential \((\Bbbk,\partial)\)-modules and algebras, and their ``modified'' versions are categories concretely equivalent to varieties of algebras in the sense of universal algebra so that each algebraic functor, that is, a functor which preserves the underlying sets, has a left adjoint. Therefore the authors focus on explicit constructions for these objects, which is far more interesting and more tricky (see, e.g., Theorem 3.5, p.~1114, Theorem~3.8, p.~1116).
Reviewer: Laurent Poinsot (Villetaneuse)Morphisms of double (quasi-)Poisson algebras and action-angle duality of integrable systemshttps://zbmath.org/1496.160452022-11-17T18:59:28.764376Z"Fairon, Maxime"https://zbmath.org/authors/?q=ai:fairon.maximeThis paper is a contribution to the theory of non-commutative Poisson structures, with applications to integrable systems. The main theoretical results concern double (quasi)Poisson structures, as introduced in [\textit{M. Van den Bergh}, Trans. Am. Math. Soc. 360, No. 11, 5711--5769 (2008; Zbl 1157.53046)].
Fusion for algebras generalizes fusion for quiver algebras corresponding to identification of vertices; Van den Bergh established that the behaviour of non-commutative Poisson structures under fusion is of significant interest.
For double Poisson structures, the author first proves that iterated fusions are independent of the choices involved and likewise for Hamiltonian algebras (i.e., in presence of a moment map).
The double quasi-Poisson case is much more delicate, since the passage of such a structure to the fusion algebra involves a correction fusion term, analysed in special cases by Van den Bergh and in full generality in [\textit{M. Fairon}, Algebr. Represent. Theory 24, No. 4, 911--958 (2021; Zbl 1480.16049)].
The main algebraic result (announced in [Zbl 1480.16049]) is a double quasi-Poisson analogue of the above, together with a version for quasi-Hamiltonian algebras (i.e., in presence of a multiplicative moment map).
The results are illustrated by examples constructed from quivers (following Van den Bergh) which give, respectively, a Hamiltonian double Poisson structure and a quasi-Hamiltonian structure. The above theorems imply that, up to isomorphism, these structures only depend upon the underlying graph of the quiver.
These results are applied to give a very conceptual explanation of action-angle duality for several examples of classical integrable systems, notably generalizations of the Calogero-Moser system and of the Ruijsenaars-Schneider systems. The associated phase spaces are constructed as quiver varieties, with Poisson structure induced by a NC-Poisson structure in the sense of [\textit{W. Crawley-Boevey}, J. Algebra 325, No. 1, 205--215 (2011; Zbl 1255.17012)]. The author exhibits action-angle coordinates, building upon [\textit{O. Chalykh} and \textit{M. Fairon}, J. Geom. Phys. 121, 413--437 (2017; Zbl 1418.70026)] and [\textit{O. Chalykh} and \textit{A. Silantyev}, J. Math. Phys. 58, No. 7, 071702, 31 p. (2017; Zbl 1370.37126)]. The action-angle duality corresponds to the reversal of arrows of the quiver.
Reviewer: Geoffrey Powell (Angers)Hypergeometry, integrability and Lie theory. Virtual conference, Lorentz Center, Leiden, the Netherlands, December 7--11, 2020https://zbmath.org/1496.170012022-11-17T18:59:28.764376ZPublisher's description: This volume contains the proceedings of the virtual conference on Hypergeometry, Integrability and Lie Theory, held from December 7--11, 2020, which was dedicated to the 50th birthday of Jasper Stokman.
The papers represent recent developments in the areas of representation theory, quantum integrable systems and special functions of hypergeometric type.
The articles of this volume will be reviewed individually.
Indexed articles:
\textit{Etingof, Pavel; Kazhdan, David}, Characteristic functions of \(p\)-adic integral operators, 1-27 [Zbl 07602313]
\textit{Garbali, Alexandr; Zinn-Justin, Paul}, Shuffle algebras, lattice paths and the commuting scheme, 29-68 [Zbl 07602314]
\textit{Kolb, Stefan}, The bar involution for quantum symmetric pairs -- hidden in plain sight, 69-77 [Zbl 07602315]
\textit{Koornwinder, Tom H.}, Charting the \(q\)-Askey scheme, 79-94 [Zbl 07602316]
\textit{Rains, Eric M.}, Filtered deformations of elliptic algebras, 95-154 [Zbl 07602317]
\textit{Regelskis, Vidas; Vlaar, Bart}, Pseudo-symmetric pairs for Kac-Moody algebras, 155-203 [Zbl 07602318]
\textit{Reshetikhin, N.; Stokman, J. V.}, Asymptotic boundary KZB operators and quantum Calogero-Moser spin chains, 205-241 [Zbl 07602319]
\textit{Rösler, Margit; Voit, Michael}, Elementary symmetric polynomials and martingales for Heckman-Opdam processes, 243-262 [Zbl 07602320]
\textit{Schomerus, Volker}, Conformal hypergeometry and integrability, 263-285 [Zbl 07602321]
\textit{Varchenko, Alexander}, Determinant of \(\mathbb{F}_p\)-hypergeometric solutions under ample reduction, 287-307 [Zbl 07602322]
\textit{Varchenko, Alexander}, Notes on solutions of KZ equations modulo \(p^s\) and \(p\)-adic limit \(s\to\infty\), 309-347 [Zbl 07602323]Toral posets and the binary spectrum propertyhttps://zbmath.org/1496.170022022-11-17T18:59:28.764376Z"Coll, Vincent E. jun."https://zbmath.org/authors/?q=ai:coll.vincent-e-jun"Mayers, Nicholas W."https://zbmath.org/authors/?q=ai:mayers.nicholas-wSummary: We introduce a family of posets which generate Lie poset subalgebras of \(A_{n-1}=\mathfrak{sl}(n)\) whose index can be realized topologically. In particular, if \(\mathcal{P}\) is such a \textit{toral poset}, then it has a simplicial realization which is homotopic to a wedge sum of \(d\) one-spheres, where \(d\) is the index of the corresponding type-A Lie poset algebra \(\mathfrak{g}_A(\mathcal{P})\). Moreover, when \(\mathfrak{g}_A(\mathcal{P})\) is Frobenius, its spectrum is \textit{binary}, that is, consists of an equal number of 0's and 1's. We also find that all Frobenius, type-A Lie poset algebras corresponding to a poset whose largest totally ordered subset is of cardinality at most three have a binary spectrum.A criterion for the strong primeness of Lie algebrashttps://zbmath.org/1496.170032022-11-17T18:59:28.764376Z"Golubkov, A. Yu."https://zbmath.org/authors/?q=ai:golubkov.artem-yu"Kudlay, A. S."https://zbmath.org/authors/?q=ai:kudlay.a-sIn the paper under review the authors prove that a Lie algebra \(L\) is strongly prime if and only if \(\operatorname{ad}_{x} \operatorname{ad}(L) \operatorname{ad}_{y} \ne \{ 0 \}\) for all nonzero elements \(x, y \in L\). Here a Lie algebra \(L\) over the associative, commutative ring \(F\) is said to be strongly prime if it is prime, meaning that the product of any two non-zero ideals in non-zero, and does not contain non-zero covers of sandwiches; an element \(x \in L\) is said to be a cover of sandwich if \(\operatorname{ad}_{x} (F \operatorname{Id}_{L} + \operatorname{ad}(L)) \operatorname{ad}_{x} = \{ 0 \}\), where \(\operatorname{Id}_{L}\) is the identity map on \(L\). \par This refines two results from the book of \textit{A. Fernández López} [Jordan structures in Lie algebras. Providence, RI: American Mathematical Society (AMS) (2019; Zbl 1441.17001)].
Reviewer: Andrea Caranti (Trento)Gelfand-Tsetlin modules for \(\mathfrak{gl}(m|n)\)https://zbmath.org/1496.170042022-11-17T18:59:28.764376Z"Futorny, Vyacheslav"https://zbmath.org/authors/?q=ai:futorny.vyacheslav-m"Serganova, Vera"https://zbmath.org/authors/?q=ai:serganova.vera-v"Zhang, Jian"https://zbmath.org/authors/?q=ai:zhang.jian.4Let the underlying field be the set of complex numbers. Let \(\mathfrak{g}\) be the Lie superalgebra \(\mathfrak{gl}(m|n)\). A chain \(\mathfrak{g}=\mathfrak{g}^{1}\supset \mathfrak{g}^{2}\supset \ldots \supset \mathfrak{g}^{m+n}\) of subalgebras of \(\mathfrak{g}\) is such that \(\mathfrak{g}^k\) is isomorphic to \(\mathfrak{gl}(p|q)\) with \(p+q=m+n-k+1 \), which forms a complete flag in \(\mathfrak{g}\) if \(\mathfrak h^k:=\mathfrak h\cap\mathfrak{g}^k\) is a Cartan subalgebra of \(\mathfrak{g}^k\). A complete flag induces a chain of Cartan subalgebras \(\mathfrak h=\mathfrak h^{1}\supset \mathfrak h^{2}\supset \ldots \supset \mathfrak h^{m+n}\).
Every complete flag \(\mathcal C\) in \(\mathfrak{g}\) defines the commutative subalgebra \(\Gamma_{\mathcal C}\) in the universal enveloping algebra \(U(\mathfrak{g})\) generated by the centers of the members of the chain. Here, \(\Gamma_{\mathcal C}\) is the Gelfand-Tsetlin subalgebra of \(U(\mathfrak{g})\) associated with the flag \(\mathcal{C}\).
Letting \(\Gamma=\Gamma_{\mathcal C}\) be a Gelfand-Tsetlin subalgebra of \(U(\mathfrak{g})\), a finitely generated module \(M\) over \(\mathfrak{g}\) is called a Gelfand-Tsetlin module with respect to \(\Gamma\) if \(M\) decomposes as the direct sum of \(M(\textbf{m})\) as a \(\Gamma\)-module, with \(\textbf{m}\) \(\in\) \(\text{Specm}(\Gamma)\), where \(M(\textbf{m}) = \{ x\in M : \textbf{m}^k x = 0 \text{ for some } k\geq 0\}\) and \(\text{Specm} (\Gamma)\) is the set of maximal ideals of \(\Gamma\).
V. Futorny, V. Serganova, and J. Zhang address the problem of classifying irreducible Gelfand-Tsetlin modules for \(\mathfrak{g}\) and show that it reduces to the classification of Gelfand-Tsetlin modules for the even part. They also give an explicit tableaux construction and the irreducibility criterion for the class of quasi typical and quasi covariant Gelfand-Tsetlin modules, which includes all essentially typical and covariant tensor finite-dimensional modules. In the quasi typical case, new irreducible representations are infinite-dimensional \(\mathfrak{g}\)-modules which are isomorphic to the parabolically induced Kac modules.
Reviewer: Mee Seong Im (Annapolis)Homogeneous bases for Demazure moduleshttps://zbmath.org/1496.170052022-11-17T18:59:28.764376Z"Kambaso, Kunda"https://zbmath.org/authors/?q=ai:kambaso.kundaThis paper studies the PBW filtration on various classes of Demazure modules over classical simple Lie algebras. The main results include a construction of a normal polytope labeling a basis for the associated graded vector space of The Demazure module with respect to the PBW filtration and a proof that the annihilating ideal of this associated graded module is a monomial ideal.
Reviewer: Volodymyr Mazorchuk (Uppsala)On isoclinic extensions of Lie algebras and nilpotent Lie algebrashttps://zbmath.org/1496.170062022-11-17T18:59:28.764376Z"Arabyani, Homayoon"https://zbmath.org/authors/?q=ai:arabyani.homayoon"Sadeghifard, Mohammad Javad"https://zbmath.org/authors/?q=ai:sadeghifard.mohammad-javadThe concepts of isoclinism and the Schur multiplier, inspired by group theory concepts, have a long history in Lie algebras. The authors find relations between these concepts. They also study pairs of nilpotent Lie algebras, finding conditions for a pair of Lie algebras to be nilpotent. Also, they look at the c-nilpotent multiplier of filiform Lie algebras for which they find bounds on the dimension.
Reviewer: Ernest L. Stitzinger (Raleigh)On Ricci negative derivationshttps://zbmath.org/1496.170072022-11-17T18:59:28.764376Z"Gutiérrez, María Valeria"https://zbmath.org/authors/?q=ai:gutierrez.maria-valeriaThe present paper is a contribution to the problem of negative Ricci curvature in the homogeneous setting. A question considered is the following: Given a nilpotent Lie algebra \(\mathfrak{n}\), which are the solvable Lie algebras with nilradical \(\mathfrak{n}\) admitting a metric with \(\mathrm{Ric}<0\)? Lauret and Will have conjectured that given a nilpotent Lie algebra the space of all digonalizable derivations such that the corresponding one-dimensional solvable extension admits a left-invariant metric with negative Ricci curvature, coincides with an open convex subset of derivations defined in terms of the moment map for the variety of nilpotent Lie algebras. The author proves the validity of this conjecture in dimensions \(\le 5\), as well as for Heisenberg Lie algebras and standard filiform Lie algebras. A related conjecture has been also posed by Nikolayevsky and Nikonorov.
Reviewer: Andreas Arvanitoyeorgos (Patras)Goldie ranks of primitive ideals and indexes of equivariant Azumaya algebrashttps://zbmath.org/1496.170082022-11-17T18:59:28.764376Z"Losev, Ivan"https://zbmath.org/authors/?q=ai:losev.ivan-v"Panin, Ivan"https://zbmath.org/authors/?q=ai:panin.ivanLet \(\mathfrak g\) be a semisimple Lie algebra. The authors establish a new relation between the Goldie rank of a primitive ideal \(J \subset U(\mathfrak g )\) and the dimension of the corresponding irreducible representation \(V\) of an appropriate finite \(W\)-algebra, see on \(W\)-algebras [\textit{A. Premet}, Adv. Math. 170, No. 1, 1--55 (2002; Zbl 1005.17007); \textit{I. Losev}, J. Am. Math. Soc. 23, No. 1, 35--59 (2010; Zbl 1246.17015)].
Namely, they show that \(\mathrm{Grk}(J )\le \dim V/d_V\), where \(d_V\) is the index of a suitable equivariant Azumaya algebra on a homogeneous space. They also compute \(d_V\) in representation theoretic terms.
Reviewer: Victor Petrogradsky (Brasília)On quantum toroidal algebra of type \(A_1\)https://zbmath.org/1496.170092022-11-17T18:59:28.764376Z"Chen, Fulin"https://zbmath.org/authors/?q=ai:chen.fulin"Jing, Naihuan"https://zbmath.org/authors/?q=ai:jing.naihuan"Kong, Fei"https://zbmath.org/authors/?q=ai:kong.fei"Tan, Shaobin"https://zbmath.org/authors/?q=ai:tan.shao-binLet \(\dot{\mathfrak{g}}\) be a finite dimensional simple Lie algebra of type \(A_1\) over \(\mathbb C\). The main result of the paper under review is the construction of a middle quantum algebra \(\mathcal U\) between the quantum affinization algebra \(\mathcal U_{\hbar}(\hat{\mathfrak{g}})\) introduced by \textit{N. Jing} [Lett. Math. Phys. 44, No. 4, 261--271 (1998; Zbl 0911.17006)] and the quantum toroidal algebra \(\mathcal U_{\hbar}(\dot{\mathfrak{g}}_{tor})\) introduced by \textit{V. Ginzburg} et al. [Math. Res. Lett. 2, No. 2, 147--160 (1995; Zbl 0914.11040)].
It is shown that \(\mathcal U\) admits a triangular decomposition and it has a deformed Drinfeld coproduct, which allows to define a (topological) Hopf algebra structure on \(\mathcal U\). A vertex representation for \(\mathcal U\) is also obtained.
Reviewer: Sonia Natale (Córdoba)Vertex representations of quantum \(N\)-toroidal algebras for type \(C\)https://zbmath.org/1496.170102022-11-17T18:59:28.764376Z"Jing, Naihuan"https://zbmath.org/authors/?q=ai:jing.naihuan"Xu, Zhucheng"https://zbmath.org/authors/?q=ai:xu.zhucheng"Zhang, Honglian"https://zbmath.org/authors/?q=ai:zhang.honglianThe paper is about the vertex representation of quantum \(N\)-toroidal algebra for type \(C\). The definition of the quantum \(N\)-toroidal algebra for type \(C_n\) via generating functions have been recalled and the Fock space and vertex operators have been constructed. The main goal of the paper, is the construction of a level-one module for the quantum \(N\)-toroidal algebra of type \(C\). This construction provides new example of realization of the algebra for non-simply laced types and a model to understand enlargement of the torus. The construction shows that \((N-1)\) copies of the affine Heisenberg algebra stand at the common underlying lattice to represent additional dimension in the torus and similarly the auxiliary Heisenberg sub-algebra also bears similar property.
Reviewer: Laure Gouba (Trieste)Yangians versus minimal \(W\)-algebras: a surprising coincidencehttps://zbmath.org/1496.170112022-11-17T18:59:28.764376Z"Kac, Victor G."https://zbmath.org/authors/?q=ai:kac.victor-g"Möseneder Frajria, Pierluigi"https://zbmath.org/authors/?q=ai:moseneder-frajria.pierluigi"Papi, Paolo"https://zbmath.org/authors/?q=ai:papi.paoloIn this paper the authors provide a Lie theoretic proof of the following \textit{surprising coincidence} mentioned in the title. The coincidence involves two fundamental algebras associated to a complex finite-dimensional simple Lie algebra \(\mathfrak{g}\) (assumed to be different from \(\mathfrak{sl}(2)\)): the Yangian \(Y\mathfrak{g}\) on one side (the unique homogeneous quantization of the current algebra \(U(\mathfrak{g}[t])\)) and the affine W-algebra \(W^k(\mathfrak{g}, \theta)\) on the other (a vertex algebra associated to a minimal nilpotent element \(e_{-\theta}\), obtained by quantum Hamiltonial reduction on the affine vertex algebra at level \(k\in\mathbb{C}\)). The Yangian \(Y\mathfrak{g}\) contains \(\mathfrak{g}\) and \textit{V. G. Drinfel'd} proved in [Sov. Math., Dokl. 32, 256--258 (1985; Zbl 0588.17015); translation from Dokl. Akad. Nauk SSSR 283, 1060--1064 (1985), Theorem 8] that the adjoint representation of \(\mathfrak{g}\) can be canonically extended to a representation \(V\) of \(Y\mathfrak{g}\) by adding an extra dimension (for this result it becomes relevant that \(\mathfrak{g}\neq\mathfrak{sl}(2)\)). It is well-known that finite-dimensional representations of Yangians give rise to rational \(R\)-matrices, that is operator-valued rational functions (in a complex parameter \(u\in\mathbb{C}\)) satisfying the Yang-Baxter equation. \textit{V. Chari} and \textit{A. Pressley} [J. Reine Angew. Math. 417, 87--128 (1991; Zbl 0726.17014), Section 5] compute the poles of the \(R\)-matrix \(R_V(u)\) corresponding to the representation \(V\), given by \(1\), \(h^\vee/2\), where \(h^\vee\) is the dual Coxeter number, and the roots of an explicit quadratic monic polynomial \(q(u)\).
The authors provide a new interpretation of the defining formulae of the representation \(V\) (and therefore of \(q(u)\)), given in terms of the grading of \(\mathfrak{g}\) associated to a minimal nilpotent element. This eventually leads to the surprising coincidence: the same polynomial appears on the W-algebra side. Namely, by a result of \textit{D. Adamović} et al. [J. Algebra 500, 117--152 (2018; Zbl 1420.17028)], the operator product expansion of quantum fields of conformal weight \(3/2\) depends on a canonical monic quadratic polynomial \(p(k)\) in the level \(k\in\mathbb{C}\), which is given precisely by \(p(k)=q(-k)\).
Reviewer: Andrea Appel (Parma)The surjectivity of the evaluation map of the affine super Yangianhttps://zbmath.org/1496.170122022-11-17T18:59:28.764376Z"Ueda, Mamoru"https://zbmath.org/authors/?q=ai:ueda.mamoruIn his previous work [``Affine super Yangian'', Preprint, \url{arXiv:1911.06666}], the author constructed a homomorphism from the affine super Yangian to the completion of the universal enveloping algebra of \(\widehat{\mathfrak{gl}}(m|n)\). In the paper under review he shows that the image of this homomorphism is dense in the completion of \(U(\widehat{\mathfrak{gl}}(m|n))\), which allows him to obtain irreducible representations of the affine super Yangian.
Reviewer: Sonia Natale (Córdoba)A quantum Mirković-Vybornov isomorphismhttps://zbmath.org/1496.170132022-11-17T18:59:28.764376Z"Webster, Ben"https://zbmath.org/authors/?q=ai:webster.ben"Weekes, Alex"https://zbmath.org/authors/?q=ai:weekes.alex"Yacobi, Oded"https://zbmath.org/authors/?q=ai:yacobi.odedMirković and Vybornov construct an isomorphism between slices to (spherical) Schubert varieties in the affine Grassmannian of \(\mathrm{PGL}_n\) on the one hand, and Slodowy slices in \(\mathfrak{gl}_N\) intersected with nilpotent orbit closures on the other. This isomorphism has important applications, such as it appears in works on the mathematical definition of the Coulomb branch associated to quiver gauge theories, the analog of the geometric Satake isomorphism for affine Kac-Moody groups, and geometric approaches to knot homologies. These varieties have quantizations corresponding to natural Poisson structures on them. The goal of this paper is to show that the Mirković-Vybornov isomorphism is the classical limit of an isomorphism of these quantizations.
B. Webster, A. Weekes, and O. Yacobi present a quantization of an isomorphism of Mirković and Vybornov, which relates the intersection of a Slodowy slice and a nilpotent orbit closure in \(\mathfrak{gl}_N\), to a slice between spherical Schubert varieties in the affine Grassmannian of \(\mathrm{PGL}_n\) (with weights encoded by the Jordan types of the nilpotent orbits). A quantization of the former variety is provided by a parabolic \(W\)-algebra and of the latter by a truncated shifted Yangian (Theorem 4.3 (c), page 66). The authors also define an explicit isomorphism between these noncommutative algebras, and show that its classical limit is a variation of the original isomorphism of Mirković and Vybornov (Theorem 4.3 (d), page 66). As a corollary, they deduce that the \(W\)-algebra is free as a left (or right) module over its Gelfand-Tsetlin subalgebra.
Reviewer: Mee Seong Im (Annapolis)Typical irreducible characters of generalized quantum groupshttps://zbmath.org/1496.170142022-11-17T18:59:28.764376Z"Yamane, Hiroyuki"https://zbmath.org/authors/?q=ai:yamane.hiroyukiIn the paper under review the author provides a counterpart, for a generalized quantum group over a field of any characteristic, of a result of \textit{V. Kac} [Lect. Notes Math. 676, 597--626 (1978; Zbl 0388.17002); Commun. Algebra 5, 889--897 (1977; Zbl 0359.17010)] on a Weyl-type character formula for the typical finite-dimensional irreducible modules of a basic classical Lie superalgebra. As a by-product, he obtains a Weyl-Kac-type character formula of the typical irreducible modules of the quantum superalgebras associated with the basic classical Lie superalgebras.
Reviewer: Sonia Natale (Córdoba)Commutative matching Rota-Baxter operators, shuffle products with decorations and matching Zinbiel algebrashttps://zbmath.org/1496.170152022-11-17T18:59:28.764376Z"Gao, Xing"https://zbmath.org/authors/?q=ai:gao.xing"Guo, Li"https://zbmath.org/authors/?q=ai:guo.li"Zhang, Yi"https://zbmath.org/authors/?q=ai:zhang.yi.5|zhang.yi.2|zhang.yi.14|zhang.yi.3|zhang.yi|zhang.yi.1|zhang.yi.10|zhang.yi.6|zhang.yi.8|zhang.yi.4|zhang.yi.12Fix a unitary commutative associative ring \(\mathbf{k}\). A Rota-Baxter algebra is a commutative associative algebra \(R\) over \(\mathbf{k}\) together with a \(\mathbf{k}\)-linear operator \(P: R \longrightarrow R\) satisfying the so-called \textit{Rota-Baxter} identity for \(f, g\) in \(R\):
\[
P(f)P(g) = P(fP(g)) + P(P(f)g).
\]
This is a special case of a more general definition in the paper. We make this simplification in the review because the main results of the paper deal with the special case. As a pioneering example, the ring \(\mathrm{Cont}(\mathbb{R})\) of continuous functions on \(\mathbb{R}\) is a Rota-Baxter algebra over \(\mathbb{R}\), with the operator \(P\) defined by the Riemann integral for \(f \in \mathrm{Cont}(\mathbb{R})\) and \(x \in \mathbb{R}\):
\[
(P(f))(x) := \int_0^x f(t)\, dt.
\]
Let \(\Omega\) be an index set. A matching Rota-Baxter algebra with respect to \(\Omega\) is a commutative associative algebra \(R\) together with a family of linear operators \(P_{\alpha}: R \longrightarrow R\) indexed by \(\alpha \in \Omega\) such that for \(x, y\) in \(R\) and \(\alpha, \beta\) in \(\Omega\) we have:
\[
P_{\alpha}(x) P_{\beta}(y) = P_{\alpha}(xP_{\beta}(y)) + P_{\beta}(P_{\alpha}(x)y).
\]
Each pair \((R, P_{\alpha})\) for a fixed \(\alpha\) forms an ordinary Rota-Baxter algebra. If \((R, P)\) is a Rota-Baxter algebra and \((g_{\alpha})_{\alpha \in \Omega}\) is a family of elements of \(R\), then we can equip \(R\) with a structure of matching Rota-Baxter algebra by setting \(P_{\alpha}(f) := P(g_{\alpha} f)\) for \(\alpha \in \Omega\) and \(f \in R\).
Let \(\mathbf{CAlg}_{\mathbf{k}}\) denote the category of commutative associative algebras over \(\mathbf{k}\) and \(\mathbf{MRBA}_{\mathbf{k},\Omega}\) denote the category of matching Rota-Baxter algebras. By definition we have a forgetful functor \(\mathcal{F}: \mathbf{MRBA}_{\mathbf{k},\Omega} \longrightarrow \mathbf{CAlg}_{\mathbf{k}}\).
One of the main results of this paper is an explicit construction, via \textit{shuffle product}, of a functor \(\mathcal{G}: \mathbf{CAlg}_{\mathbf{k}} \longrightarrow \mathbf{MRBA}_{\mathbf{k},\Omega}\) left adjoint to \(\mathcal{F}\). In more details, for \(A\) a commutative associative algebra, \(\mathcal{G}(A)\) is the tensor product algebra of \(A\) with the shuffle algebra associated to the \(\mathbf{k}\)-module \(\mathbf{k}\Omega \otimes A\), which is naturally a matching Rota-Baxter algebra. The authors also extend this result to the relative setting. Fix an object \(X\) of \(\mathbf{MRBA}_{\mathbf{k},\Omega}\) and let \(\mathcal{C}_X\) denote the category of morphisms \(X \longrightarrow Y\) in \(\mathbf{MRBA}_{\mathbf{k},\Omega}\). Then the forgetful functor \(\mathcal{C}_X \longrightarrow \mathbf{CAlg}_{\mathbf{k}}\) sending a morphism \(X \longrightarrow Y\) to \(\mathcal{F}(Y)\) is shown to admit an explicit left adjoint.
Reviewer: Huafeng Zhang (Villeneuve d'Ascq)2-local derivations on generalized Witt algebrashttps://zbmath.org/1496.170162022-11-17T18:59:28.764376Z"Ayupov, Shavkat"https://zbmath.org/authors/?q=ai:ayupov.sh-a"Kudaybergenov, Karimbergen"https://zbmath.org/authors/?q=ai:kudaybergenov.karimbergen-k"Yusupov, Baxtiyor"https://zbmath.org/authors/?q=ai:yusupov.baxtiyorA linear map \(d: L \to L\) on a Lie algebra \(L\) is called a \(2\)-local derivation, if for every \(x,y \in L\), there exists a derivation \(d_{x,y}\) of \(L\) such that the values of \(d_{x,y}\) and \(d\) at \(x\) and \(y\) coincide. It is proved that any \(2\)-local derivation of various classes of generalized Witt algebras, as well as of their so-called Borel subalgebras (roughly, subalgebras spanned by positive components in the multigrading), is a derivation.
Reviewer: Pasha Zusmanovich (Ostrava)Polynomial representations of \(\mathrm{GL}(m|n)\)https://zbmath.org/1496.170172022-11-17T18:59:28.764376Z"Flicker, Yuval Z."https://zbmath.org/authors/?q=ai:flicker.yuval-zThe paper under review develops a theory of polynomial representations of the super general linear group \(\mathrm{GL}(m|n,A)\), defined over an arbitrary commutative superalgebra \(A\). The methods used adapt and parallel Green's approach to the usual Schur algebra via comodules, as presented in \S 2 of [\textit{J. A. Green}, Polynomial representations of \(\mathrm{GL}_n\). With an appendix on Schensted correspondence and Littelmann paths by K. Erdmann, J. A. Green and M. Schocker. 2nd corrected and augmented edition. Berlin: Springer (2007; Zbl 1108.20044)]. Thus, rather than work directly with super modules for \(\mathrm{GL}(m|n,A)\), the author first defines, for each suitable sub-super coalgebra \(D\) of the super algebra of finitary functions \(\mathrm{GL}(m|n,A)\), a module category \(\mathrm{mod}_D(A\mathrm{GL}(m|n,A))\) of representations of the super group algebra \(A\mathrm{GL}(m|n,A)\) such that the coefficient functions of the representing matrices lie in \(D\). There is an equivalence of categories
\[
\mathrm{mod}_D(A\mathrm{GL}(m|n,A)) \simeq \mathrm{com}(D)
\]
between this category and the category of super \(D\)-comodules. Taking \(D\) to be the super coalgebra of polynomial functions \(\mathrm{GL}(m|n,A)\) of degree \(r\) and dualizing, the author obtains a superalgebra \(S_A(m|n,r)\) that is the super analogue of the usual Schur algebra. A key result on the Schur algebra is that if \(F\) is an infinite field then category of polynomial representations of \(\mathrm{GL}(n,F)\) of polynomial degree \(r\) is equivalent to the category of representations of the Schur algebra \(S_F(n,r)\), defined over the field \(F\). The author proves the analogous result for representations of \(S_A(m|n,r)\) in Proposition 4.2.
The main result of this paper, described as `super Schur duality' is Theorem 5.1. In it \(A\) is a commutative superalgebra over an infinite field \(F\) and \(E_A = A^{m|n}\) is a free rank \(m|n\) \(A\)-module, generated by \(m\) even elements and \(n\) odd elements. The free \(A\)-module \(E_A^{\otimes r}\) is acted on by \(\mathrm{GL}(m|n,A)\) with coefficient functions of polynomial degree \(r\). It may therefore be regarded as a representation of \(S_A(m|n,r)\). The symmetric group \(S_r\) acts on \(E_A^{\otimes r}\) by permuting the factors (with signs coming from the super structure). Theorem 5.1 states that the natural action map
\[
S_A(m|n,r) \rightarrow \mathrm{End}_A(E_A^{\otimes r}) \tag{\(\star\)}
\]
is injective, and its image is precisely those \(A\)-endomorphisms of \(E_A^{\otimes r}\) which commute with the action of the symmetric group \(S_r\). This is the super version of Schur's theorem (see [loc. cit., Theorem 2.6c]) that \(S_F(n,r) \cong \mathrm{End}_{S_r}(V^{\otimes r})\), where \(V\) is the natural \(\mathrm{GL}_n(F)\)-module. As is the case with Schur's theorem, the main force of the result is that \emph{every} \(S_r\) endomorphism of the tensor algebra comes from an element of the super Schur algebra, and so is induced by the action of a suitable linear combination of elements in the super group algebra \(A\mathrm{GL}(m|n,A)\). An important corollary is that if \(F\) has infinite characteristic or prime characteristic \(p > r\) then \(S_A(m|n,r)\) is semisimple.
The author begins with a brief but useful survey of other approaches to Schur-Weyl duality, emphasising that the main novel feature in his paper is to work with the supergroup \(\mathrm{GL}(m|n)\) rather than its super Lie algebra \(\mathfrak{gl}(m|n)\). In this connection we mention that modules for \(S_A(m|n,r)\) are direct sums of the special class of covariant \(\mathfrak{gl}(m|n)\)-modules: see Chapter 3 of Moens' Ph.D.~thesis [Supersymmetric Schur functions and Lie superalgebra representations. Universiteit Gent (2007)] for an excellent introduction. In general, and in contrast to the case for \(S_A(m|n,r)\), modules for \(\mathfrak{gl}(m|n)\) are not completely reducible. Another reference one might add to the author's list is [\textit{D. Benson} and \textit{S. Doty}, Arch. Math. 93, No. 5, 425--435 (2009; Zbl 1210.20039)], which shows (amongst other results) that the Schur algebra analogue of (\(\star\)) holds over any field \(F\) such that \(|F| > r\).
The paper under review includes all the results needed to perform \(p\)-modular reduction on the category of representations of the Super Schur algebra \(S(m|n,r)\). The author remarks that `It seems to us such a modular theory is needed for a geometric theory'.
Reviewer: Mark Wildon (Egham)On cohomologies and algebraic \(K\)-theory of Lie \(p\)-superalgebrashttps://zbmath.org/1496.170182022-11-17T18:59:28.764376Z"Rakviashvili, Giorgi"https://zbmath.org/authors/?q=ai:rakviashvili.giorgiSummary: An enveloping associative superalgebra \(\Lambda [L,\alpha ,\beta ]\) and its groups of cohomologies are defined and it is proved that there exists Frobenius multiplication of the Quillen algebraic \(K\)-functors of \(\Lambda [L,\alpha ,\beta ]\). These results generalize corresponding results for Lie \(p\)-algebras which were proved by the author earlier.Krichever-Novikov type algebras: definitions and resultshttps://zbmath.org/1496.170192022-11-17T18:59:28.764376Z"Schlichenmaier, Martin"https://zbmath.org/authors/?q=ai:schlichenmaier.martinThe author is one of the main experts in Krichever-Novikov (KN)-type algebras and their applications. He has written a book on this topic [Krichever-Novikov type algebras. Theory and applications. Berlin: de Gruyter (2014; Zbl 1347.17001)], and a good introduction to them in [Proc. Symp. Pure Math. 92, 181--220 (2016; Zbl 1427.17034)], with a number of references.
The recent work has a lot in common with those. This article collects the main definitions and results for interested non-experts, and refers to older works for details. It also considers the multipoint setting, and contains the author's newer results from [Commun. Algebra 45, No. 2, 776--821 (2017; Zbl 1406.17029)].
If somebody wants to have a general view on KN-algebras without getting into details, this is a good source for that.
For the entire collection see [Zbl 1492.17001].
Reviewer: Alice Fialowski (Budapest)The structure of parafermion vertex operator algebras \(K(osp(1|2n),k)\)https://zbmath.org/1496.170202022-11-17T18:59:28.764376Z"Jiang, Cuipo"https://zbmath.org/authors/?q=ai:jiang.cuipo"Wang, Qing"https://zbmath.org/authors/?q=ai:wang.qing.1|wang.qing.3|wang.qing.2|wang.qing.4|wang.qingFix positive integers \(n\) and \(k\). Consider the simple Lie superalgebra \(\mathfrak{g} = osp(1|2n)\). Let \(\mathfrak{h}\) be its Cartan subalgebra. Let \(V(k, 0)\) be the vertex operator algebra associated to the universal vacuum \(\hat{\mathfrak{g}}\)-module, \(L(k,0)\) be its simple quotient. Let \(M_{\hat{\mathfrak{h}}}(k, 0)\) be the Heisenberg vertex operator algebra. The paper studies the structure of three vertex operator algebras.
\begin{itemize}
\item[1.] The vertex operator subalgebra \(V(k, 0)(0) = \{v\in V(k, 0): h(0) v = 0, \forall h\in \mathfrak{h}\}\) of \(V(k, 0)\).
\item[2.] The commutant vertex operator algebra \(N(osp(1|2n), k)\) of \(M_{\mathfrak{h}}(k, 0)\) in \(V(k, 0)\).
\item[3.] The parafermion vertex operator algebra \(K(osp(1|2), k)\) associated to \(L(k, 0)\), which is the simple quotient of \(N(osp(1|2), k)\).
\end{itemize}
The paper gives explicit formulas for the generators of these three vertex operator algebras. Besides, the paper also gives explicit formulas for the generators of the unique maximal ideal in \(N(osp(1|2n), k)\). Remarkably, these formulas shows that \(K(osp(1|2), k)\) and \(K(sl_2, 2k)\) are the buidling blocks of \(K(osp(1|2n), k)\). They will be useful in the study of the representation theory of \(K(osp(1|2n),k)\).
Reviewer: Fei Qi (Winnipeg)Vertex algebras associated with hypertoric varietieshttps://zbmath.org/1496.170212022-11-17T18:59:28.764376Z"Kuwabara, Toshiro"https://zbmath.org/authors/?q=ai:kuwabara.toshiroThe functors known as classical and quantum Hamiltonian reduction enjoy an algebraic realisation in terms of BRST cohomology and a geometric one building on nilpotent orbits, arc spaces and Slodowy slices. There is therefore a lot of interest in better understanding the relationships between these two approaches.
This article considers the quantum affine version of the reduction that produces hypertoric varieties. A quantisation of these varieties has already been constructed [\textit{I. M. Musson} and \textit{M. Van den Bergh}, Invariants under tori of rings of differential operators and related topics. Providence, RI: American Mathematical Society (AMS) (1998; Zbl 0928.16019)] and is known as a quantised hypertoric algebra. Here, an affinisation of this, meaning a vertex algebra, is proposed by carefully analysing the jet bundle over the hypertoric variety. It is moreover identified as a coset of a number of symplectic bosons (\(\beta\gamma\) systems) by a number of free bosons (Heisenberg vertex algebras). Finally, the Zhu algebra of this vertex algebra is shown to be at least a subalgebra of the corresponding quantised hypertoric algebra.
Heisenberg cosets of \(\beta\gamma\) systems have received a lot of attention in the literature, see for example [\textit{A. R. Linshaw}, J. Pure Appl. Algebra 213, No. 5, 632--648 (2009; Zbl 1230.17023)]. In the final section, this work is used to identify certain special cases of the proposed affinisation of the quantised hypertoric algebra. In particular, one particularly interesting special case turns out to give the subregular W-algebra of \(\mathfrak{sl}_n\) at non-admissible level \(-n+1\).
Reviewer: David Ridout (Melbourne)\(\mathfrak{L}\)-prolongations of graded Lie algebrashttps://zbmath.org/1496.170222022-11-17T18:59:28.764376Z"Marini, Stefano"https://zbmath.org/authors/?q=ai:marini.stefano"Medori, Costantino"https://zbmath.org/authors/?q=ai:medori.costantino"Nacinovich, Mauro"https://zbmath.org/authors/?q=ai:nacinovich.mauroAuthors' abstract: In this paper we translate the necessary and sufficient conditions of Tanaka's theorem on the finiteness of effective prolongations of a fundamental graded Lie algebras into computationally effective criteria, involving the rank of some matrices that can be explicitly constructed. Our results would apply to geometries, which are defined by assigning a structure algebra on the contact distribution.
Reviewer: V. V. Gorbatsevich (Moskva)An introduction to pre-Lie algebrashttps://zbmath.org/1496.170232022-11-17T18:59:28.764376Z"Bai, Chengming"https://zbmath.org/authors/?q=ai:bai.chengmingAn pre-Lie algebra is a vector space with a binary operation \((x,y)\mapsto xy\) satisfying
\[
(xy)z-x(yz)=(yx)z-y(xz)
\]
The examples of pre-Lie algebras are left-invariant affine structures on Lie groups, deformation complexes of algebras and right-symmetric algebras, rooted tree algebras (which are free pre-Lie algebra), also pre-Lie algebras are naturally related to symplectic Lie algebras and vertex algebras.
In the paper under review these motivating examples are considered in details. In Section 2, we will introduce some background and the different motivation of introducing the notion of pre-Lie algebra. Then the basic properties and construction are explained. A relation between pre-Lie algebras and classical Yang-Baxter equation is explained. Finally the authors puts ``pre-Lie algebras into a bigger framework as one of the algebraic structures of the Lie analogues of Loday algebras. There is an operadic interpretation of these algebraic structures which is related to Manin black products.''
For the entire collection see [Zbl 1492.17001].
Reviewer: Dmitry Artamonov (Moskva)The algebra of \(2 \times 2\) upper triangular matrices as a commutative algebra: gradings, graded polynomial identities and Specht propertyhttps://zbmath.org/1496.170242022-11-17T18:59:28.764376Z"Morais, Pedro"https://zbmath.org/authors/?q=ai:morais.pedro"da Silva Souza, Manuela"https://zbmath.org/authors/?q=ai:souza.manuela-da-silvaLet \(K\) be a field and let \(UT_2(K)\) be the vector space of the \(2\times 2\) upper triangular matrices over \(K\). The authors consider the new multiplication \(a\circ b=ab+ba\) on \(UT_2(K)\) in the case when \(K\) is infinite and of characteristic 2. Thus \(UT_2(K)\) becomes a Lie algebra. The authors describe all group gradings on this algebra. Furthermore they deduce the graded identities satisfied by each one of the gradings. Moreover they prove that the corresponding ideals of graded identities satisfy the Specht property. The authors also obtain that there are non-isomorphic gradings that satisfy the same graded identities. Recall that in [\textit{P. Koshlukov} and \textit{F. Yukihide Yasumura}, Linear Algebra Appl. 534, 1--12 (2017; Zbl 1416.17023)] it was proved that in characteristic 0, a grading on the Jordan algebra \(UT_n(K)\) is uniquely determined by the graded identities it satisfies. For the associative algebra \(UT_n(K)\) the analogous result was obtained in [\textit{O. M. Di Vincenzo} et al., J. Algebra 275, No. 2, 550--566 (2004; Zbl 1066.16047)].
Reviewer: Plamen Koshlukov (Campinas)Bernstein graph algebrashttps://zbmath.org/1496.170252022-11-17T18:59:28.764376Z"Ward, Harold N."https://zbmath.org/authors/?q=ai:ward.harold-nA simply connected graph \(G=(V,E)\), with \(|V|\geq 3\), defines a Bernstein superalgebra over a field of characteristic distinct from two. Its basis is the disjoint union of \(V\) and \(E\). In addition, if \(\{x,y\}\) is the edge connecting two vertices \(x\) and \(y\), then the product of the algebra is described by bilinearity as \(x\{x,y\}=\{x,y\}x=y\), \(y\{x,y\}=\{x,y\}y=x\), \(xy=0\) and \(\{x,y\}\{u,v\}=0\), for all \(\{x,y\}\), \(\{u,v\}\in E\). Any other vertex-edge product is zero. \textit{A. Grishkov} and \textit{R. Costa} proved in [J. Math. Sci., New York 93, No. 6, 877--882 (1999; Zbl 0979.17012)] that graphs with isomorphic algebras are isomorphic. In their proof, they used techniques from algebraic groups, which restrict implicitly the base field. In this paper, this result is reproved for any base field of characteristic distinct from two. In addition, the automorphisms of the Bernstein superalgebra are described.
Reviewer: Raúl M. Falcón (Sevilla)Two-dimensional topological theories, rational functions and their tensor envelopeshttps://zbmath.org/1496.180182022-11-17T18:59:28.764376Z"Khovanov, Mikhail"https://zbmath.org/authors/?q=ai:khovanov.mikhail-g"Ostrik, Victor"https://zbmath.org/authors/?q=ai:ostrik.victor"Kononov, Yakov"https://zbmath.org/authors/?q=ai:kononov.yakovThis paper works over a field \(\boldsymbol{k}\), occasionally specializing to a characteristic \(0\) field. The authors consider \(\boldsymbol{k}\)-linear symmetric monoidal categories called \textit{tensor categories}.
This paper is concerned with the following nine categories.
\begin{itemize}
\item \(\mathrm{Cob}_{2}\): Oriented \(2\)D cobordisms between one-manifolds
\item \(\mathrm{VCob}_{\alpha}\): Viewable cobordisms
\item \(\mathrm{SCob}_{\alpha}\): Skein category, denoted \(\mathrm{PCob}_{\alpha }\)
\item \(\mathrm{Cob}_{\alpha}\): Gligible quotient of \(\mathrm{SCob}_{\alpha} \) by the kernels of trace forms
\item \(\mathrm{DCob}_{\alpha}\): Deligne category
\item \(\underline{\mathrm{DCob}}_{\alpha}\): Gligible quotient of the Deligne category
\item \(\mathrm{SCob}_{\alpha}^{\oplus}\): The finite additive closure of \(\mathrm{SCob}_{\alpha}^{{}}\)
\item \(\mathrm{Cob}_{\alpha}^{\oplus}\): The finite additive closure of \(\mathrm{Cob}_{\alpha}\)
\end{itemize}
The nine categories are connected by functors as follows.
\[
\begin{array} [c]{ccccccccc} \mathrm{Cob}_{2} & \rightarrow & \mathrm{VCob}_{\alpha} & \rightarrow & \mathrm{SCob}_{\alpha}^{{}} & \rightarrow & \mathrm{SCob}_{\alpha}^{\oplus} & \rightarrow & \mathrm{DCob}_{\alpha}\\
& & & & \downarrow & & \downarrow & & \downarrow\\
& & & & \mathrm{Cob}_{\alpha} & \rightarrow & \mathrm{Cob}_{\alpha} ^{\oplus} & \rightarrow & \underline{\mathrm{DCob}}_{\alpha} \end{array}
\]
The four rightmost categories are additive and the three categories to the left of them are \(\boldsymbol{k}\)-linear and pre-additive, while \(\mathrm{Cob}_{2}\) is neither pre-additive nor \(\boldsymbol{k}\)-linear. All eight categories are rigid symmetic monoidal. The bottom three categories are gligible quotients of the respective categories above them, and their hom spaces carry non-degenerate bilinear forms. Category \(\mathrm{DCob}_{\alpha} \) is the analogue of the Deligne category \(\mathrm{Rep}(S_{t})\) and specializes to it when the sequence \(\alpha\) is constant,
\[
\alpha(t)=(t,t,\dots),\quad Z_{\alpha(t)}=\frac{t}{1-T},\quad t\in\boldsymbol{k}
\]
Category \(\underline{\mathrm{DCob}}_{\alpha}\) is the analogue of the quotient \underline{\(\mathrm{Rep}\)}\((S_{t})\) of \(\mathrm{Rep} (S_{t})\) by negligible morphisms, and specializes to it when the sequence \(\alpha\) is constant.
This paper investigates generalized Deligne categories \(\mathrm{DCob}_{\alpha }\), their quotients \(\underline{\mathrm{DCob}}_{\alpha}\) as well as categories \(\mathrm{SCob}_{\alpha}^{{}}\) and \(\mathrm{Cob}_{\alpha}\) for other rational series \(\alpha\), which are referred to as \textit{tensor envelopes} of \(\alpha\).
The synopsis of the paper goes as follows.
\begin{itemize}
\item[\S 2] discusses basic properties of tensor envelopes. \S 2.1 points out the the scaling
\[
Z(T)\mapsto\lambda^{-1}Z(\lambda T)
\]
for an invertible \(\lambda=\mu^{2}\) does not change the categories considered. \S 2.2 explains that any commutative Frobenius algebra object in a pre-additive tensor category gives rise to a power series \(\alpha\) with coefficients in the commutative ring \(\mathrm{End}(\boldsymbol{1} )\) of endomorphisms of the unit object. \S 2.3 recalls the universal property of \(\mathrm{Cob}_{\alpha}\). \S 2.4 investigates direct sum decompositions of commutative Frobenius algebra objects which mirror partial decompositions of their rational generating series.
\item[\S 3] contains key semisimplicity and abelian realization criteria for the tensor envelopes of \(\alpha\), including Theorem 3.2 and Theorem 3.7, both of which characterize admitting an abelian realization. In particular, the authors classify series \(\alpha\) with the semisimple category \(\underline {\mathrm{DCob}}_{\alpha}\).
\item[\S 4] reviews properties of the endomorphism ring of the one-circle object in categories \(\mathrm{SCob}_{\alpha}\) and \(\mathrm{Cob}_{\alpha}\).
\item[\S 5] describes the structure of the gligible category \(\mathrm{Cob} _{\alpha}\) for the constant function (series \(\alpha=(\beta ,0,0,\dots)\)). Theorem 5.1 claims that the dimension of the state space \(A(n)\) of \(n\) circles for this function [\textit{M. Khovanov}, ``Universal construction of topological theories in two dimensions'', Preprint, \url{arXiv:2007.03361}] is the Catalin number, for \(\boldsymbol{k}\) of characteristic \(0\). A monoidal equivalence between the Karoubi envelope \(\underline{\mathrm{DCob}}_{\alpha} \) of \(\mathrm{Cob}_{\alpha}\), and a suitable category of finite-dimensional representations of the Lie superalgebra \(osp(1\mid2)\) is established in \S 5.5.
\item[\S 6] consists of two sections. \S 6.1 investigates Gram determinants of a natural spanning set of surfaces for the function \(\beta/(1-\gamma T)\), where tensor envelopes correspond to the Deligne category [\textit{M. Khovanov} and \textit{R. Sazdanovic}, ``Bilinear pairings on two-dimensional cobordisms and generalizations of the Deligne category'', Preprint, \url{arXiv:2007.11640}]. These are rank one theories. \S 6.2 gives determinant computations for various rank two theories.
\item[\S 7] considers the case of a polynomial generating function, beyond the constant function case studied in \S 5. When the function is linear, associated tensor envelopes can be expressed via the unoriented Brauer category and its gligible quotient, due to the presence of a commutative Frobenius object in the Brauer category with a linear generating function, as can be seen in \S 7.1. \S 7.2 provides numerical data for the Gram determinants in categories when the generating function is a polynomial of degree two or three. \S 7.3 considers arbitrary degree polynomials, where a conjectural basis in the state space of \(n\) circles for the theory is established, and some properties of the Gram determinant for the set of vectors is established.
\item[\S 8] explains how to enrich category \(\mathrm{Cob}_{2}\) of two-dimensional oriented cobordisms by adding codimension two defects (dots). Presence of the handle cobordism allows of adding relations intertwining the handle cobordism with dot decorations. Going from less general to more general examples, dots may be viewed as fractional handles, elements of a commutative monoid, or elements of a commutative algebra.
\end{itemize}
The works [\textit{J. Flake} et al., ``Indecomposable objects in Khovanov-Sazdanovic's generalizations of Deligne's interpolation categories'', Preprint, \url{arXiv:2106.05798}; \textit{E. Meir}, ``Interpolations of monoidal categories and algebraic structures by invariant theory'', Preprint, \url{arXiv:2105.04622}] are related to this one.
Reviewer: Hirokazu Nishimura (Tsukuba)Higher central charges and Witt groupshttps://zbmath.org/1496.180192022-11-17T18:59:28.764376Z"Ng, Siu-Hung"https://zbmath.org/authors/?q=ai:ng.siu-hung"Rowell, Eric C."https://zbmath.org/authors/?q=ai:rowell.eric-c"Wang, Yilong"https://zbmath.org/authors/?q=ai:wang.yilong"Zhang, Qing"https://zbmath.org/authors/?q=ai:zhang.qing.4|zhang.qing|zhang.qing.1|zhang.qing.3|zhang.qing.2Let \(\mathcal{C}\) be a fusion category. Using the number theoretic properties of the categorical dimension and the Frobenius-Perron dimension of \(\mathcal{C}\) respectively, the paper under review defines two notions of signature of \(\mathcal{C}\), defining homomorphisms from the (super-)Witt group \(\mathcal{W}\) of non-degenerate braided fusion categories to the group of maps from the absolute Galois group \(\operatorname{Gal}(\overline{\mathbb{Q}})\) to \(\left\{ 1,-1 \right\}\). In the case of Frobenius-Perron dimension, the signature homomorphism generalizes to the setting of fusion categories over a symmetric fusion category studied in [\textit{A. Davydov} et al., Sel. Math., New Ser. 19, No. 1, 237--269 (2013; Zbl 1345.18005)].
If \(\mathcal{C}\) is modular, the paper gives a description of higher charges of \(\mathcal{C}\), introduced in [\textit{S. Ng} et al., Sel. Math., New Ser. 25, No. 4, Paper No. 53, 32 p. (2019; Zbl 1430.18015)], in terms of the signature of \(\mathcal{C}\). Viewing the central charge construction for \(\mathcal{C}\) as a function \(\Psi_{\mathcal{C}}\) from \(\operatorname{Gal}(\overline{\mathbb{Q}})\), the signature description is used to show that the assignment \(\mathcal{C} \mapsto \Psi_{\mathcal{C}}\) gives a group homomorphism from the Witt group of pseudounitary modular categories to the group of functions from \(\operatorname{Gal}(\overline{\mathbb{Q}})\) to \(\bigcup_{n=1}^{\infty} \mu_{n}\), where \(\mu_{n}\) is the group of \(n\)th roots of unity.
Further, the paper determines the signatures of certain infinite families of categories \(\mathcal{C}_{r} := \mathfrak{so}(2r+1)_{2r+1}\) of integrable highest weight modules of level \(2r+1\) over the affinization of \(\mathfrak{so}(2r+1)\). Using these results, it is shown that, for an Ising modular category \(\mathcal{I}\), the equation \(x^{2} = [I]\) has infinitely many solutions in the quotient \(\mathcal{W}/\mathcal{W}_{\operatorname{pt}}\) of \(\mathcal{W}\) by the pointed part of \(\mathcal{W}\). This is shown to imply that for the super-Witt group \(s\mathcal{W}\), the torsion subgroup \(s\mathcal{W}_{2}\) generated by the completely anisotropic \(s\)-simple braided fusion categories is of infinite rank. This confirms a conjecture made in [\textit{A. Davydov} et al., Sel. Math., New Ser. 19, No. 1, 237--269 (2013; Zbl 1345.18005)].
Reviewer: Mateusz Stroiński (Uppsala)On webs in quantum type \(C\)https://zbmath.org/1496.180202022-11-17T18:59:28.764376Z"Rose, David E. V."https://zbmath.org/authors/?q=ai:rose.david-e-v"Tatham, Logan C."https://zbmath.org/authors/?q=ai:tatham.logan-cThe authors give a linear pivotal category \(\mathbf{Web}(\mathfrak{sp}_{6}) \) defined by diagrams and relations, and conjecture its equivalence to the full subcategory \(\mathbf{FundRep}(U_q(\mathfrak{sp}_{6})) \) of finite-dimensional representations of \( \mathfrak{sp}_{6} \) tensor-generated by fundamental representations. This is a step towards generalizing to type \(C\) case of the main open problem from Kuperberg's Spider for rank 2 Lie algebras [\textit{G. Kuperberg} Commun. Math. Phys. 180, No. 1, 109--151 (1996; Zbl 0870.17005)]. They prove a number of results that support the conjecture. Namely, they construct a functor from \(\mathbf{Web}(\mathfrak{sp}_{6}) \) to \(\mathbf{FundRep}(U_q(\mathfrak{sp}_{6})) \) that is full and essentially surjective (they prove that the well-known surjection from the BMW algebra to the representation category factors through the web category). Moreover they prove that all \(\Hom\)-spaces in \(\mathbf{Web}(\mathfrak{sp}_{6}) \) are finite-dimensional and that the endomorphism algebra of the tensor unit in \(\mathbf{Web}(\mathfrak{sp}_{6}) \) is one-dimensional. As a consequence, the authors give a new approach to the quantum \( \mathfrak{sp}_{6} \) link invariants in the same lines as the Kauffman bracket description of the Jones polynomial. In this paper it is also given a thickening of \(\mathbf{Web}(\mathfrak{sp}_{6}) \) to a category constructed using ladders that is one of the ingredients is proving the results above.
Reviewer's remark: The conjecture that is the subject of this paper is a consequence of the results in \url{arXiv:2103.14997} for \(sp_{2n}\) by the authors together with \textit{E. Bodish} and \textit{B. Elias}.
Reviewer: Pedro Vaz (Louvain-la-Neuve)Affine Beilinson-Bernstein localization at the critical level for \(\mathrm{GL}_2\)https://zbmath.org/1496.220102022-11-17T18:59:28.764376Z"Raskin, Sam"https://zbmath.org/authors/?q=ai:raskin.samThe paper under review proves an important special case of a conjecture of Frenkel and Gaitsgory on an affine analogue of the Beilinson-Bernstein localization theorem.
To provide a context of the main theorem, let us first discuss the Beilinson--Bernstein theorem. Let \(G\) be a reductive group, \(B\) be a Borel subgroup, and \(\mathfrak{g}\) and \(\mathfrak{b}\) be their Lie algebras, respectively. One of the central problems in the subject of representation theory is to classify representations of \(\mathfrak{g}\). As the center \(Z(\mathfrak{g})\) of the universal enveloping algebra \(U(\mathfrak g)\) acts through a character, called a central character, one may as well fix the central character. Then one is led to the study of the category \(\mathfrak g\text{-mod}_0\) of representations of \(U(\mathfrak{g})\) with the central character being the same as the trivial representation.
Consider the flag variety \(G/B\) and the category \(D(G/B)\) of D-modules on it. For an object \(M \in D(G/B)\), its space \(\Gamma (G/B,M)\) of global sections has an induced action of \(\mathfrak{g}\) and one can check that it has the same central character as the trivial representation. Then the celebrated Beilinson-Bernstein localization theorem says that the functor
\[
\Gamma \colon D(G/B)\to \mathfrak{g}\text{-mod}_0
\]
is in fact an equivalence of abelian and derived categories.
For an affine analogue, consider an affine Kac-Moody algebra \(\widehat{\mathfrak{g}}_\kappa \) where \(\kappa\) is a level, or a symmetric invariant bilinear form on \(\mathfrak{g}\). Feigin and Frenkel proved that the center \(Z(\widehat{\mathfrak{g}}_{\kappa }):= Z(U(\widehat{\mathfrak{g}}_\kappa ))\) of the universal enveloping algebra \(U(\widehat{\mathfrak{g}}_\kappa )\) is trivial unless \(\kappa\) is the critical level. Moreover, they identified the center at the critical level in terms of the Langlands dual group \(\check{G}\) and exhibited the following commutative diagrams: \begin{center} \begin{tikzpicture} \draw [-, thick] (0,1.5) to (0,0.5) ; \draw [-, thick] (3,1.5) to (3,0.5) ; \draw [->>, thick] (1,2) to (2,2) ; \draw [->>, thick] (1,0) to (2,0) ; \node at (-0.5,1) {\( \cong \)}; \node at (3.5,1) {\( \cong \)}; \node at (0,0) {\(\text{Fun}(\text{Op}_{\check{G}} )\)}; \node at (3,0) {\(\text{Fun}( \text{Op}_{\check{G}}^{\text{reg}})\)}; \node at (0,2) {\(Z(\widehat{\mathfrak{g}}_{\text{crit}})\)}; \node at (3,2) {\(\text{End}(\mathbb{V}_{\text{crit}})\)}; \end{tikzpicture} \end{center} where \(\mathbb{V}_{\text{crit}} := \text{Ind}^{\widehat{\mathfrak{g}}_{\text{crit}} }_{\mathfrak g[[t]]}\mathbb C\) is the vacuum module, \( \text{Op}_{\check{G}}\) is the space of \(\check{G}\)-opers on the punctured disk, and \( \text{Op}_{\check{G}}^{\text{reg}}\) is the space of \(\check{G}\)-opers on the disk. Now we are interested in understanding the category \( \widehat{\mathfrak{g}}_{\text{crit}}\text{-mod}_{\text{reg}} \) of \(\widehat{\mathfrak{g}}_{\text{crit}} \)-modules where \(Z(\widehat{\mathfrak{g}}_{\text{crit}} )\) acts through the quotient \(\text{Fun}( \text{Op}_{\check{G}}^{\text{reg}})\) of \(\text{Fun}(\text{Op}_{\check{G}} )\).
Just like the finite-dimensional case, we have a functor
\[
\Gamma \colon D_{\text{crit}}(\text{Gr}_G) \to \widehat{\mathfrak{g}}_{\text{crit}}\text{-mod}_{\text{reg}} ,
\]
where \(\text{Gr}_G:=G(K)/G(O)\) is the affine Grassmannian of \(G\) understood as a flag variety of the loop group \(G(K)\). However, this has no chance of being an equivalence; the delta function \(\delta_1\) goes to the vacuum module \(\mathbb{V}_{\text{crit}}\), but the endomorphisms of the delta function are trivial whereas the endomorphisms of the vacuum module are big, as mentioned before.
Frenkel and Gaitsgory's paper ``Local geometric Langlands correspondence and affine Kac-Moody algebras'' suggested to instead consider the following functor
\[
\Gamma^{\text{Hecke}} \colon D_{\text{crit}}(\text{Gr}_G) \otimes_{\text{Rep}(\check{G})} \text{QCoh}( \text{Op}_{\check{G}}^{\text{reg}}) \to \widehat{\mathfrak{g}}_{\text{crit}}\text{-mod}_{\text{reg}}
\]
where
\begin{itemize}
\item the geometric Satake functor \(\text{Rep}(\check{G}) \to D_{\text{crit}}(\text{Gr}_G)^{G(O)}\) is a symmetric monoidal functor and \(D_{\text{crit}}(\text{Gr}_G)^{G(O)}\) acts on \(D_{\text{crit}}(\text{Gr}_G)\);
\item as the space \( \text{Op}_{\check{G}}^{\text{reg}}\) is the moduli space of flat \(\check{G}\)-bundles on the formal disk with an extra structure, the forgetful map \( \text{Op}_{\check{G}}^{\text{reg}}\to \text{Flat}_{\check{G}}(D) = B\check{G} \) gives a symmetric monoidal functor \(\text{Rep}(\check{G})\to \text{QCoh}( \text{Op}_{\check{G}}^{\text{reg}})\).
\end{itemize}
Then their conjecture is that \(\Gamma^{\text{Hecke}}\) is an equivalence of abelian and derived categories.
The main theorem of the current paper proves the conjecture for a group \(G=GL_2\) as well as simple groups of rank 1. Let us give an idea of the proof.
\begin{itemize}
\item Frenkel and Gaitsgory proved that \(\Gamma^{\text{Hecke}}\) is fully faithful and that \(\Gamma^{\text{Hecke}}\) is an equivalence on \(I^0\)-equivariant objects, where \(I^0\) is the radical of the Iwahori subgroup \(I\);
\item Earlier results of Raskin's ``W-algebras and Whittaker categories'' and Frenkel-Gaitsgory-Vilonen's ``Whittaker patterns in the geometry of moduli spaces of bundles on curves'' showed that \(\Gamma^{\text{Hecke}}\) is an equivalence on Whittaker objects;
\item The new result of the current paper is that for \(G=PGL_2\), any \(G(K)\)-category \(C\) with a level \(\kappa\) is generated under \(G(K)\) by the \(I^0\)-equivariant objects and the Whittaker objects.
\end{itemize}
Note that the first two points work for any reductive group \(G\). Then by observing that the essential image of \(\Gamma^{\text{Hecke}}\) is a \(G(K)\)-category, one can prove the result. With some more work, this can also be used to prove t-exactness.
Later results of \textit{D. Yang} and \textit{S. Raskin} prove the conjecture for a general reductive group \(G\) [``Affine Beilinson-Bernstein localization at the critical level'', Preprint, \url{arXiv:2203.13885}], using categorical Moy-Prasad theory developed by \textit{D. Yang} [``Categorical Moy-Prasad theory Yang'', Preprint, \url{arXiv:2104.12917}] as a main new ingredient.
Reviewer: Philsang Yoo (New Haven)The Bergmann-Shilov boundary of a bounded symmetric domainhttps://zbmath.org/1496.460712022-11-17T18:59:28.764376Z"Mackey, M."https://zbmath.org/authors/?q=ai:mackey.michael-t"Mellon, P."https://zbmath.org/authors/?q=ai:mellon.paulineThe authors study the infinite-dimensional analogs of determining sets and Bergman-Shilov boundary for holomorphic maps in the setting of bounded symmetric domains in complex Banach spaces. Their approach relies upon \(\mathrm{JB}^*\)-triple algebraic techniques, based on the fact that any such domain is biholomorphically equivalent to the unit ball \(B\) of some \(\mathrm{JB}^*\)-triple \(E\) and any biholomorphic automorphism \(g\in\mathrm{Aut}(B)\) of \(B\) extends continuously to a neighborhood of the closure \(\overline{B}\).
After a brief introduction to \(\mathrm{JB}^*\)-theory providing a geometrically useful selection of Jordan-triple identities, they prove that the family \(\Gamma\) of the extreme points of \(\overline{B}\) resp. the family \(\Gamma_1\) of unitary tripotents is invariant under the maps \(g\in\mathrm{Aut}(B)\). Actually, the interesting results are achieved for the cases when \(\Gamma\ne\emptyset\) or \(\Gamma_1\ne\emptyset\). In particular, under such hypothesis we have a Russo-Dye type theorem, that is, \(\overline{B}\) is the closed convex hull of \(\Gamma\) resp. \(\Gamma_1\). Moreover, given any map \(f \in\mathrm{Hol}(\overline{B},X)\) into a complex Banach space \(X\) (\(f:\overline{B}\to X\) is continuous and holomorphic when restricted to \(B\)), we have \(f(\overline{B}) \subset \overline{\mathrm{co}}\big( f(\Gamma)\big)\) with \(\Vert f\Vert_{\overline{B}} = \Vert f\Vert_{\Gamma}\) resp. \(\Vert f\Vert_{\overline{B}} = \Vert f\Vert_{\Gamma_1}\) (for the sup-norms \(\Vert f \Vert_\Delta =\sup_{x\in\Delta} \Vert f(x) \Vert\)).
The major part of the paper is focused on the case of finte rank \(\mathrm{JB}^*\)-triples. The approach does not use their classification stating that we can regard \(E\) as a finite direct sum of reflexive Cartan factors, instead a unifying algebraic treatment is developed along the lines of some ideas in [\textit{W. Kaup} and \textit{J. Sauter}, Manuscr. Math. 101, No. 3, 351--360 (2000; Zbl 0981.32012)] on the structure of boundary components. The main result concerns the Bergman-Shilov boundary: If the underlying \(\mathrm{JB}^*\)-triple is of finite rank, then \(\Gamma\) is the smallest closed subset \(\Lambda\) of \(\overline{B}\) such that \(\Vert f\Vert_{\overline{B}} = \Vert f\Vert_{\Lambda}\) for all \(f\in\mathrm{Hol}(\overline{B},\mathbb{C})\).
Reviewer: László Stachó (Szeged)From deformation theory of wheeled props to classification of Kontsevich formality mapshttps://zbmath.org/1496.550122022-11-17T18:59:28.764376Z"Andersson, Assar"https://zbmath.org/authors/?q=ai:andersson.assar"Merkulov, Sergei"https://zbmath.org/authors/?q=ai:merkulov.sergei-aAuthors' abstract: We study the homotopy theory of the wheeled prop controlling Poisson structures on formal graded finite-dimensional manifolds and prove, in particular, that the Grothendieck-Teichmüller group acts on that wheeled prop faithfully and homotopy nontrivially. Next, we apply this homotopy theory to the study of the deformation complex of an arbitrary Kontsevich formality map and compute the full cohomology group of that deformation complex in terms of the cohomology of a certain graph complex introduced earlier by \textit{M. Kontsevich} [Math. Phys. Stud. 20, 139--156 (1997; Zbl 1149.53325)] and studied by \textit{T. Willwacher} [Invent. Math. 200, No. 3, 671--760 (2015; Zbl 1394.17044)].
Reviewer: Iakovos Androulidakis (Athína)Generalised point vortices on a planehttps://zbmath.org/1496.810662022-11-17T18:59:28.764376Z"Galajinsky, Anton"https://zbmath.org/authors/?q=ai:galajinsky.anton-vSummary: A three-vortex system on a plane is known to be minimally superintegrable in the Liouville sense. In this work, integrable generalisations of the three-vortex planar model, which involve root vectors of simple Lie algebras, are proposed. It is shown that a generalised system, which is governed by a positive definite Hamiltonian, admits a natural integrable extension by spin degrees of freedom. It is emphasised that the \(n\)-vortex planar model and plenty of its generalisations enjoy the nonrelativistic scale invariance, which gives room for possible holographic applications.The noncommutative space of light-like worldlineshttps://zbmath.org/1496.810692022-11-17T18:59:28.764376Z"Ballesteros, Angel"https://zbmath.org/authors/?q=ai:ballesteros.angel"Gutierrez-Sagredo, Ivan"https://zbmath.org/authors/?q=ai:gutierrez-sagredo.ivan"Herranz, Francisco J."https://zbmath.org/authors/?q=ai:herranz.francisco-joseSummary: The noncommutative space of light-like worldlines that is covariant under the light-like (or null-plane) \(\kappa\)-deformation of the (3+1) Poincaré group is fully constructed as the quantization of the corresponding Poisson homogeneous space of null geodesics. This new noncommutative space of geodesics is five-dimensional, and turns out to be defined as a quadratic algebra that can be mapped to a non-central extension of the direct sum of two Heisenberg-Weyl algebras whose noncommutative parameter is just the Planck scale parameter \(\kappa^{-1}\). Moreover, it is shown that the usual time-like \(\kappa\)-deformation of the Poincaré group does not allow the construction of the Poisson homogeneous space of light-like worldlines. Therefore, the most natural choice in order to model the propagation of massless particles on a quantum Minkowski spacetime seems to be provided by the light-like \(\kappa\)-deformation.Spin Hurwitz theory and Miwa transform for the Schur Q-functionshttps://zbmath.org/1496.810772022-11-17T18:59:28.764376Z"Mironov, A."https://zbmath.org/authors/?q=ai:mironov.andrei-d"Morozov, A."https://zbmath.org/authors/?q=ai:morozov.alexei-yurievich"Zhabin, A."https://zbmath.org/authors/?q=ai:zhabin.aSummary: Schur functions are the common eigenfunctions of generalized cut-and-join operators which form a closed algebra. They can be expressed as differential operators in time-variables and also through the eigenvalues of auxiliary \(N \times N\) matrices \(X\), known as Miwa variables. Relevant for the cubic Kontsevich model and also for spin Hurwitz theory is an alternative set of Schur Q-functions. They appear in representation theory of the Sergeev group, which is a substitute of the symmetric group, related to the queer Lie superalgebras \(\mathfrak{q}(N)\). The corresponding spin \(\hat{\mathcal{W}}\)-operators were recently found in terms of time-derivatives, but a substitute of the Miwa parametrization remained unknown, which is an essential complication for the matrix model technique and further developments. We demonstrate that the Miwa representation, in this case, involves a fermionic matrix \(\Psi\) in addition to \(X\), but its realization using supermatrices is \textit{not} quite naive.Super Hirota bilinear equations for the super modified BKP hierarchyhttps://zbmath.org/1496.811122022-11-17T18:59:28.764376Z"Chen, Huizhan"https://zbmath.org/authors/?q=ai:chen.huizhanSummary: In this paper, the super modified BKP (SmBKP) hierarchy is constructed from the perspective of the neutral free superfermions by using highest weight representations of the infinite-dimensional Lie superalgebra \(\mathfrak{b}_{\infty|\infty}(\mathfrak{g})\). Based upon this, the corresponding super Hirota bilinear identity of the SmBKP hierarchy is obtained by using the super Boson-Fermion correspondence of type B, and some specific examples of super Hirota bilinear equations are given. The super bilinear identity with respect to super wave and adjoint wave functions is also constructed. At last, we also give a class of solutions other than group orbit by the neutral free superfermions.Emergent behaviors of discrete Lohe aggregation flowshttps://zbmath.org/1496.820122022-11-17T18:59:28.764376Z"Choi, Hyungjun"https://zbmath.org/authors/?q=ai:choi.hyungjun"Ha, Seung-Yeal"https://zbmath.org/authors/?q=ai:ha.seung-yeal"Park, Hansol"https://zbmath.org/authors/?q=ai:park.hansolSummary: The Lohe sphere model and the Lohe matrix model are prototype continuous aggregation models on the unit sphere and the unitary group, respectively. These models have been extensively investigated in recent literature. In this paper, we propose several discrete counterparts for the continuous Lohe type aggregation models and study their emergent behaviors using the Lyapunov function method. For suitable discretization of the Lohe sphere model, we employ a scheme consisting of two steps. In the first step, we solve the first-order forward Euler scheme, and in the second step, we project the intermediate state onto the unit sphere. For this discrete model, we present a sufficient framework leading to the complete state aggregation in terms of system parameters and initial data. For the discretization of the Lohe matrix model, we use the Lie group integrator method, Lie-Trotter splitting method and Strang splitting method to propose three discrete models. For these models, we also provide several analytical frameworks leading to complete state aggregation and asymptotic state-locking.