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On the formality of the little disks operad in positive characteristicMar 21 2019Using a variant of the Boardman-Vogt tensor product, we construct an action of the Grothendieck-Teichm\"uller group on the completion of the little n-disks operad $E_n$. This action is used to establish a partial formality theorem for $E_n$ with mod $p$ ... More
Representations of Simple Hom-Lie algebrasMar 21 2019The purpose of this paper is to study representations of simple multiplicative Hom-Lie algebras. First, we provide a new proof using Killing form for characterization theorem of simple Hom-Lie algebras given by Chen and Han, then discuss the representations ... More
Research topics in finite groups and vertex algebrasMar 21 2019We suggest a few projects for studying vertex algebras with emphasis on finite group viewpoints.
Slightly trivial extensions of a fusion categoryMar 21 2019We introduce and study the notion of slightly trivial extensions of a fusion category which can be viewed as the first level of complexity of extensions. We also provide two examples of slightly trivial extensions which arise from rank $3$ fusion categories. ... More
On generalized near-group fusion categoriesMar 21 2019In this paper, we study the structure of a generalized near-group fusion category and classified it when it is slightly degenerate.
Trace decategorification of tensor product algebrasMar 20 2019We show that in ADE type the trace of Webster's categorification of a tensor product of irreducibles for the quantum group is isomorphic to a tensor product of Weyl modules for the current algebra $\dot{U}(\mathfrak{g}[t])$. This extends a result of Beliakova, ... More
Towers of solutions of qKZ equations and their applications to loop modelsMar 20 2019Cherednik's type A quantum affine Knizhnik-Zamolodchikov (qKZ) equations form a consistent system of linear $q$-difference equations for $V_n$-valued meromorphic functions on a complex $n$-torus, with $V_n$ a module over the $\mathrm{GL}_n$-type extended ... More
Integral forms in vertex operator algebras, a surveyMar 20 2019We give a brief survey of recent work on integral forms in vertex operator algebras (VOAs).
Determinants for integral forms in lattice type vertex operator algebrasMar 19 2019We prove a determinant formula for the standard integral form of a lattice vertex operator algebra.
Frobenius bimodules and flat-dominant dimensionsMar 19 2019We establish relations between Frobenius parts and between flat-dominant dimensions of algebras linked by Frobenius bimodules. This is motivated by the Nakayama conjecture and an approach of Martinez-Villa to the Auslander-Reiten conjecture on stable ... More
Generators for Coulomb branches of quiver gauge theoriesMar 18 2019We study the Coulomb branches of $3d$ $\mathcal{N}=4$ quiver gauge theories, focusing on the generators for their quantized coordinate rings. We show that there is a surjective map from a shifted Yangian onto the quantized Coulomb branch, once the deformation ... More
BiHom-Novikov algebras and infinitesimal BiHom-bialgebrasMar 18 2019We introduce and study infinitesimal BiHom-bialgebras, BiHom-Novikov algebras, BiHom-Novikov-Poisson algebras, and find some relations among these concepts. Our main result is to show how to obtain a left BiHom-pre-Lie algebra from an infinitesimal BiHom-bialgebra ... More
Free Field Realizations from the Higgs BranchMar 18 2019We present free field realizations for the associated vertex operator algebras of a number of four-dimensional $\mathcal{N}=2$ superconformal field theories. Our constructions utilize an exceptionally small set of chiral bosons whose number matches the ... More
A Dolbeault-Dirac Spectral Triple for Quantum Projective SpaceMar 18 2019The notion of a K\"ahler structure for a differential calculus was recently introduced by the second author as a framework in which to study the noncommutative geometry of the quantum flag manifolds. It was subsequently shown that any covariant positive ... More
Affine Screening Operators, Affine Laumon Spaces, and Conjectures Concerning Non-Stationary Ruijsenaars FunctionsMar 18 2019Based on the screened vertex operators associated with the affine screening operators, we introduce the formal power series f^{hat{gl}_N}(x,p|s,kappa|q,t) which we call the non-stationary Ruijsenaars function. We identify it with the generating function ... More
Quadratic differential algebras generated by Euclidean spacesMar 17 2019We define a class of quadratic differential algebras which are generated as differential graded algebras by the elements of an Euclidean space. Such a differential algebra is a differential calculus over the quadratic algebra of its elements of differential ... More
Characteristic (Fedosov-)class of a twist constructed by Drinfel'dMar 15 2019In a seminal paper Drinfel'd explained how to associate to every classical r-matrix for a Lie algebra $\lie g$ a twisting element based on $\mathcal{U}(\lie g)[[\hbar]]$, or equivalently a left invariant star product of the corresponding symplectic structure ... More
On slightly degenerate fusion categoriesMar 15 2019In this paper, we first show for a slightly degenerate pre-modular fusion category $\mathcal{C}$ that squares of dimensions of simple objects divide half of the dimension of $\mathcal{C}$, and that slightly degenerate fusion categories of FP-dimensions ... More
On slightly degenerate fusion categoriesMar 15 2019Mar 21 2019In this paper, we first show for a slightly degenerate pre-modular fusion category $\mathcal{C}$ that squares of dimensions of simple objects divide half of the dimension of $\mathcal{C}$, and that slightly degenerate fusion categories of FP-dimensions ... More
On the Bosonization of the Super Jordan PlaneMar 14 2019Let $H$ and $K$ be the bosonizations of the Jordan and super Jordan plane by the group algebra of a cyclic group; the algebra $K$ projects onto an algebra $L$ that can be thought of as the quantum Borel of $\mathfrak{sl}(2)$ at $-1$. The finite-dimensional ... More
Generalized Macdonald Functions on Fock Tensor Spaces and Duality Formula for Changing Preferred DirectionMar 14 2019An explicit formula is obtained for the generalized Macdonald functions on the $N$-fold Fock tensor spaces, calculating a certain matrix element of a composition of several screened vertex operators. As an application, we prove the factorization property ... More
An Introduction to Nonassociative PhysicsMar 13 2019We give a pedagogical introduction to the nonassociative structures arising from recent developments in quantum mechanics with magnetic monopoles, in string theory and M-theory with non-geometric fluxes, and in M-theory with non-geometric Kaluza-Klein ... More
Strands algebras and Ozsváth-Szabó's Kauffman-states functorMar 13 2019We define new differential graded algebras A(n,k,S) in the framework of Lipshitz-Ozsv\'ath-Thurston's and Zarev's strands algebras from bordered Floer homology. The algebras A(n,k,S) are meant to be strands models for Ozsv\'ath-Szab\'o's algebras B(n,k,S); ... More
Generators, relations, and homology for Ozsváth-Szabó's Kauffman-states algebrasMar 13 2019We give a generators-and-relations description of differential graded algebras recently introduced by Ozsv\'ath and Szab\'o for the computation of knot Floer homology. We also compute the homology of these algebras and determine when they are formal.
Towards a mathematical formalism for classifying phases of matterMar 13 2019We propose a unified mathematical framework for classifying phases of matter. The framework is based on different types of combinatorial structures with a notion of locality called lattices. A tensor lattice is a local prescription that associates tensor ... More
The colored Jones polynomial and Kontsevich-Zagier series for double twist knots, IIMar 12 2019Let $K_{(m,p)}$ denote the family of double twist knots where $2m-1$ and $2p$ are non-zero integers denoting the number of half-twists in each region. Using a result of Takata, we prove a formula for the colored Jones polynomial of $K_{(-m,-p)}$ and $K_{(-m,p)}$. ... More
Noncommutative gauge theories on D-branes in non-geometric backgroundsMar 12 2019We investigate the noncommutative gauge theories arising on the worldvolumes of D-branes in non-geometric backgrounds obtained by T-duality from twisted tori. We revisit the low-energy effective description of D-branes on three-dimensional T-folds, examining ... More
Fusion Rules for the Lattice Vertex Operator Algebra $V_L$Mar 12 2019For a positive-definite, even, integral lattice $L$, the lattice vertex operator algebra $V_L$ is known to be rational and $C_2$-cofinite, and thus the fusion products of its modules always exist. The fusion product of two untwisted irreducible $V_L$-modules ... More
Isotropic cuspidal functions in the Hall algebra of a quiverMar 11 2019From the structure of the category of representations of an affine cycle-free quiver, we determine an explicit linear form on the space of regular cuspidal functions over a finite field: its kernel is exactly the space of cuspidal functions. Moreover, ... More
Compact quantum groups generated by their toriMar 09 2019Associated to any closed subgroup $G\subset U_N^+$ is a family of toral subgroups $T_Q\subset G$, indexed by the unitary matrices $Q\in U_N$. The family $\{T_Q|Q\in U_N\}$ is expected to encode the main properties of $G$, and there are several conjectures ... More
A note on the half-liberation operationMar 09 2019We propose a new approach to the half-liberation question, for the compact groups $T_N\subset G_N\subset U_N$, where $T_N=\mathbb Z_2^N$. Indeed, we can construct a quantum group $T_N^*\subset G_N^*\subset U_N^*$, simply by setting $G_N^*=<G_N,T_N^*>$. ... More
Conformal embeddings in affine vertex superalgebrasMar 09 2019This paper is a natural continuation of our previous work on conformal embeddings of vertex algebras [6], [7], [8]. Here we consider conformal embeddings in simple affine vertex superalgebra $V_k(\mathfrak g)$ where $\mathfrak g=\mathfrak g_{\bar 0}\oplus ... More
Schur-Weyl type duality for quantized gl(1|1),the Burau representation of braid groups and invariants of tangled graphsMar 08 2019We show that the Schur-Weyl type duality between $gl(1|1)$ and $GL_n$ gives a natural representation-theoretic setting for the relation between reduced and non-reduced Burau representations.
Gauge Theory and Boundary IntegrabilityMar 08 2019We study the mixed topological / holomorphic Chern-Simons theory of Costello, Witten and Yamazaki on an orbifold $(\Sigma\times{\mathbb C})/{\mathbb Z}_2$, obtaining a description of lattice integrable systems in the presence of a boundary. By performing ... More
Candidate for the crystal $B(-\infty)$ for the queer Lie superalgebraMar 08 2019It is shown that the direct limit of the semistandard decomposition tableau model for polynomial representations of the queer Lie superalgebra exists, which is believed to be the crystal for the upper half of the corresponding quantum group. An extension ... More
Restricted shifted Yangians and restricted finite $W$-algebrasMar 07 2019We study the truncated shifted Yangian $Y_{n,l}(\sigma)$ over an algebraically closed field $\mathbb{k}$ of characteristic $p > 0$, which is known to be isomorphic to the finite $W$-algebra $U(\mathfrak{g}, e)$ associated to a corresponding nilpotent ... More
Quantization of Magnetic Poisson StructuresMar 07 2019We describe three perspectives on higher quantization, using the example of magnetic Poisson structures which embody recent discussions of nonassociativity in quantum mechanics with magnetic monopoles and string theory with non-geometric fluxes. We survey ... More
Fock space representation of the circle quantum groupMar 07 2019In [ arXiv:1711.07391 ] we have defined quantum groups $\mathbf{U}_\upsilon(\mathfrak{sl}(\mathbb{R}))$ and $\mathbf{U}_\upsilon(\mathfrak{sl}(S^1))$, which can be interpreted as continuous generalizations of the quantum groups of the Kac-Moody Lie algebras ... More
Untwisting twisted spectral triplesMar 06 2019We examine the index data associated to twisted spectral triples and higher order spectral triples. In particular, we show that a Lipschitz regular twisted spectral triple can always be `logarithmically dampened' through functional calculus, to obtain ... More
The Tian-Todorov Theorem Of Cyclic $A_\infty$-AlgebrasMar 05 2019Let $A$ be a finite-dimensional smooth unital cyclic $A_\infty$-algebra. Assume furthermore that $A$ satisfies the Hodge-to-de-Rham degeneration property. In this short note, we prove the non-commutative analogue of the Tian-Todorov theorem: the deformation ... More
The Tian-Todorov Theorem Of Cyclic $A_\infty$-AlgebrasMar 05 2019Mar 07 2019Let $A$ be a finite-dimensional smooth unital cyclic $A_\infty$-algebra. Assume furthermore that $A$ satisfies the Hodge-to-de-Rham degeneration property. In this short note, we prove the non-commutative analogue of the Tian-Todorov theorem: the deformation ... More
The Bogomolov-Tian-Todorov Theorem Of Cyclic $A_\infty$-AlgebrasMar 05 2019Mar 13 2019Let $A$ be a finite-dimensional smooth unital cyclic $A_\infty$-algebra. Assume furthermore that $A$ satisfies the Hodge-to-de-Rham degeneration property. In this short note, we prove the non-commutative analogue of the Bogomolov-Tian-Todorov theorem: ... More
Multi-tribracketsMar 05 2019We introduce multi-tribrackets, algebraic structures for region coloring of diagrams of knots and links with different operations at different kinds of crossings. In particular we consider the case of component multi-tribrackets which have different tribracket ... More
Quantization of continuum Kac-Moody algebrasMar 04 2019Continuum Kac-Moody algebras have been recently introduced by the authors and O. Schiffmann. These are Lie algebras governed by a continuum root system, which can be realized as uncountable colimits of Borcherds-Kac-Moody algebras. In this paper, we prove ... More
A graphical categorification of the two-variable Chebyshev polynomials of the second kindMar 04 2019We show that the $A_2$ clasps in the Karoubi envelope of $A_2$ spider satisfy the recursive formula of the two-variable Chebyshev polynomials of the second kind associated with a root system of type $A_2$. The $A_2$ spider is a diagrammatic description ... More
Quantum cluster algebras via Hall algebras of morphismsMar 03 2019We realize the quantum cluster algebra with principal coefficients as a subquotient of certain Hall algebra involving the category of morphisms between projectives.
A construction of lower-bounded generalized twisted modules for a grading-restricted vertex (super)algebraMar 02 2019We give a general, direct and explicit construction of lower-bounded generalized twisted modules satisfying a universal property for a grading-restricted vertex (super)algebra $V$ associated to an automorphism $g$ of $V$. In particular, when $g$ is the ... More
Tree series and pattern avoidance in syntax treesMar 02 2019A syntax tree is a planar rooted tree where internal nodes are labeled on a graded set of generators. There is a natural notion of occurrence of contiguous pattern in such trees. We describe a way, given a set of generators and a set of patterns, to enumerate ... More
On double quantum affinization: 1. Type $\mathfrak a_1$Mar 01 2019We define the double quantum affinization $\ddot{\mathrm{U}}_q(\mathfrak a_1)$ of type $\mathfrak{a}_1$ as a topological Hopf algebra. We prove that it admits a subalgebra $\ddot{\mathrm{U}}_q'(\mathfrak a_1)$ whose completion is (bicontinuously) isomorphic ... More
A bound for crystallographic arrangementsMar 01 2019A crystallographic arrangement is a set of linear hyperplanes satisfying a certain integrality property and decomposing the space into simplicial cones. Crystallographic arrangements were completely classified in a series of papers by Heckenberger and ... More
Boundary matrices for the higher spin six vertex modelMar 01 2019Mar 12 2019In this paper we consider solutions to the reflection equation related to the higher spin stochastic six vertex model. The corresponding higher spin $R$-matrix is associated with the affine quantum algebra $U_q(\widehat{sl(2)})$. The explicit formulas ... More
Boundary matrices for the higher spin six vertex modelMar 01 2019In this paper we consider solutions to the reflection equation related to the higher spin stochastic six vertex model. The corresponding higher spin $R$-matrix is associated with the affine quantum algebra $U_q(\widehat{sl(2)})$. The explicit formulas ... More
Isomorphism between the R-matrix and Drinfeld presentations of quantum affine algebra: type CMar 01 2019Mar 15 2019An explicit isomorphism between the $R$-matrix and Drinfeld presentations of the quantum affine algebra in type $A$ was given by Ding and I. Frenkel (1993). We show that this result can be extended to types $B$, $C$ and $D$ and give a detailed construction ... More
Isomorphism between the R-matrix and Drinfeld presentations of quantum affine algebra: type CMar 01 2019An explicit isomorphism between the $R$-matrix and Drinfeld presentations of the quantum affine algebra in type $A$ was given by Ding and I. Frenkel (1993). We show that this result can be extended to types $B$, $C$ and $D$ and give a detailed construction ... More
Geometry of the Maurer-Cartan equation near degenerate Calabi-Yau varietiesFeb 28 2019In this paper, we provide a construction of a dgBV algebra PV(X) associated to a possibly degenerate Calabi-Yau variety X equipped with local deformation data. This can be regarded as a singular version of the Kodaira-Spencer dgLa. We work in an abstract ... More
Principal subspaces for the affine Lie algebras in types $D$, $E$ and $F$Feb 27 2019We consider the principal subspaces of certain level $k\geqslant 1$ integrable highest weight modules and generalized Verma modules for the untwisted affine Lie algebras in types $D$, $E$ and $F$. Generalizing the approach of G. Georgiev we construct ... More
The Hall Algebras of AnnuliFeb 27 2019We refine and prove the central conjecture of our first paper for annuli with at least two marked intervals on each boundary component by computing the derived Hall algebras of their Fukaya categories.
Rank two false theta functions and Jacobi forms of negative definite matrix indexFeb 27 2019In this paper, we study a family of rank two false theta series associated to the root lattice of type $A_2$. We show that these functions appear as Fourier coefficients of a meromorphic Jacobi form of negative definite matrix index. Hypergeometric $q$-series ... More
A cobordism category attached to Khovanov-Rozansky link homologies based on operadsFeb 27 2019We consider colored operads and their actions on categories. As a special example we construct a cobordism category with a colored operad action arising from oriented planar arc diagrams. This is used to construct an invariant of oriented tangle diagrams ... More
A cobordism category attached to Khovanov-Rozansky link homologies based on operadsFeb 27 2019Mar 15 2019We consider colored operads and their actions on categories. As a special example we construct a cobordism category with a colored operad action arising from oriented planar arc diagrams. This is used to construct an invariant of oriented tangle diagrams ... More
Characterization of Hopf QuasigroupsFeb 26 2019In this paper, we first discuss some properties of the Galois linear maps. We provide some equivalent conditions for Hopf algebras and Hopf (co)quasigroups as its applications. Then let $H$ be a Hopf quasigroup with bijective antipode and $G$ be the set ... More
Mixed cohomology of Lie superalgebrasFeb 26 2019We investigate a new cohomology of Lie superalgebras, which may be compared to a de Rham cohomology of Lie supergroups involving both differential and integral forms. It is defined by a BRST complex of Lie superalgebra modules, which is formulated in ... More
Twist vertex operators for twisted modulesFeb 26 2019We introduce and study twist vertex operators for a (lower-bounded generalized) twisted modules for a grading-restricted vertex (super)algebra. We prove duality, weak associativity, a Jacobi identity, a generalized commutator formula, generalized weak ... More
Existence and rigidity of quantum isometry groups for compact metric spacesFeb 26 2019We prove the existence of a quantum isometry groups for new classes of metric spaces: (i) geodesic metrics for compact connected Riemannian manifolds (possibly with boundary) and (ii) metric spaces admitting a uniformly distributed probability measure. ... More
Classification of spin models for Yang-Baxter planar algebrasFeb 24 2019In this paper, we classify all spin models for singly-generated Yang-Baxter planar algebras in terms of certain highly regular graphs. Using Liu's classification of singly generated Yang-Baxter planar algebras, this classifies all spin models for the ... More
Unitary and non-unitary $N=2$ minimal modelsFeb 22 2019The unitary $N = 2$ superconformal minimal models have a long history in string theory and mathematical physics, while their non-unitary (and logarithmic) cousins have recently attracted interest from mathematicians. Here, we give an efficient and uniform ... More
Kac-Paljutkin Quantum Group as a Quantum Subgroup of the Quantum SU(2)Feb 21 2019We show that the Kac-Paljutkin Hopf algebra appears as a quotient of $C(SU_{-1}(2))$, which means that the corresponding quantum group $G_{KP}$ can be regarded as a quantum subgroup of $SU_{-1}(2)$. We combine the fact that corepresentation category of ... More
The $q$-Bannai-Ito algebra and multivariate $(-q)$-Racah and Bannai-Ito polynomialsFeb 21 2019The Gasper and Rahman multivariate $(-q)$-Racah polynomials appear as connection coefficients between bases diagonalizing different abelian subalgebras of the recently defined higher rank $q$-Bannai-Ito algebra $\mathcal{A}_n^q$. Lifting the action of ... More
On $q$-Schur algebras corresponding to Hecke algebras of type BFeb 20 2019In this paper the authors investigate the $q$-Schur algebras of type B that were constructed earlier using coideal subalgebras for the quantum group of type A. The authors present a coordinate algebra type construction that allows us to realize these ... More
Quantum toric degeneration of quantum flag and Schubert varietiesFeb 20 2019We show that certain homological regularity properties of graded connected algebras, such as being AS-Gorenstein or AS-Cohen-Macaulay, can be tested by passing to associated graded rings. In the spirit of noncommutative algebraic geometry, this can be ... More
One example of a chiral Lie groupFeb 20 2019We quantize the Khesin-Zakharevich Poisson-Lie group of pseudo-differential symbols.
On the 2-head of the colored Jones polynomial for pretzel knotsFeb 19 2019In this paper, we prove a formula for the 2-head of the colored Jones polynomial for an infinite family of pretzel knots. Following Hall, the proof utilizes skein-theoretic techniques and a careful examination of higher order stability properties for ... More
Crossed $S$-matrices and Fourier matrices for Coxeter groups with automorphismFeb 19 2019We study crossed $S$-matrices for braided $G$-crossed categories and reduce their computation to a submatrix of the de-equivariantization. We study the more general case of a category containing the symmetric category $\mathrm{Rep}(A,z)$ with $A$ a finite ... More
$6A$-Algebra and its representationsFeb 19 2019In this paper, we study the structure and representation of a $6A$-algebra which is a vertex operator algebra generated by two Ising vectors $e,f$ with inner product $\left\langle e,f\right\rangle =\frac{5}{2^{10}}.$ In particular, we prove the uniqueness ... More
$6A$-Algebra and its representationsFeb 19 2019Feb 28 2019In this paper, we study the structure and representation of a $6A$-algebra which is a vertex operator algebra generated by two Ising vectors $e,f$ with inner product $\left\langle e,f\right\rangle =\frac{5}{2^{10}}.$ In particular, we prove the uniqueness ... More
Differential graded algebra over quotients of skew polynomial rings by normal elementsFeb 18 2019Differential graded algebra techniques have played a crucial role in the development of homological algebra, especially in the study of homological properties of commutative rings carried out by Serre, Tate, Gulliksen, Avramov, and others. In this article, ... More
Rank-finiteness for G-crossed braided fusion categoriesFeb 16 2019We establish rank-finiteness for the class of $G$-crossed braided fusion categories, generalizing the recent result for modular categories and including the important case of braided fusion categories. This necessitates a study of slightly degenerate ... More
Refined scattering diagrams and theta functions from asymptotic analysis of Maurer-Cartan equationsFeb 15 2019We further develop the asymptotic analytic approach to the study of scattering diagrams. We do so by analysing the asymptotic behavior of Maurer-Cartan elements of a differential graded Lie algebra constructed from a (not-necessarily tropical) monoid-graded ... More
A quantum categorification of the Alexander polynomialFeb 15 2019Using a modified foam evaluation, we give a categorification of the Alexander polynomial of a knot. We also give a purely algebraic version of this knot homology which makes it appear as the infinite page of a spectral sequence starting at the reduced ... More
Deep Learning the Hyperbolic Volume of a KnotFeb 14 2019Feb 20 2019An important conjecture in knot theory relates the large-$N$, double scaling limit of the colored Jones polynomial $J_{K,N}(q)$ of a knot $K$ to the hyperbolic volume of the knot complement, $\text{Vol}(K)$. A less studied question is whether $\text{Vol}(K)$ ... More
Deep Learning the Hyperbolic Volume of a KnotFeb 14 2019An important conjecture in knot theory relates the large-$N$, double scaling limit of the colored Jones polynomial $J_{K,N}(q)$ of a knot $K$ to the hyperbolic volume of the knot complement, $\text{Vol}(K)$. A less studied question is whether $\text{Vol}(K)$ ... More
Donaldson-Thomas invariants from tropical disksFeb 14 2019We prove that the quantum DT invariants associated to quivers with genteel potential can be expressed in terms of certain refined counts of tropical disks. This is based on a quantum version of Bridgeland's description of cluster scattering diagrams in ... More
Donaldson-Thomas invariants from tropical disksFeb 14 2019Mar 03 2019We prove that the quantum DT invariants associated to quivers with genteel potential can be expressed in terms of certain refined counts of tropical disks. This is based on a quantum version of Bridgeland's description of cluster scattering diagrams in ... More
Morita Bicategories of Algebras and Duality InvolutionsFeb 13 2019The notion of a weak duality involution on a bicategory was recently introduced by Shulman in [arXiv:1606.05058]. We construct a weak duality involution on the fully dualisable part of $\text{Alg}$, the Morita bicategory of finite-dimensional k-algebras. ... More
Noncommutative cyclic isolated singularitiesFeb 13 2019The question of whether a noncommutative graded quotient singularity $A^G$ is isolated depends on a subtle invariant of the $G$-action on $A$, called the pertinency. We prove a partial dichotomy theorem for isolatedness, which applies to a family of noncommutative ... More
Categorical Saito theory, I: A comparison resultFeb 12 2019In this paper, we present an explicit cyclic minimal $A_\infty$ model for the category of matrix factorizations $\MF(W)$ of an isolated hypersurface singularity. The key observation is to use Kontsevich's deformation quantization technique. Pushing this ... More
Set-theoretical solutions of the pentagon equation on groupsFeb 12 2019Let $M$ be a set. A set-theoretical solution of the pentagon equation on $M$ is a map $s:M\times M\longrightarrow M\times M$ such that \begin{equation*} s_{23}\, s_{13}\, s_{12}=s_{12}\, s_{23}, \end{equation*} where $s_{12}=s\times id_M$, $s_{23}=id_M ... More
Universal enveloping Poisson conformal algebrasFeb 08 2019Lie conformal algebras are useful tools for studying vertex operator algebras and their representations. In this paper, we establish close relations between Poisson conformal algebras and representations of Lie conformal algebras. We also calculate explicitly ... More
Plethora of cluster structures on $GL_n$Feb 08 2019We continue the study of multiple cluster structures in the rings of regular functions on $GL_n$, $SL_n$ and $\operatorname{Mat}_n$ that are compatible with Poisson-Lie and Poisson-homogeneous structures. According to our initial conjecture, each class ... More
Stable maps, Q-operators and category OFeb 07 2019We define and construct stable maps on tensor products of representations in the category O of the Borel subalgebra of an arbitrary untwisted quantum affine algebra. Our purely representation-theoretical construction is based on the study of the action ... More
W algebra, Cosets and VOA for 4d N = 2 SCFT from M5 branesFeb 07 2019We identify vertex operator algebras (VOAs) of a class of Argyres-Douglas (AD) matters with two types of non-abelian flavor symmetries. They are the $W$ algebra defined using nilpotent orbit with partition $[q^m,1^s]$. Gauging above AD matters, we can ... More
Cyclotomic expansions of HOMFLY-PT colored by rectangular Young diagramsFeb 06 2019Feb 14 2019We conjecture a closed form expression of HOMFLY-PT invariants of double twist knots colored by rectangular Young diagrams where the twist is encoded in interpolation Macdonald polynomials. We also put forth a conjecture of cyclotomic expansions of HOMFLY-PT ... More
Cyclotomic expansions of HOMFLY-PT colored by rectangular Young diagramsFeb 06 2019We conjecture a closed form expression of HOMFLY-PT invariants of double twist knots colored by rectangular Young diagrams where the twist is encoded in interpolation Macdonald polynomials. We also put forth a conjecture of cyclotomic expansions of HOMFLY-PT ... More
Hopf Ore ExtensionsFeb 06 2019Brown, O'Hagan, Zhang, and Zhuang gave a set of conditions on an automorphism $\sigma$ and a $\sigma$-derivation $\delta$ of a Hopf $k$-algebra $R$ for when the skew polynomial extension $T=R[x, \sigma, \delta]$ of $R$ admits a Hopf algebra structure ... More
alpha-type Chevalley-Eilenberg cohomology of Hom-Lie algebras and bialgebrasFeb 06 2019The purpose of this paper is to define an $\alpha$-type cohomology, which we call $\alpha$-type Chevalley-Eilenberg cohomology, for Hom-Lie algebras. We relate it to the known Chevalley-Eilenberg cohomology and provide explicit computations for some examples. ... More
Dimension and Trace of the Kauffman Bracket Skein AlgebraFeb 06 2019Feb 27 2019Let $F$ be a finite type surface and $\zeta$ a complex root of unity. The Kauffman bracket skein algebra $K_{\zeta}(F)$ is an important object in both classical and quantum topology as it has relations to the character variety, the Teichm\"uller space, ... More
Dimension and Trace of the Kauffman Bracket Skein AlgebraFeb 06 2019Let $F$ be a finite type surface and $\zeta$ a complex root of unity. The Kauffman bracket skein algebra $K_{\zeta}(F)$ is an important object in both classical and quantum topology as it has relations to the character variety, the Teichm\"uller space, ... More
Quantum affine algebras and cluster algebrasFeb 04 2019This article is an extended version of the minicourse given by the second author at the summer school of the conference "Interactions of quantum affine algebras with cluster algebras, current algebras and categorification", held in June 2018 in Washington. ... More
The braided group of a square-free solution of the Yang-Baxter equation and its group algebraFeb 03 2019Set-theoretic solutions of the Yang--Baxter equation form a meeting-ground of mathematical physics, algebra and combinatorics. Such a solution $(X,r)$ consists of a set $X$ and a bijective map $r:X\times X\to X\times X$ which satisfies the braid relations. ... More
The alternating PBW basis for the positive part of $U_q(\widehat{\mathfrak{sl}}_2)$Feb 02 2019The positive part $U^+_q$ of $U_q(\widehat{\mathfrak{sl}}_2)$ has a presentation with two generators $A,B$ that satisfy the cubic $q$-Serre relations. We introduce a PBW basis for $U^+_q$, said to be alternating. Each element of this PBW basis commutes ... More