Recent zbMATH articles in MSC 17Bhttps://www.zbmath.org/atom/cc/17B2021-04-16T16:22:00+00:00WerkzeugTotally symmetric self-complementary plane partitions and the quantum Knizhnik-Zamolodchikov equation: a conjecture.https://www.zbmath.org/1456.822442021-04-16T16:22:00+00:00"Di Francesco, P."https://www.zbmath.org/authors/?q=ai:francesco.p-di|di-francesco.paolo|di-francesco.philippeThe Bethe ansatz in a periodic box-ball system and the ultradiscrete Riemann theta function.https://www.zbmath.org/1456.821452021-04-16T16:22:00+00:00"Kuniba, Atsuo"https://www.zbmath.org/authors/?q=ai:kuniba.atsuo"Sakamoto, Reiho"https://www.zbmath.org/authors/?q=ai:sakamoto.reihoAnalogues of centralizer subalgebras for fiat 2-categories and their 2-representations.https://www.zbmath.org/1456.180162021-04-16T16:22:00+00:00"Mackaay, Marco"https://www.zbmath.org/authors/?q=ai:mackaay.marco"Mazorchuk, Volodymyr"https://www.zbmath.org/authors/?q=ai:mazorchuk.volodymyr"Miemietz, Vanessa"https://www.zbmath.org/authors/?q=ai:miemietz.vanessa"Zhang, Xiaoting"https://www.zbmath.org/authors/?q=ai:zhang.xiaotingFinitary 2-categories are higher representation-theoretic analogues of finite-dimensional algebras, and the basic classification problem in higher representation theory is that of simple transitive 2-representations of a given 2-category \(\mathcal{C}\). That is, it turns out that simple transitive 2-representations are exhausted by the class of cell 2-representations, and a certain subquotient of the 2-category of Soergel bimodules over the coinvariant algebra, of type \(B_2\), is a non-elementary example. It was then applied to study simple transitive 2-representations for all small quotients of Soergel bimodules associated to finite Coxeter systems.
The main result of this manuscript says for a fiat 2-category \(\mathcal{C}\) and its 2-subcategory \(\mathcal{A}\), there is a bijection between certain classes of simple transitive 2-representations of \(\mathcal{C}\) and \(\mathcal{A}\). This reduces the problem of classification of simple transitive 2-representations for fiat 2-categories to that for fiat 2-categories with only one non-identity left, right, and two-sided cell. As an application, the authors Mackaay, Mazorchuk, Miemietz, and Zhang classify simple transitive 2-representations of various categories of Soergel bimodules, in particular, completing the classification in types \(B_3\) and \(B_4\).
Reviewer: Mee Seong Im (West Point)Fock representations of \(Q\)-deformed commutation relations.https://www.zbmath.org/1456.812662021-04-16T16:22:00+00:00"Bożejko, Marek"https://www.zbmath.org/authors/?q=ai:bozejko.marek"Lytvynov, Eugene"https://www.zbmath.org/authors/?q=ai:lytvynov.eugene-w"Wysoczański, Janusz"https://www.zbmath.org/authors/?q=ai:wysoczanski.januszSummary: We consider Fock representations of the \(Q\)-deformed commutation relations \(\partial_s \partial_t^{\dagger} = Q(s, t) \partial_t^{\dagger} \partial_s + \delta(s, t)\) for \(s, t \in T\). Here \(T : = \mathbb{R}^d\) (or more generally \(T\) is a locally compact Polish space), the function \(Q : T^2 \rightarrow \mathbb{C}\) satisfies \(| Q(s, t) | \leq 1\) and \(Q(s, t) = \overline{Q(t, s)}\), and \(\int_{T^2} h(s) g(t) \delta(s, t) \sigma(d s) \sigma(d t) : = \int_T h(t) g(t) \sigma(d t)\), \(\sigma\) being a fixed reference measure on \(T\). In the case, where \(| Q(s, t) | \equiv 1\), the \(Q\)-deformed commutation relations describe a generalized statistics studied by Liguori and Mintchev. These generalized statistics contain anyon statistics as a special case (with \(T = \mathbb{R}^2\) and a special choice of the function \(Q\)). The related \(Q\)-deformed Fock space \(\mathcal{F}(\mathcal{H})\) over \(\mathcal{H} : = L^2(T \rightarrow \mathbb{C}, \sigma)\) is constructed. An explicit form of the orthogonal projection of \(\mathcal{H}^{\otimes n}\) onto the \(n\)-particle space \(\mathcal{F}_n(\mathcal{H})\) is derived. A scalar product in \(\mathcal{F}_n(\mathcal{H})\) is given by an operator \(\mathcal{P}_n \geq 0\) in \(\mathcal{H}^{\otimes n}\) which is strictly positive on \(\mathcal{F}_n(\mathcal{H})\). We realize the smeared operators \(\partial_t^{\dagger}\) and \(\partial_t\) as creation and annihilation operators in \(\mathcal{F}(\mathcal{H})\), respectively. Additional \(Q\)-commutation relations are obtained between the creation operators and between the annihilation operators. They are of the form \(\partial_s^{\dagger} \partial_t^{\dagger} = Q(t, s) \partial_t^{\dagger} \partial_s^{\dagger}\), \(\partial_s \partial_t = Q(t, s) \partial_t \partial_s\), valid for those \(s, t \in T\) for which \(|Q(s,t)| = 1\).{
\copyright 2017 American Institute of Physics}Perfect integrability and Gaudin models.https://www.zbmath.org/1456.822942021-04-16T16:22:00+00:00"Lu, Kang"https://www.zbmath.org/authors/?q=ai:lu.kang|lu.kang.1Summary: We suggest the notion of perfect integrability for quantum spin chains and conjecture that quantum spin chains are perfectly integrable. We show the perfect integrability for Gaudin models associated to simple Lie algebras of all finite types, with periodic and regular quasi-periodic boundary conditions.Bethe eigenvectors of higher transfer matrices.https://www.zbmath.org/1456.823012021-04-16T16:22:00+00:00"Mukhin, E."https://www.zbmath.org/authors/?q=ai:mukhin.evgeny"Tarasov, V."https://www.zbmath.org/authors/?q=ai:tarasov.vitaly-o"Varchenko, A."https://www.zbmath.org/authors/?q=ai:varchenko.alexander-nQuantum character varieties and braided module categories.https://www.zbmath.org/1456.170102021-04-16T16:22:00+00:00"Ben-Zvi, David"https://www.zbmath.org/authors/?q=ai:ben-zvi.david"Brochier, Adrien"https://www.zbmath.org/authors/?q=ai:brochier.adrien"Jordan, David"https://www.zbmath.org/authors/?q=ai:jordan.david-andrewSummary: We compute quantum character varieties of arbitrary closed surfaces with boundaries and marked points. These are categorical invariants \(\int_S\mathcal{A}\) of a surface \(S\), determined by the choice of a braided tensor category \(\mathcal{A}\), and computed via factorization homology. We identify the algebraic data governing marked points and boundary components with the notion of a \textit{braided module category} for \(\mathcal{A}\), and we describe braided module categories with a generator in terms of certain explicit algebra homomorphisms called \textit{quantum moment maps}. We then show that the quantum character variety of a decorated surface is obtained from that of the corresponding punctured surface as a quantum Hamiltonian reduction. Characters of braided \(\mathcal{A}\)-modules are objects of the torus category \(\int_{T^2}\mathcal{A}\). We initiate a theory of character sheaves for quantum groups by identifying the torus integral of \(\mathcal{A}=\mathrm{Rep}_{q}G\) with the category \(\mathcal{D}_q(G/G)\)-mod of equivariant quantum \(\mathcal{D}\)-modules. When \(G=GL_n\), we relate the mirabolic version of this category to the representations of the spherical double affine Hecke algebra \(\mathbb{SH}_{q,t}\).A Lax type operator for quantum finite \(W\)-algebras.https://www.zbmath.org/1456.170082021-04-16T16:22:00+00:00"De Sole, Alberto"https://www.zbmath.org/authors/?q=ai:de-sole.alberto"Kac, Victor G."https://www.zbmath.org/authors/?q=ai:kac.victor-g"Valeri, Daniele"https://www.zbmath.org/authors/?q=ai:valeri.danieleIn [Adv. Math. 327, 173--224 (2018; Zbl 1398.17007)] the authors for any nilpotent \(N \times N\) matrix \( f\) associated an \(r_1\times r_1\), where \(r_1\) is the multiplicity of the largest Jordan block of \(f\), a matrix \(L(z)\) with entries in \(U(gl_N)((z^{-1}))\). It was proved that 1) the matrix elements of this matrix belong to quantum finite \(W\)-algebra associated to the nilpotent element \(f\); 2) \(L(z)\) is an operator of Yangian type in this quantum finite \(W\)-algebra.
In the present paper the first result is generalized to the case of an arbitrary reductive Lie algebra \(g\). And the second result is generalized to the case of a simple Lie algebra of classical typ
Reviewer: Dmitry Artamonov (Moskva)Generalized dualities and higher derivatives.https://www.zbmath.org/1456.830932021-04-16T16:22:00+00:00"Codina, Tomas"https://www.zbmath.org/authors/?q=ai:codina.tomas"Marqués, Diego"https://www.zbmath.org/authors/?q=ai:marques.diegoSummary: Generalized dualities had an intriguing incursion into Double Field Theory (DFT) in terms of local \(O(d,d)\) transformations. We review this idea and use the higher derivative formulation of DFT to compute the first order corrections to generalized dualities. Our main result is a unified expression that can be easily specified to any generalized T-duality (abelian, non-abelian, Poisson-Lie, etc.) or deformations such as Yang-Baxter, in any of the theories captured by the bi-parametric deformation (bosonic, heterotic strings and HSZ theory), in any supergravity scheme related by field redefinitions. The prescription allows further extensions to higher orders. As a check we recover some previously known particular examples.Integrals of motion for critical dense polymers and symplectic fermions.https://www.zbmath.org/1456.813942021-04-16T16:22:00+00:00"Nigro, Alessandro"https://www.zbmath.org/authors/?q=ai:nigro.alessandroCluster realization of \(\mathcal{U}_q(\mathfrak{g})\) and factorizations of the universal \(R\)-matrix.https://www.zbmath.org/1456.170112021-04-16T16:22:00+00:00"Ip, Ivan C. H."https://www.zbmath.org/authors/?q=ai:ip.ivan-chi-hoGiven a simple, finite dimensional, complex Lie algebra \(\mathfrak{g}\), the paper under review constructs an embedding of the corresponding quantum algebra \(U_q(\mathfrak{g})\) into a certain quantum torus algebra \(\mathcal{D}_\mathfrak{g}\) (modulo some central elements). The algebra \(\mathcal{D}_\mathfrak{g}\) has the associated quiver which is shown to be related, via mutations, to quivers describing the cluster structure of the moduli space of framed local system on a disk with 3 marked points on its boundary. Additionally, the paper provides a factorization of the universal \(R\)-matrix into quantum dilogarithms of cluster monomials, and interprets the conjugation by the R-matrix in terms of a sequence of quiver mutations.
Reviewer: Volodymyr Mazorchuk (Uppsala)On the domain wall partition functions of level-1 affine \(\mathrm{so}(n)\) vertex models.https://www.zbmath.org/1456.822502021-04-16T16:22:00+00:00"Dow, A."https://www.zbmath.org/authors/?q=ai:dow.alan-s"Foda, O."https://www.zbmath.org/authors/?q=ai:foda.omarGeneral solution of an exact correlation function factorization in conformal field theory.https://www.zbmath.org/1456.814022021-04-16T16:22:00+00:00"Simmons, Jacob J. H."https://www.zbmath.org/authors/?q=ai:simmons.jacob-j-h"Kleban, Peter"https://www.zbmath.org/authors/?q=ai:kleban.peterAuslander-Reiten quiver and representation theories related to KLR-type Schur-Weyl duality.https://www.zbmath.org/1456.160102021-04-16T16:22:00+00:00"Oh, Se-jin"https://www.zbmath.org/authors/?q=ai:oh.se-jinSummary: We introduce new partial orders on the sequence positive roots and study the statistics of the poset by using Auslander-Reiten quivers for finite type ADE. Then we can prove that the statistics provide interesting information on the representation theories of KLR-algebras, quantum groups and quantum affine algebras including Dorey's rule, bases theory for quantum groups, and denominator formulas between fundamental representations. As applications, we prove Dorey's rule for quantum affine algebras \(U_q(E_{6,7,8}^{(1)})\) and partial information of denominator formulas for \(U_q(E_{6,7,8}^{(1)})\). We also suggest conjecture on complete denominator formulas for \(U_q(E_{6,7,8}^{(1)})\).Bose-Einstein condensation in a gas of Fibonacci oscillators.https://www.zbmath.org/1456.825742021-04-16T16:22:00+00:00"Algin, Abdullah"https://www.zbmath.org/authors/?q=ai:algin.abdullahLogarithmic minimal models.https://www.zbmath.org/1456.812172021-04-16T16:22:00+00:00"Pearce, Paul A."https://www.zbmath.org/authors/?q=ai:pearce.paul-a"Rasmussen, Jørgen"https://www.zbmath.org/authors/?q=ai:rasmussen.jorgen-h|rasmussen.jorgen|rasmussen.jorgen-born"Zuber, Jean-Bernard"https://www.zbmath.org/authors/?q=ai:zuber.jean-bernardA new path description for the \({\mathcal M} (k+1,2k+3)\) models and the dual \({\mathcal Z}_k\) graded parafermions.https://www.zbmath.org/1456.813872021-04-16T16:22:00+00:00"Jacob, P."https://www.zbmath.org/authors/?q=ai:jacob.punnoose|jacob.p-r|jacob.pierre-e|jacob.pierre|jacob.p-j|jacob.patrick"Mathieu, P."https://www.zbmath.org/authors/?q=ai:mathieu.phillipe|mathieu.paulette|mathieu.phillippe|mathieu.philippe|mathieu.pierre|mathieu.pierre.2|mathieu.p-pThe Virasoro fusion kernel and Ruijsenaars' hypergeometric function.https://www.zbmath.org/1456.814002021-04-16T16:22:00+00:00"Roussillon, Julien"https://www.zbmath.org/authors/?q=ai:roussillon.julienSummary: We show that the Virasoro fusion kernel is equal to Ruijsenaars' hypergeometric function up to normalization. More precisely, we prove that the Virasoro fusion kernel is a joint eigenfunction of four difference operators. We find a renormalized version of this kernel for which the four difference operators are mapped to four versions of the quantum relativistic hyperbolic Calogero-Moser Hamiltonian tied with the root system \(BC_1\). We consequently prove that the renormalized Virasoro fusion kernel and the corresponding quantum eigenfunction, the (renormalized) Ruijsenaars hypergeometric function, are equal.Virasoro blocks and quasimodular forms.https://www.zbmath.org/1456.813672021-04-16T16:22:00+00:00"Das, Diptarka"https://www.zbmath.org/authors/?q=ai:das.diptarka"Datta, Shouvik"https://www.zbmath.org/authors/?q=ai:datta.shouvik"Raman, Madhusudhan"https://www.zbmath.org/authors/?q=ai:raman.madhusudhanSummary: We analyse Virasoro blocks in the regime of heavy intermediate exchange (\(h_p \rightarrow \infty\)). For the 1-point block on the torus and the 4-point block on the sphere, we show that each order in the large-\(h_p\) expansion can be written in closed form as polynomials in the Eisenstein series. The appearance of this structure is explained using the fusion kernel and, more markedly, by invoking the modular anomaly equations via the 2d/4d correspondence. The existence of these constraints allows us to develop a faster algorithm to recursively construct the blocks in this regime. We then apply our results to find corrections to averaged heavy-heavy-light OPE coefficients.Left-symmetric algebra structures on the planar Galilean conformal algebra.https://www.zbmath.org/1456.170122021-04-16T16:22:00+00:00"Chi, Lili"https://www.zbmath.org/authors/?q=ai:chi.lili"Sun, Jiancai"https://www.zbmath.org/authors/?q=ai:sun.jiancaiCup-product for equivariant Leibniz cohomology and Zinbiel algebras.https://www.zbmath.org/1456.170032021-04-16T16:22:00+00:00"Mukherjee, Goutam"https://www.zbmath.org/authors/?q=ai:mukherjee.goutam"Saha, Ripan"https://www.zbmath.org/authors/?q=ai:saha.ripanThe goal of the the article under review is the introduction of equivariant Leibniz cohomology and a Zinbiel product on it.
Recall that a (right) \textit{Leibniz algebra} is is a \(k\)-vector space \({\mathfrak g}\) with a \(k\)-bilinear bracket \([,]\) such that for all \(x,y,z\in{\mathfrak g}\)
\[[x,[y,z]]=[[x,y],z]-[x,z],y].\]
Jean-Louis Loday introduced Leibniz algebras to study the failure of periodicity in algebraic K-theory. He noticed that the Chevalley-Eilenberg coboundary operator lifts to tensor powers (when putting the bracket in the \(i\)th place) to give the coboundary operator \(d:{\mathfrak g}^{\otimes n}\to {\mathfrak g}^{\otimes(n-1)}\) for the homology of Leibniz algebras
The authors consider linear actions of a finite group \(G\) by automorphisms on a Leibniz algebra \({\mathfrak g}\). They develop in detail a differential geometric example (where they naturally switch to \textit{left} Leibniz algebras), namely the Leibniz algebroid structure on \(\Lambda^{n-1}T^*M\) for a Nambu-Poisson manifold \(M\) of order \(n\) with a smooth action of a finite group \(G\).
Then follows the definition of the equivariant cohomology. For this, the authors use Bredon's equivariant cohomology set-up. Namely, to \(G\), one associates a category \(O_G\) whose objects are the left cosets \(G/H\) for \(H\) running through all subgroups \(H\) of \(G\), and morphisms \(G/H\to G/K\) being \(G\)-maps (for the \(G\)-action on \(G/H\) given by left-translation). An \(O_G\)-module is then a contravariant functor \(O_G\to k\)-mod. In the symmetric monoidal category of \(O_G\)-modules, one can consider associative commutative algebras \(A\), but also (right) Leibniz algebras. In fact, the data of a Leibniz algebra \({\mathfrak g}\) in \(k\)-modules equipped with the action of a finite group \(G\) gives rise to a Leibniz algebra in \(O_G\)-modules.
The equivariant cohomology is then defined with the help of standard complexes. On the one hand, one can transpose the setting of the Loday standard complex for Leibniz cohomology into the category of \(O_G\)-modules. On the other hand, for an associative commutative \(O_G\)-algebra \(A\), one can consider in the collection of
\[S^n({\mathfrak g},A):=\bigoplus_{H < G}CL^n({\mathfrak g},A(G/H))\]
the subcomplex of invariant cochains. The authors show in their Theorem 4.5 that both complexes compute the same cohomology, the \textit{\(G\)-equivariant Leibniz cohomology} of \({\mathfrak g}\).
The last section is then devoted to the construction to the graded Zinbiel cup product on equivariant Leibniz cohomology.
Reviewer: Friedrich Wagemann (Nantes)Complexity measures from geometric actions on Virasoro and Kac-Moody orbits.https://www.zbmath.org/1456.813712021-04-16T16:22:00+00:00"Erdmenger, Johanna"https://www.zbmath.org/authors/?q=ai:erdmenger.johanna"Gerbershagen, Marius"https://www.zbmath.org/authors/?q=ai:gerbershagen.marius"Weigel, Anna-Lena"https://www.zbmath.org/authors/?q=ai:weigel.anna-lenaSummary: We further advance the study of the notion of computational complexity for 2d CFTs based on a gate set built out of conformal symmetry transformations. Previously, it was shown that by choosing a suitable cost function, the resulting complexity functional is equivalent to geometric (group) actions on coadjoint orbits of the Virasoro group, up to a term that originates from the central extension. We show that this term can be recovered by modifying the cost function, making the equivalence exact. Moreover, we generalize our approach to Kac-Moody symmetry groups, finding again an exact equivalence between complexity functionals and geometric actions. We then determine the optimal circuits for these complexity measures and calculate the corresponding costs for several examples of optimal transformations. In the Virasoro case, we find that for all choices of reference state except for the vacuum state, the complexity only measures the cost associated to phase changes, while assigning zero cost to the non-phase changing part of the transformation. For Kac-Moody groups in contrast, there do exist non-trivial optimal transformations beyond phase changes that contribute to the complexity, yielding a finite gauge invariant result. Moreover, we also show that our Virasoro complexity proposal is equivalent to the on-shell value of the Liouville action, which is a complexity functional proposed in the context of path integral optimization. This equivalence provides an interpretation for the path integral optimization proposal in terms of a gate set and reference state. Finally, we further develop a new proposal for a complexity definition for the Virasoro group that measures the cost associated to non-trivial transformations beyond phase changes. This proposal is based on a cost function given by a metric on the Lie group of conformal transformations. The minimization of the corresponding complexity functional is achieved using the Euler-Arnold method yielding the Korteweg-de Vries equation as equation of motion.Reduction of Nambu-Poisson manifolds by regular distributions.https://www.zbmath.org/1456.530672021-04-16T16:22:00+00:00"Das, Apurba"https://www.zbmath.org/authors/?q=ai:das.apurbaSummary: The version of Marsden-Ratiu reduction theorem for Nambu-Poisson manifolds by a regular distribution has been studied by \textit{R. Ibáñez} et al. [Rep. Math. Phys. 42, No. 1-2, 71--90 (1998; Zbl 0931.37024)]. In this paper we show that the reduction is always ensured unless the distribution is zero. Next we extend the more general Falceto-Zambon Poisson reduction theorem for Nambu-Poisson manifolds. Finally, we define gauge transformations of Nambu-Poisson structures and show that these transformations commute with the reduction procedure.Duality and symmetry in chiral Potts model.https://www.zbmath.org/1456.823192021-04-16T16:22:00+00:00"Roan, Shi-shyr"https://www.zbmath.org/authors/?q=ai:roan.shishyrRelaxing unimodularity for Yang-Baxter deformed strings.https://www.zbmath.org/1456.831032021-04-16T16:22:00+00:00"Hronek, Stanislav"https://www.zbmath.org/authors/?q=ai:hronek.stanislav"Wulff, Linus"https://www.zbmath.org/authors/?q=ai:wulff.linusSummary: We consider so-called Yang-Baxter deformations of bosonic string sigma- models, based on an \(R\)-matrix solving the (modified) classical Yang-Baxter equation. It is known that a unimodularity condition on \(R\) is sufficient for Weyl invariance at least to two loops (first order in \(\alpha^\prime)\). Here we ask what the necessary condition is. We find that in cases where the matrix \((G + B)_{ mn }\), constructed from the metric and \(B\)-field of the undeformed background, is degenerate the unimodularity condition arising at one loop can be replaced by weaker conditions. We further show that for non-unimodular deformations satisfying the one-loop conditions the Weyl invariance extends at least to two loops (first order in \(\alpha^\prime)\). The calculations are simplified by working in an \(O(D,D)\)-covariant doubled formulation.Lie symmetry analysis and similarity solutions for the Camassa-Choi equations.https://www.zbmath.org/1456.351662021-04-16T16:22:00+00:00"Paliathanasis, Andronikos"https://www.zbmath.org/authors/?q=ai:paliathanasis.andronikosSummary: The method of Lie symmetry analysis of differential equations is applied to determine exact solutions for the Camassa-Choi equation and its generalization. We prove that the Camassa-Choi equation is invariant under an infinity-dimensional Lie algebra, with an essential five-dimensional Lie algebra. The application of the Lie point symmetries leads to the construction of exact similarity solutions.Vertex operators, solvable lattice models and metaplectic Whittaker functions.https://www.zbmath.org/1456.820972021-04-16T16:22:00+00:00"Brubaker, Ben"https://www.zbmath.org/authors/?q=ai:brubaker.ben"Buciumas, Valentin"https://www.zbmath.org/authors/?q=ai:buciumas.valentin"Bump, Daniel"https://www.zbmath.org/authors/?q=ai:bump.daniel"Gustafsson, Henrik P. A."https://www.zbmath.org/authors/?q=ai:gustafsson.henrik-p-aThis paper discusses two mechanisms by which the quantum groups \(U_q (\hat{\mathfrak{g}})\), for a simple Lie algebra or superalgebra \(\mathfrak{g}\), produce families of special functions with a number of interesting properties related to functional equations, branching rules and unexpected algebraic relations. The first mechanism uses solvable lattice models associated to finite-dimensional modules of \(U_q (\hat{\mathfrak{g}})\). The second mechanism uses actions of Heisenberg and Clifford algebras on a fermionic Fock space, exploiting the boson-fermion correspondence arising in connection with soliton theory, dating back to [\textit{M. Jimbo} and \textit{T. Miwa}, Publ. Res. Inst. Math. Sci. 19, 943--1001 (1983; Zbl 0557.35091)] and pushed forward by \textit{T. Lam} [Math. Res. Lett. 13, No. 2--3, 377--392 (2006; Zbl 1160.05056)] and especially by [\textit{M. Kashiwara} et al., Sel. Math., New Ser. 1, No. 4, 787--805 (1995; Zbl 0857.17013)]. These two points of view provide new insight into the theory of metaplectic Whittaker functions for the general linear group and relate them to LLT polynomials (known also as ribbon symmetric functions). The main theorem of the paper considers two solvable lattice models, named Gamma ice and Delta, and details in Section 4 their row transfer matrices. In this study, metaplectic ice models are exploited, whose partition functions are metaplectic Whittaker functions. In the process, the authors introduce new symmetric functions termed metaplectic symmetric functions and explain how they are related to Whittaker functions. It is explained that half vertex operators agree with Lam's construction, and this interpretation allows for many new identities for metaplectic symmetric and Whittaker functions, including Cauchy identities. While both metaplectic symmetric functions and LLT polynomials [\textit{A. Lascoux} et al., J. Math. Phys. 38, No. 2, 1041--1068 (1997; Zbl 0869.05068)] can be related to vertex operators on the quantum Fock space, only metaplectic symmetric functions are connected to solvable lattice models. A number of links with the existing literature is identified as well.
Reviewer: Piotr Garbaczewski (Opole)Classifications of \(\Gamma\)-colored \(d\)-complete posets and upper \(P\)-minuscule Borel representations.https://www.zbmath.org/1456.051762021-04-16T16:22:00+00:00"Strayer, Michael C."https://www.zbmath.org/authors/?q=ai:strayer.michael-cSummary: The \(\Gamma\)-colored \(d\)-complete posets correspond to certain Borel representations that are analogous to minuscule representations of semisimple Lie algebras. We classify \(\Gamma\)-colored \(d\)-complete posets which specifies the structure of the associated representations. We show that finite \(\Gamma\)-colored \(d\)-complete posets are precisely the dominant minuscule heaps of \textit{J. R. Stembridge} [J. Algebra 235, No. 2, 722--743 (2001; Zbl 0973.17034)]. These heaps are reformulations and extensions of the colored \(d\)-complete posets of \textit{R. A. Proctor} [J. Algebra 213, No. 1, 272--303 (1999; Zbl 0969.05068)]. We also show that connected infinite \(\Gamma\)-colored \(d\)-complete posets are precisely order filters of the connected full heaps of \textit{R. M. Green} [Combinatorics of minuscule representations. Cambridge: Cambridge University Press (2013; Zbl 1320.17005)].On LA-Courant algebroids and Poisson Lie 2-algebroids.https://www.zbmath.org/1456.580042021-04-16T16:22:00+00:00"Jotz Lean, M."https://www.zbmath.org/authors/?q=ai:jotz.madeleineT.J. Courant discovered a skew-symmetric Lie bracket on \(TM \oplus T^* M\). The more general structure of a Courant algebroid, links the study of constrained Hamiltonian systems with generalised complex geometry. They were studied extensively throughout the 1990s by Zhang-Ju Liu, Alan Weinstein and Ping Xu, as well as Severa and Roytenberg. The associated integrability problem is an open question to this day.
To this end, it is important to understand better these structures. Courant algebroids are ``higher'' geometric structures. This can be made precise in the following ways: Roytenberg and Severa (independently) understood them in a graded sense, namely they described them as symplectic Lie 2-algebroids. On the other hand, Courant's example fits into \textit{K. C. H. Mackenzie}'s study of multiple structures, in particular it is an example of a double Lie algebroid [J. Reine Angew. Math. 658, 193--245 (2011; Zbl 1246.53112)]. \textit{D. Li-Bland} in his PhD thesis [LA-Courant algebroids and their applications. \url{arXiv:1204.2796}] introduced a more general class of Courant algebroids (LA-Courant algebroids) which are Courant algebroid structures in the category of vector bundles. They too fit in the multiple structures studied by Mackenzie.
The paper under review studies the correspondence between LA-Courant algebroids and Poisson Lie 2-algebroids (the latter generalize symplectic Lie 2-algebroids), using the author's earlier results on split Lie 2-algebroids and self-dual 2-representations.
Reviewer: Iakovos Androulidakis (Athína)Root cones and the resonance arrangement.https://www.zbmath.org/1456.520282021-04-16T16:22:00+00:00"Gutekunst, Samuel C."https://www.zbmath.org/authors/?q=ai:gutekunst.samuel-c"Mészáros, Karola"https://www.zbmath.org/authors/?q=ai:meszaros.karola"Petersen, T. Kyle"https://www.zbmath.org/authors/?q=ai:petersen.t-kyleSummary: We study the connection between triangulations of a type \(A\) root polytope and the resonance arrangement, a hyperplane arrangement that shows up in a surprising number of contexts. Despite an elementary definition for the resonance arrangement, the number of resonance chambers has only been computed up to the \(n=8\) dimensional case. We focus on data structures for labeling chambers, such as sign vectors and sets of alternating trees, with an aim at better understanding the structure of the resonance arrangement, and, in particular, enumerating its chambers. Along the way, we make connections with similar (and similarly difficult) enumeration questions. With the root polytope viewpoint, we relate resonance chambers to the chambers of polynomiality of the Kostant partition function. With the hyperplane viewpoint, we clarify the connections between resonance chambers and threshold functions. In particular, we show that the base-2 logarithm of the number of resonance chambers is asymptotically \(n^2\).An overview and supplements to the theory of functional relations for zeta-functions of root systems.https://www.zbmath.org/1456.111702021-04-16T16:22:00+00:00"Komori, Yasushi"https://www.zbmath.org/authors/?q=ai:komori.yasushi"Matsumoto, Kohji"https://www.zbmath.org/authors/?q=ai:matsumoto.kohji"Tsumura, Hirofumi"https://www.zbmath.org/authors/?q=ai:tsumura.hirofumiSummary: We give an overview of the theory of functional relations for zeta-functions of root systems, and show some new results on functional relations involving zeta-functions of root systems of types \(B_r, D_r, A_3\) and \(C_2\). To show those new results, we use two different methods. The first method, for \(B_r, D_r, A_3\), is via generating functions, which is based on the symmetry with respect to Weyl groups, or more generally, on our theory of lattice sums of certain hyperplane arrangements. The second method for \(C_2\) is more elementary, using partial fraction decompositions.
For the entire collection see [Zbl 1446.11004].Generalized symplectic Schur functions and SUC hierarchy.https://www.zbmath.org/1456.370742021-04-16T16:22:00+00:00"Huang, Fang"https://www.zbmath.org/authors/?q=ai:huang.fang"Wang, Na"https://www.zbmath.org/authors/?q=ai:wang.naThe authors define a generalization of
symplectic Schur functions and their vertex operator realizations.
They obtain a series of
nonlinear partial differential equations of infinite order, called
symplectic universal character hierarchy; they regard it as an extension of the symplectic
Kadomtsev-Petviashvili (KP) hierarchy.
This paper is organized as
follows. Section 1 is an introduction to the subject. In Sections
2 and 3, the authors recall some results on Schur functions in general
and in relation with solutions of the KP hierarchy. In Section 3,
they define an integrable system whose tau function can be
obtained from the symplectic Schur function. In Section 4, the
authors define the generalized symplectic Schur function, which is
an extension of the symplectic Schur function and construct its
vertex operator realization. They prove that these functions can be identified as
vacuum expectation values of fermionic operators. They give
the boson-fermion correspondence for the generalized symplectic
Schur functions and construct a modified integrable
system, thus proving that the generalized symplectic Schur functions
are solutions of it.
Reviewer: Ahmed Lesfari (El Jadida)Podleś spheres for the braided quantum \(\mathrm{SU}(2)\).https://www.zbmath.org/1456.580062021-04-16T16:22:00+00:00"Sołtan, Piotr M."https://www.zbmath.org/authors/?q=ai:soltan.piotr-mikolajPodleś quantum sphere \(\mathsf{S}^2_q=\mathrm{SU}_q(2)/\mathbb{T}\), \(q\) is a real number with \(0<|q|\le 1\) is extended to the quotient of braded quantum \(\mathrm{SU}_q(2)\), \(q\) is a complex number with \(0<|q)\le 1\) [\textit{P. Kasprzak} et al., J. Noncommut. Geom. 10, No. 4, 1611--1625 (2016; Zbl 1358.81128)]. That is define (braided) quantum sphere \(\mathbb{S}^2_q\) by \(\mathbb{S}^2_q=\mathrm{SU}_q(2)/\mathbb{T}\), \(q\) a complex number \(0<|q|\le 1\).
Let \(\mathbb{S}^2_q\) be the braided quantum sphere and \(\mathsf{S}^2_q\) be the Pofleś' quantum sphere Then it is shown \(\mathbb{S}^2_q=\mathsf{S}^2_{|q|}\) (\S4. \S7, Cor. 7.3). An axiomatic definition of braided quantum sphere is also given (\S6. Remark 6.1, \S7. Def.7.2).
To define and study braded quantum \(\mathrm{SU}_q(2)\) use braided tensor product \(A\boxtimes B\). It is explained in\S2 together with related topics following [loc. cit.] and [\textit{R. Meyer} et al., Int. J. Math. 25, No. 2, Article ID 1450019, 37 p. (2014; Zbl 1315.46076)]. Braided quantum \(\mathrm{SU}_q(2)\) is explained in \S3, and shwo the algebra of functions \(C(\mathrm{SU}_q(2)\) is the uiversal \(C^\ast\)-algebras generated by two elements \(\alpha, \gamma\), together with their relations ((3.1),(3.2)) Braided quantum sphere \(\mathbb{S}^2_q\) is defined as the quotient \(\mathrm{SU}_q(2)/\mathbb{T}\) in \S4. It is shown \(C(\mathbb{S}^2_q)\) is the closed unital \(\ast\)-subalgebra of \(C(\mathrm{SU}_q(2))\) generated by \(\alpha\gamma^\ast\) and \(\gamma^\ast\gamma\) (Cor.4.3). From this corollary and [\textit{P. Podleś}, Lett. Math. Phys. 14, 193--202 (1987; Zbl 0634.46054)] \(\mathbb{S}^2_q=\mathsf{S}^2_{|q|}\) is derived.
To obtain axiomatic definition \(\mathbb{S}^2_q\), that says a compact quantum space with \(\mathbb{T}\)-action is a braided quantum sphere, if and only if there exists \(\Gamma:\mathbb{C}(\mathbb{X})\to C(\mathrm{SU}_q(2))\boxtimes_\zeta C(\mathbb{X})\);
\[(\Gamma\boxtimes_\zeta\circ\Gamma)=(\mathrm{id})\circ\Gamma=(\mathrm{id}\boxtimes_\zeta \Delta_{\mathrm{SU}_q(2)})\circ\Gamma, \]
with 4 conditions, stated in the beginning of \S6, the three dimensional irreducible representation of \(\mathrm{SU}_q(2)\) found as an irreducible subrepresentation of the tensor square of the fundamental representation is defined and studied in \S5. Then the four conditions found in \S6, from discussions in \S5 and \S4. \S7, the last Section, show that \(C^\ast\)-algebra defined by relations in \S6 indeed carry and action of the braided \(\mathrm{SU}_q(2)\) and that they coincide with Podleś sphere (Def.7,2. Cor. 7.3).
Some hard calculus in the study of the tensor product of the fundamental representation are given in Appendix.
Reviewer: Akira Asada (Takarazuka)Color Lie rings and PBW deformations of skew group algebras.https://www.zbmath.org/1456.170162021-04-16T16:22:00+00:00"Fryer, S."https://www.zbmath.org/authors/?q=ai:fryer.sian"Kanstrup, T."https://www.zbmath.org/authors/?q=ai:kanstrup.tina"Kirkman, E."https://www.zbmath.org/authors/?q=ai:kirkman.ellen-e"Shepler, A. V."https://www.zbmath.org/authors/?q=ai:shepler.anne-v"Witherspoon, S."https://www.zbmath.org/authors/?q=ai:witherspoon.sarah-jThe authors study color Lie rings over finite group algebras and the corresponding universal enveloping algebras [\textit{M. Scheunert}, J. Math. Phys. 20, 712--720 (1979; Zbl 0423.17003)]. More precisely, they consider such rings that arise from finite abelian groups acting diagonally on a finite dimensional vector space over a field of characteristic 0 and prove that their universal enveloping algebras can be presented as quantum Drinfeld orbifold algebras [\textit{P. Shroff}, Commun. Algebra 43, No. 4, 1563--1570 (2015; Zbl 1332.16020)]. The proof is mainly based on the theory of PBW deformations and related tools (see, e.g. [\textit{P. Shroff} and \textit{S. Witherspoon}, J. Algebra Appl. 15, No. 3, Article ID 1650049, 15 p. (2016; Zbl 1345.16025)]. As an application they show that these algebras are braided Hopf algebras.
Reviewer: Aleksandr G. Aleksandrov (Moskva)The maximum dimension of a Lie nilpotent subalgebra of \(\mathbb{M}_n(F)\) of index \(m\).https://www.zbmath.org/1456.160272021-04-16T16:22:00+00:00"Szigeti, J."https://www.zbmath.org/authors/?q=ai:szigeti.jeno"Den Berg, J. van"https://www.zbmath.org/authors/?q=ai:van-den-berg.rob"van Wyk, L."https://www.zbmath.org/authors/?q=ai:van-wyk.leon"Ziembowski, M."https://www.zbmath.org/authors/?q=ai:ziembowski.michalThis paper under review is an attempt to answer a conjecture posed by \textit{J. Szigeti} and \textit{L. Van Wyk} in [Commun. Algebra 43, No. 11, 4783--4796 (2015; Zbl 1333.16003)]. The statement of this conjecture is rendered less cumbersome if expressed in terms of a function \(M(\ell, n)\) of positive integer arguments \(\ell\) and \(n\), defined as follows:
\[
\begin{aligned}
M(\ell, n) = & \max \Bigg\{\, \frac{1}{2} \left(n^2- \sum \limits _{i=1}^{\ell} k_i^2\right)+1 \, | \, k_1, k_2, \cdots, k_{\ell}\,\\
&
\text{are nonnegative integers such that}\, \sum_{i=1}^{\ell} k_i=n \, \Bigg\}
\end{aligned}
\]
Szigeti and van Wyk's conjecture is the following: if \(F\) is any field and \(R\) any \(F\)-subalgebra of the algebra
\(\mathbb{M}_n(F)\) of \(n\times n\) matrices over \(F\) with Lie nilpotence index \(m\), then
\[
\dim_F R\leq M(m+1, n).
\]
This conjecture is eventually solved in the current joint work by the four authors. The case \(m=1\) reduces to a classical theorem of \textit{I. Schur} [J. Reine Angew. Math. 130, 66--76 (1905; JFM 36.0140.01)], later generalized by \textit{N. Jacobson} [Bull. Am. Math. Soc. 50, 431--436 (1944; Zbl 0063.03016)] to all fields, which asserts that if \(F\) is an algebraically closed field of characteristic zero and \(R\) is any commutative F-subalgebra of \(\mathbb{M}_n(F)\), then \(\dim_FR\leq \lfloor \frac{n^2}{4} \rfloor+1\). Examples constructed from block upper triangular matrices show that the upper bound of \(M(m+1, n)\) cannot be lowered for any choice of \(m\) and \(n\). An explicit formula for \(M(m+1, n)\) is also derived simultaneously.
Reviewer: Wei Feng (Beijing)Code algebras, axial algebras and VOAs.https://www.zbmath.org/1456.170152021-04-16T16:22:00+00:00"Castillo-Ramirez, Alonso"https://www.zbmath.org/authors/?q=ai:castillo-ramirez.alonso"McInroy, Justin"https://www.zbmath.org/authors/?q=ai:mcinroy.justin-f"Rehren, Felix"https://www.zbmath.org/authors/?q=ai:rehren.felixSummary: Inspired by code vertex operator algebras (VOAs) and their representation theory, we define code algebras, a new class of commutative non-associative algebras constructed from binary linear codes. Let \(C\) be a binary linear code of length \(n\). A basis for the code algebra \(A_C\) consists of \(n\) idempotents and a vector for each non-constant codeword of \(C\). We show that code algebras are almost always simple and, under mild conditions on their structure constants, admit an associating bilinear form. We determine the Peirce decomposition and the fusion law for the idempotents in the basis, and we give a construction to find additional idempotents, called the \(s\)-map, which comes from the code structure. For a general code algebra, we classify the eigenvalues and eigenvectors of the smallest examples of the \(s\)-map construction, and hence show that certain code algebras are axial algebras. We give some examples, including that for a Hamming code \(H_8\) where the code algebra \(A_{H_8}\) is an axial algebra and embeds in the code VOA \(V_{H_8}\).Fusion hierarchies, \(T\)-systems, and \(Y\)-systems of logarithmic minimal models.https://www.zbmath.org/1456.813912021-04-16T16:22:00+00:00"Morin-Duchesne, Alexi"https://www.zbmath.org/authors/?q=ai:morin-duchesne.alexi"Pearce, Paul A."https://www.zbmath.org/authors/?q=ai:pearce.paul-a"Rasmussen, Jørgen"https://www.zbmath.org/authors/?q=ai:rasmussen.jorgen-h|rasmussen.jorgen|rasmussen.jorgen-bornShifted derived Poisson manifolds associated with Lie pairs.https://www.zbmath.org/1456.530652021-04-16T16:22:00+00:00"Bandiera, Ruggero"https://www.zbmath.org/authors/?q=ai:bandiera.ruggero"Chen, Zhuo"https://www.zbmath.org/authors/?q=ai:chen.zhuo"Stiénon, Mathieu"https://www.zbmath.org/authors/?q=ai:stienon.mathieu"Xu, Ping"https://www.zbmath.org/authors/?q=ai:xu.pingSummary: We study the shifted analogue of the ``Lie-Poisson'' construction for \(L_\infty\) algebroids and we prove that any \(L_\infty\) algebroid naturally gives rise to shifted derived Poisson manifolds. We also investigate derived Poisson structures from a purely algebraic perspective and, in particular, we establish a homotopy transfer theorem for derived Poisson algebras. As an application, we prove that, given a Lie pair \((L, A)\), the space \(\mathrm{tot}\Omega^{\bullet}_A(\Lambda^\bullet (L/A))\) admits a degree \((+1)\) derived Poisson algebra structure with the wedge product as associative multiplication and the Chevalley-Eilenberg differential \(d_A^{\mathrm{Bott}}:\Omega^{\bullet}_A(\Lambda^\bullet(L/A))\rightarrow\Omega^{\bullet+1}_A(\Lambda^\bullet (L/A))\) as unary \(L_\infty\) bracket. This degree \((+1)\) derived Poisson algebra structure on \(\mathrm{tot}\Omega^{\bullet}_A(\Lambda^\bullet(L/A))\) is unique up to an isomorphism having the identity map as first Taylor coefficient. Consequently, the Chevalley-Eilenberg hypercohomology \(\mathbb{H}(\mathrm{tot}\Omega^{\bullet}_A(\Lambda^\bullet (L/A)),d_A^{\mathrm{Bott}})\) admits a canonical Gerstenhaber algebra structure.Fractional supersymmetric quantum mechanics and lacunary Hermite polynomials.https://www.zbmath.org/1456.812062021-04-16T16:22:00+00:00"Bouzeffour, F."https://www.zbmath.org/authors/?q=ai:bouzeffour.fethi"Garayev, M."https://www.zbmath.org/authors/?q=ai:garayev.mubariz-tapdigogluSummary: We consider a realization of fractional supersymmetric of quantum mechanics of order \(r\), where the Hamiltonian and supercharges involve reflection operators. It is shown that the Hamiltonian has \(r\)-fold degenerate spectrum and the eigenvalues of hermitian supercharges are zeros of the associated Hermite polynomials of Askey and Wimp. Also it is shown that the associated eigenfunctions involve lacunary Hermite polynomials.Recursion operators and hierarchies of mKdV equations related to the Kac-Moody algebras \(D_4^{(1)}\), \(D_4^{(2)}\), and \(D_4^{(3)}\).https://www.zbmath.org/1456.370722021-04-16T16:22:00+00:00"Gerdjikov, V. S."https://www.zbmath.org/authors/?q=ai:gerdzhikov.vladimir-stefanov"Stefanov, A. A."https://www.zbmath.org/authors/?q=ai:stefanov.aleksander-a"Iliev, I. D."https://www.zbmath.org/authors/?q=ai:iliev.ilya-d|iliev.iliya-dimov|iliev.ilija-d"Boyadjiev, G. P."https://www.zbmath.org/authors/?q=ai:boyadjiev.g-p"Smirnov, A. O."https://www.zbmath.org/authors/?q=ai:smirnov.alexander-o"Matveev, V. B."https://www.zbmath.org/authors/?q=ai:matveev.vladimir-b"Pavlov, M. V."https://www.zbmath.org/authors/?q=ai:pavlov.maxim-vSummary: We construct three nonequivalent gradings in the algebra \(D_4\simeq so(8)\). The first is the standard grading obtained with the Coxeter automorphism \(C_1=S_{\alpha_2}S_{\alpha_1}S_{\alpha_3}S_{\alpha_4}\) using its dihedral realization. In the second, we use \(C_2=C_1R\), where \(R\) is the mirror automorphism. The third is \(C_3=S_{\alpha_2}S_{\alpha_1}T\), where \(T\) is the external automorphism of order 3. For each of these gradings, we construct a basis in the corresponding linear subspaces \(\mathfrak{g}^{(k)} \), the orbits of the Coxeter automorphisms, and the related Lax pairs generating the corresponding modified Korteweg-de Vries (mKdV) hierarchies. We find compact expressions for each of the hierarchies in terms of recursion operators. Finally, we write the first nontrivial mKdV equations and their Hamiltonians in explicit form. For \(D_4^{(1)} \), these are in fact two mKdV systems because the exponent 3 has the multiplicity two in this case. Each of these mKdV systems consists of four equations of third order in \(\partial_x\). For \(D_4^{(2)} \), we have a system of three equations of third order in \(\partial_x\). For \(D_4^{(3)}\), we have a system of two equations of fifth order in \(\partial_x\).Some combinatorial coincidences for standard representations of affine Lie algebras.https://www.zbmath.org/1456.170142021-04-16T16:22:00+00:00"Primc, Mirko"https://www.zbmath.org/authors/?q=ai:primc.mirkoThe study of combinatorial bases of Feigin-Stoyanovsky's type subspaces on the one hand and of standard modules for affine Lie algebras on the other has been very important in the last few decades. The author shows that a basis for \(W_{B_2^{(1)}}(k\Lambda_0)\) and \(L_{A_1^{(1)}}(k\Lambda_0)\) have the same combinatorial description of the difference conditions for colored partitions parametrizing monomial bases. Likewise, \(W_{C_{2\ell}^{(1)}}(k\Lambda_0)\) and \(L_{C_{\ell}^{(1)}}(k\Lambda_0)\) have the same combinatorial description of the leading terms of relations. So one could think of exploiting these types of coincidences to overcome some difficulties in the study of standard modules.
For the entire collection see [Zbl 1433.17002].
Reviewer: Stefano Capparelli (Roma)Chiral algebra, localization, modularity, surface defects, and all that.https://www.zbmath.org/1456.813682021-04-16T16:22:00+00:00"Dedushenko, Mykola"https://www.zbmath.org/authors/?q=ai:dedushenko.mykola"Fluder, Martin"https://www.zbmath.org/authors/?q=ai:fluder.martinThe authors study Lagrangian \(\mathcal{N} = 2\) superconformal field theories in four dimensions.
By employing supersymmetric localization on a rigid background of the form \(S^3 \times S^1_y\) they explicitly localize a given Lagrangian superconformal field theory and obtain the corresponding two-dimensional vertex operator algebra VOA (chiral algebra) on the torus \(S^1\times S^1_y\subset S^3\times S^1_y\). To derive the VOA the authors define the appropriate rigid supersymmetric \(S^3 \times S^1_y\) background reproducing the superconformal index. They analyze the supersymmetry algebra and classify the possible fugacities and their preserved subalgebras. Although the minimal amount of supersymmetry needed to retain the VOA construction is \(\mathfrak{su}(1|1)_\ell\times \mathfrak{su}(1|1)_r\) it appears that it is possible to turn on fugacities preserving an \(\mathfrak{su}(1|1)_\ell\times \mathfrak{su}(2|1)_r\) subalgebra which can be further broken to the minimal one by defects. Specifically, discrete fugacities \(M,N \in \mathbb{Z}\) can be turned on. The authors argue that these deformations do not affect the VOA construction but change the complex structure of the
torus and affect the boundary conditions (spin structure) upon going around one of the cycles, \(S^1_y\)
The authors address the two-dimensional theory corresponding to the localization of the \(\mathcal{N} = 2\) vector multiplets and hypermultiplets. In the latter case they show that the remnant classical piece in the localization precisely reduces to the two-dimensional symplectic boson theory on the boundary torus \(S^1\times S^1_y\). The authors show that in the presence of flavor holonomies, which appear as mass-like central charges in the supersymmetry algebra, vertex operators charged under the flavor symmetries fail to remain holomorphic while the sector that remains holomorphic is formed by flavor-neutral operators.
The authors study the modular properties of the four-dimensional Schur index. They introduce formal partition functions \(Z^{(\nu_1,\nu_2)}_{(m,n)}\), which are defined as the partition function in the given spin structure \((\nu_1,\nu_2)\), but with the modified contour of the holonomy integral in the localization formula, labeled by two integers \(m\) and \(n\). The authors suggest that the objects \(Z^{(\nu_1,\nu_2)}_{(m,n)}\) furnish an infinite-dimensional projective representation of \(\mathrm{SL}(2,\mathbb{Z})\).
Finally the authors comment on the flat \(\Omega\)-background underlying the chiral algebra.
Reviewer: Farhang Loran (Isfahan)Fusion algebra of critical percolation.https://www.zbmath.org/1456.812182021-04-16T16:22:00+00:00"Rasmussen, Jørgen"https://www.zbmath.org/authors/?q=ai:rasmussen.jorgen|rasmussen.jorgen-born|rasmussen.jorgen-h"Pearce, Paul A."https://www.zbmath.org/authors/?q=ai:pearce.paul-aCohomologically rigid solvable Leibniz algebras with nilradical of arbitrary characteristic sequence.https://www.zbmath.org/1456.170042021-04-16T16:22:00+00:00"Mamadaliev, U. Kh."https://www.zbmath.org/authors/?q=ai:mamadaliev.u-kh"Omirov, B. A."https://www.zbmath.org/authors/?q=ai:omirov.bakhrom-aThe authors describe the \((n + s)\)-dimensional solvable Leibniz algebras whose nilradical has characteristic sequence \((m_1,\ldots, m_s)\), where \(m_1+\ldots+ m_s=n\). Moreover, the authors prove that such Leibniz algebras \(L\) have the trivial center, every derivation of \(L\) is inner, and the second cohomology group \(HL^2(L,L)\) is trivial.
Reviewer: Alexandre P. Pojidaev (Novosibirsk)Quasi-particle bases of principal subspaces of affine Lie algebras.https://www.zbmath.org/1456.170132021-04-16T16:22:00+00:00"Butorac, Marijana"https://www.zbmath.org/authors/?q=ai:butorac.marijanaThis paper is a survey of recent results obtained by the author on the construction, by means of quasi-particles, of combinatorial bases of principal subspaces of the generalized Verma module \(N(k\Lambda_0)\) and of the standard module \(L(k\Lambda_0)\) for the algebras \(D_4^{(1)}\), \(C_\ell^{(1)}\), \(G_2^{(1)}\), \(B_\ell^{(1)}\). Characters of principal subspaces are also obtained.
For the entire collection see [Zbl 1433.17002].
Reviewer: Stefano Capparelli (Roma)Vertex algebras and 4-manifold invariants.https://www.zbmath.org/1456.580182021-04-16T16:22:00+00:00"Dedushenko, Mykola"https://www.zbmath.org/authors/?q=ai:dedushenko.mykola"Gukov, Sergei"https://www.zbmath.org/authors/?q=ai:gukov.sergei"Putrov, Pavel"https://www.zbmath.org/authors/?q=ai:putrov.pavelFor the entire collection see [Zbl 1408.14005].Dirac operator on the quantum fuzzy four-sphere \(S_{q F}^4\).https://www.zbmath.org/1456.811712021-04-16T16:22:00+00:00"Lotfizadeh, M."https://www.zbmath.org/authors/?q=ai:lotfizadeh.mSummary: \(q\)-deformed fuzzy Dirac and chirality operators on quantum fuzzy four-sphere \(S_{q F}^4\) are studied in this article. Using the \(q\)--deformed fuzzy Ginsparg-Wilson algebra, the \(q\)--deformed fuzzy Dirac and chirality operators in an instanton and no-instanton sector are studied. In addition, gauged Dirac and chirality operators in both cases have also been constructed. It has been shown that in each step, our results have a correct commutative limit in the limit case when \(q \rightarrow 1\) and the noncommutative parameter \(l\) tends to infinity.
{\copyright 2021 American Institute of Physics}Sklyanin-like algebras for \((q\)-)linear grids and \((q\)-)para-Krawtchouk polynomials.https://www.zbmath.org/1456.812652021-04-16T16:22:00+00:00"Bergeron, Geoffroy"https://www.zbmath.org/authors/?q=ai:bergeron.geoffroy"Gaboriaud, Julien"https://www.zbmath.org/authors/?q=ai:gaboriaud.julien"Vinet, Luc"https://www.zbmath.org/authors/?q=ai:vinet.luc"Zhedanov, Alexei"https://www.zbmath.org/authors/?q=ai:zhedanov.alexei-sSummary: S-Heun operators on linear and \(q\)-linear grids are introduced. These operators are special cases of Heun operators and are related to Sklyanin-like algebras. The continuous Hahn and big \(q\)-Jacobi polynomials are functions on which these S-Heun operators have natural actions. We show that the S-Heun operators encompass both the bispectral operators and Kalnins and Miller's structure operators. These four structure operators realize special limit cases of the trigonometric degeneration of the original Sklyanin algebra. Finite-dimensional representations of these algebras are obtained from a truncation condition. The corresponding representation bases are finite families of polynomials: the para-Krawtchouk and \(q\)-para-Krawtchouk ones. A natural algebraic interpretation of these polynomials that had been missing is thus obtained. We also recover the Heun operators attached to the corresponding bispectral problems as quadratic combinations of the S-Heun operators.
{\copyright 2021 American Institute of Physics}The ubiquitous ``c '': from the Stefan-Boltzmann law to quantum information*.https://www.zbmath.org/1456.813632021-04-16T16:22:00+00:00"Cardy, John"https://www.zbmath.org/authors/?q=ai:cardy.john-lLocal smooth conjugations of Frobenius endomorphisms.https://www.zbmath.org/1456.390042021-04-16T16:22:00+00:00"Kalnitsky, V. S."https://www.zbmath.org/authors/?q=ai:kalnitskij.v-s"Petrov, A. N."https://www.zbmath.org/authors/?q=ai:petrov.andrei-n|petrov.a-n|petrov.alexander-nSummary: A generalization of the Böttcher equation is considered. It turned out that the parametrized Poisson integral, as a function of its parameters, satisfies an equation of the type described. The structure theorem for splitting maps of Frobenius endomorphisms in a ring and in an algebra over it is proved. The real field case is considered. The generalized Böttcher equation is solved for classical two-dimensional algebras and for the Poisson algebra.Linear Batalin-Vilkovisky quantization as a functor of \(\infty \)-categories.https://www.zbmath.org/1456.180182021-04-16T16:22:00+00:00"Gwilliam, Owen"https://www.zbmath.org/authors/?q=ai:gwilliam.owen"Haugseng, Rune"https://www.zbmath.org/authors/?q=ai:haugseng.runeThe authors consider a categorical construction of linear Batalin-Vilkovisky quantization in a derived setting.
The basic example that is the starting point for this article is the Weyl quantization, sending a symplectic vector space \(\mathbb R^{2n}\) to the Weyl algebra on \(2n\) generators. One can factor this construction as taking a vector space with a skew-symmetric form first to its Heisenberg Lie algebra and then to its universal envelopping algebra. The specalization at \(\hbar = 0\) of this universal envelopping algebra is a Poisson algebra and the specializiation at \(\hbar = 1\) is its quantizaiton.
The authors consider a special case of the shifted derived versions of this problem: Their starting point are chain complexes equipped with a 1-shifted symmetric pairing. Following the article we will call them quadratic modules for short.
They then construct \(\infty\)-categorical versions of both the Heisenberg Lie algebra (which is actually a shifted \(L_\infty\)-algebra) of a quadratic module, and the universal enveloping \(BD\)-algebra of a shifted Lie algebra. Both of these appear to be of independent interest.
The universal enveloping \(BD\)-algebra is a so-called Beilinson-Drinfeld algebra, a \(k[\hbar]\)-algebra over a certain operad that specialises to a shifted Poisson algebra at \(\hbar = 0\) and to an \(E_0\)-algebra at \(\hbar = 1\). (An \(E_0\)-algebra is just a pointed chain complex, but this is the correct edge case of the notion of \(E_n\)-algebras. The classical, unshifted case involves an unshifted Poisson algebra and an \(E_1\)-algebra (i.e.\ an associative algebra) as specializiations.)
Thus the authors are able to construct linear BV quantization as a symmetric monoidal \(\infty\)-functor from quadratic algebras to \(BD\)-algebras.
The proofs involve a mixture of categorical techniques (model, simplicial and \(\infty\)).
One upside of the \(\infty\)-categorical approach is that by using Lurie's descent theorem the author can consider linear BV quantization for sheaves of quadratic modules on derived stacks. Thus they are able to show that the graded vector bundle \(V \oplus V^\vee[1]\) with its obvious quadratic form quantizes to a line bundle. This is an explicit example of the BV formalism ``behaving like a determinant'', an idea the authors credit to K. Costello. The paper also provides an example that the behaviour for more general 1-shifted symplectic modules is more complicated and the quantization need only be invertible in the formal neighbourhood of a point.
The paper under review contains some interesting discussions in the introduction: Section 1.3 considers higher BV quantizations (which should arise from more general \((1-n)\)-shifted skew-symmetric forms) and a possible application to quantization of AKSZ field theories. Section 1.4 discusses the physical perspective on linear BV quantizations, providing useful context and motivation.
Reviewer: Julian Holstein (Hamburg)Ergodic actions of compact quantum groups from solutions of the conjugate equations.https://www.zbmath.org/1456.370112021-04-16T16:22:00+00:00"Pinzari, Claudia"https://www.zbmath.org/authors/?q=ai:pinzari.claudia"Roberts, John E."https://www.zbmath.org/authors/?q=ai:roberts.john-eliasSummary: We use a tensor \(C^{\ast}\)-category with conjugates and two quasitensor functors into the category of Hilbert spaces to define a \({}^{\ast}\)-algebra depending functorially on this data. If one of them is tensorial, we can complete in the maximal \(C^{\ast}\)-norm. A particular case of this construction allows us to begin with solutions of the conjugate equations and associate ergodic actions of quantum groups on the \(C^{\ast}\)-algebra in question. The quantum groups involved are \(A_{u}(Q)\) and \(B_{u}(Q)\).On free Gelfand-Dorfman-Novikov-Poisson algebras and a PBW theorem.https://www.zbmath.org/1456.170012021-04-16T16:22:00+00:00"Bokut, L. A."https://www.zbmath.org/authors/?q=ai:bokut.leonid-a"Chen, Yuqun"https://www.zbmath.org/authors/?q=ai:chen.yuqun"Zhang, Zerui"https://www.zbmath.org/authors/?q=ai:zhang.zeruiGelfand-Dorfman-Novikov algebras (GDN algebras, known also as Novikov algebras) were introduced independently in [\textit{I. M. Gel'fand} and \textit{I. Ya. Dorfman}, Funkts. Anal. Prilozh. 13, No. 4, 13--30 (1979; Zbl 0428.58009); translation in Funct. Anal. Appl. 13, 248--262 (1980; Zbl 0437.58009)]
in connection with Hamiltonian operators in the formal calculus of variations and in [\textit{A. A. Balinskii} and \textit{S. P. Novikov}, Sov. Math., Dokl. 32, 228--231 (1985; Zbl 0606.58018); translation from Dokl. Akad. Nauk SSSR 283, 1036--1039 (1985)] in connection with linear Poisson brackets of hydrodynamic type.
These algebras satisfy the identities of left symmetry
\[ x\circ(y\circ z)-(x\circ y)\circ z = y\circ (x\circ z)-(y\circ x)\circ z\]
and right commutativity
\[ (x\circ y)\circ z = (x\circ z)\circ y. \]
GDN-Poisson algebras were introduced in [\textit{X. Xu}, J. Algebra 185, No. 3, 905--934 (1996; Zbl 0863.17003); J. Algebra 190, No. 2, 253--279 (1997; Zbl 0872.17030)]. These are GDN algebras with an additional operation \(\cdot\) which equips the algebra with the structure of a commutative associative algebra with the compatibility conditions
\[ (x\cdot y)\circ z = x\cdot (y\circ z)\quad\text{and}\quad
(x\circ y)\cdot z-x\circ (y\cdot z) = (y\circ x)\cdot z-y\circ (x\cdot z). \]
In the paper under review the authors construct a linear basis of the free GDN-Poisson algebra.
Then they define the notion of a special GDN-Poisson admissible algebra. This is a differential algebra with two commutative associative products and some extra identities.The authors prove that any GDN-Poisson algebra can be embedded into its universal enveloping special GDN-Poisson admissible algebra.
Finally, they establish that any GDN-Poisson algebra with the identity
\[ x\circ(y\cdot z)=(x\circ y)\cdot z+(x\circ z)\cdot y\]
is isomorphic to a commutative associative differential algebra both as GDN-Poisson algebra and as a commutative associative differential algebra.
Reviewer: Vesselin Drensky (Sofia)Contact and Frobenius solvable Lie algebras with abelian nilradical.https://www.zbmath.org/1456.170092021-04-16T16:22:00+00:00"Alvarez, M. A."https://www.zbmath.org/authors/?q=ai:alvarez.maria-alejandra"Rodríguez-Vallarte, M. C."https://www.zbmath.org/authors/?q=ai:rodriguez-vallarte.maria-c"Salgado, G."https://www.zbmath.org/authors/?q=ai:salgado.gilThe authors obtain that complex Frobenius Lie algebras are decomposable, while in the real case, there are exactly two that are indecomposable. Families of both Frobenius and contact solvable Lie algebras are characterized under certain conditions. When these Lie algebras have abelian nilradical, conditions are determined for which they are double extensions of Lie algebras of codimension 2. It is shown that these algebras have a natural Z\(_2\) grading.
Reviewer: Ernest L. Stitzinger (Raleigh)An application of cubical cohomology to Adinkras and supersymmetry representations.https://www.zbmath.org/1456.814292021-04-16T16:22:00+00:00"Doran, Charles F."https://www.zbmath.org/authors/?q=ai:doran.charles-f"Iga, Kevin M."https://www.zbmath.org/authors/?q=ai:iga.kevin-m"Landweber, Gregory D."https://www.zbmath.org/authors/?q=ai:landweber.gregory-dSummary: An Adinkra is a class of graphs with certain signs marking its vertices and edges, which encodes off-shell representations of the super Poincaré algebra. The markings on the vertices and edges of an Adinkra are cochains for cubical cohomology. This article explores the cubical cohomology of Adinkras, treating these markings analogously to characteristic classes on smooth manifolds.On \(\mathcal A^{(1)}_{n-1}, \mathcal B^{(1)}_n, \mathcal C^{(1)}_n, \mathcal D^{(1)}_n, \mathcal A^{(2)}_{2n}, \mathcal A^{(2)}_{2n-1}\), and \(\mathcal D^{(2)}_{n+1}\) reflection \(K\)-matrices.https://www.zbmath.org/1456.822952021-04-16T16:22:00+00:00"Malara, R."https://www.zbmath.org/authors/?q=ai:malara.r"Lima-Santos, A."https://www.zbmath.org/authors/?q=ai:lima-santos.antonioIrreducibility criterion for a finite-dimensional highest weight representation of the \(sl_2\) loop algebra and the dimensions of reducible representations.https://www.zbmath.org/1456.822412021-04-16T16:22:00+00:00"Deguchi, Tetsuo"https://www.zbmath.org/authors/?q=ai:deguchi.tetsuo