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Yang-Baxter random fields and stochastic vertex modelsMay 16 2019Bijectivization refines the Yang-Baxter equation into a pair of local Markov moves which randomly update the configuration of the vertex model. Employing this approach, we introduce new Yang-Baxter random fields of Young diagrams based on spin $q$-Whittaker ... More

Finite quotients of powers of an elliptic curveMay 16 2019Let $E$ be an elliptic curve. When the symmetric group $\Sigma_{g+1}$ of order $(g+1)!$ acts on $E^{g+1}$ in the natural way, the subgroup $E_0^{g+1}$, consisting of those $(g+1)$-tuples whose coordinates sum to zero, is stable under the action of $\Sigma_{g+1}$. ... More

Legendrian Rack Invariants of Legendrian KnotsMay 15 2019We define a new algebraic structure called Legendrian racks or racks with Legendrian structure, motivated by the front-projection Reidemeister moves for Legendrian knots. We provide examples of Legendrian racks and use these algebraic structures to define ... More

Temperley-Lieb, Brauer and Racah algebras and other centralizers of su(2)May 15 2019In the spirit of the Schur-Weyl duality, we study the connections between the Racah algebra and the centralizers of tensor products of three (possibly different) irreducible representations of su(2). As a first step we show that the Racah algebra always ... More

The PBW basis of $U_{q,\bar{q}}(\ddot{\mathfrak{gl}}_n)$May 15 2019We consider the PBW basis of the type A quantum toroidal algebra developed by the author, and prove commutation relations between its generators akin to the ones studied by Burban-Schiffmann for n=1. This gives rise to a new presentation of the quantum ... More

Factorizations of skew bracesMay 14 2019We introduce strong left ideals of skew braces and prove that they produce non-trivial decomposition of set-theoretic solutions of the Yang-Baxter equation. We study factorization of skew left braces through strong left ideals and we prove analogs of ... More

On Borel subalgebras of quantum groupsMay 14 2019For a quantum group, we study those right coideal subalgebras, for which all irreducible representations are one-dimensional. If a right coideal subalgebra is maximal with this property, then we call it a Borel subalgebra. Besides the positive part of ... More

On a Poincaré polynomial from Khovanov homology and Vassiliev invariantsMay 14 2019We introduce a Poincar\'{e} polynomial with two-variable $t$ and $x$ for knots, derived from Khovanov homology, where the specialization $(t, x)$ $=$ $(1, -1)$ is a Vassiliev invariant of order $n$. Since for every $n$, there exist non-trivial knots with ... More

The Hadamard product in a crossed product C*-algebraMay 14 2019We show that for a C*-algebra A and a discrete group G with an action of G on A, the reduced crossed product C*-algebra possesses a natural generalization of the convolution product, which we suggest should be named the Hadamard product. We show that ... More

A geometric $q$-character formula for snake modulesMay 13 2019Let $\mathscr{C}$ be the category of finite dimensional modules over the quantum affine algebra $U_q(\widehat{\mathfrak{g}})$ of a simple complex Lie algebra ${\mathfrak{g}}$. Let $\mathscr{C}^-$ be the subcategory introduced by Hernandez and Leclerc. ... More

A mathematical theory of gapless edges of 2d topological orders IMay 13 2019This is the first part of a two-part work on a unified mathematical theory of gapped and gapless edges of 2d topological orders. It was known that a 2d topological order is described by a pair $(\mathcal{C}, c)$, where $\mathcal{C}$ is a unitary modular ... More

The center of monoidal bicategories in 3+1D Dijkgraaf-Witten TheoryMay 12 2019In this work, for a finite group $G$ and a 4-cocycle $\omega \in Z^4(G,\mathbf{k}^\times)$, we compute explicitly the center of the monoidal bicategory $\operatorname{2Vec}_G^{\omega}$ of $\omega$-twisted $G$-graded 1-categories of finite dimensional ... More

Real moduli space of stable rational curves revistedMay 11 2019We give a description of the operad formed by the real locus of the moduli space of stable genus zero curves with marked points $\overline{{\mathcal M}_{0,{n+1}}}({\mathbb R})$ in terms of a homotopy quotient of an operad of associative algebras. We use ... More

Special classes of homomorphisms between generalized Verma modules for ${\mathcal U}_q(su(n,n))$May 11 2019We study homomorphisms between quantized generalized Verma modules $M(V_{\Lambda})\stackrel{\phi_{\Lambda,\Lambda_1}}{\rightarrow}M(V_{\Lambda_1})$ for ${\mathcal U}_q(su(n,n))$. There is a natural notion of degree for such maps, and if the map is of ... More

Algebras of Variable Coefficient Quantized Differential OperatorsMay 11 2019In the framework of (vector valued) quantized holomorphic functions defined on non-commutative spaces, ``quantized hermitian symmetric spaces'', we analyze what the algebras of quantized differential operators with variable coefficients should be. It ... More

Pointed Hopf algebras over non abelian groups with decomposable braidings, IMay 10 2019We describe all finite-dimensional pointed Hopf algebras whose infinitesimal braiding is a fixed Yetter-Drinfeld module decomposed as the sum of two simple objects: a point and the one of transpositions of the symmetric group in three letters. We give ... More

Defect 2 spin blocks of symmetric groups and canonical basis coefficientsMay 10 2019This paper addresses the decomposition number problem for spin representations of symmetric groups in odd characteristic. Our main aim is to find a combinatorial formula for decomposition numbers in blocks of defect $2$, analogous to Richards's formula ... More

Gluing two affine Yangians of $\mathfrak{gl}_1$May 08 2019We construct a four-parameter family of affine Yangian algebras by gluing two copies of the affine Yangian of $\mathfrak{gl}_1$. Our construction allows for gluing operators with arbitrary (integer or half integer) conformal dimension and arbitrary (bosonic ... More

Quantised Painlevé monodromy manifolds, Sklyanin and Calabi-Yau algebrasMay 07 2019In this paper we study quantum del Pezzo surfaces belonging to a certain class. In particular we introduce the generalised Sklyanin-Painlev\'e algebra and characterise its PBW/PHS/Koszul properties. This algebra contains as limiting cases the generalised ... More

Examples of finite-dimensional pointed Hopf algebras in positive characteristicMay 07 2019We present new examples of finite-dimensional Nichols algebras over fields of positive characteristic. The corresponding braided vector spaces are not of diagonal type, admit a realization as Yetter-Drinfeld modules over finite abelian groups and are ... More

Modularity and value distribution of quantum invariants of hyperbolic knotsMay 06 2019We obtain an exact modularity relation for the $q$-Pochhammer symbol. Using this formula, we show that Zagier's modularity conjecture holds for hyperbolic knots $K\neq 7_2$ with at most seven crossings. For $K=4_1$, we also prove a complementary reciprocity ... More

New symmetries for the $U_q(sl_N)$ 6-j symbols from the Eigenvalue conjectureMay 06 2019In the present paper we discuss the eigenvalue conjecture, suggested in 2012, in the particular case of $U_q(sl_2)$. The eigenvalue conjecture provides a certain symmetry for Racah coefficients and we prove that \textbf{the eigenvalue conjecture is provided ... More

The regular representation of $U_v(\mathfrak{gl}_{m|n})$May 06 2019Using quantum differential operators, we construct a super representation of $U_v(\mathfrak{gl}_{m|n})$ on a certain polynomial superalgebra. We then extend the representation to its formal power series algebra which contains a $U_v(\mathfrak{gl}_{m|n})$-submodule ... More

Representations and Modules of Rota-Baxter AlgebrasMay 04 2019We give a broad study of representation and module theory of Rota-Baxter algebras, motivated by Rota-Baxter matrix representations in the renormalization of quantum field theory and by geometric connections. Regular-singular decompositions of Rota-Baxter ... More

Toeplitz Quantization of a Free $ * $-AlgebraMay 02 2019In this note we quantize the free $ * $-algebra generated by finitely many variables, which is a new example of the theory of Toeplitz quantization of $ * $-algebras as developed previously by the author. This is achieved by defining Toeplitz operators ... More

Hopf-Frobenius algebras and a new Drinfeld doubleMay 02 2019The ZX-calculus and related theories are based on so-called interacting Frobenius algebras, where a pair of $\dag$-special commutative Frobenius algebras jointly form a pair of Hopf algebras. In this setting we introduce a generalisation of this structure, ... More

A quantized Riemann-Hilbert problem in Donaldson-Thomas theoryMay 02 2019We introduce Riemann-Hilbert problems determined by refined Donaldson-Thomas theory. They involve piecewise holomorphic maps from the complex plane to the group of automorphisms of a quantum torus algebra. We study the simplest case in detail and use ... More

Classifying Module Categories for Generalized Temperley-Lieb-Jones *-2-CategoriesMay 01 2019Generalized Temperley-Lieb-Jones (TLJ) 2-categories associated to weighted bidirected graphs were introduced in unpublished work of Morrison and Walker. We introduce unitary modules for these generalized TLJ 2-categories as strong *-pseudofunctors into ... More

Higher Arity Self-Distributive Operations in Cascades and their CohomologyMay 01 2019We investigate constructions and relations of higher arity self-distributive operations and their cohomology. We study the categories of mutually distributive structures both in the binary and ternary settings and their connections through functors. This ... More

Can tangle calculus be applicable to hyperpolynomials?May 01 2019We make a new attempt at the recently suggested program to express knot polynomials through topological vertices, which can be considered as a possible approach to the tangle calculus: we discuss the Macdonald deformation of the relation between the convolution ... More

On canonical bases of Letzter algebra $\mathbf U^{\imath}(\mathfrak{sl}_2)$Apr 30 2019Let $\mathbf U^{\imath}\equiv\mathbf U^{\imath} (\mathfrak{sl}_2)$ be Letzter's coideal subalgebra of quantum $\mathfrak{sl}_2$ corresponding to the symmetric pair $(\mathfrak{sl}_2(\mathbb C),\mathbb C)$. As a subalgebra of quantum $\mathfrak{sl}_2$, ... More

The Kontsevich graph orientation morphism revisitedApr 30 2019The orientation morphism $Or\colon\Gamma\mapsto\dot{P}$ associates differential-polynomial flows $\dot{P}=Q(P)$ on spaces of bi-vectors $P$ on finite-dimensional affine manifolds $N^d$ with (sums of) finite unoriented graphs $\Gamma$ with ordered sets ... More

Hopf algebras with enough quotientsApr 30 2019A family of algebra maps $H\to A_i$ whose common domain is a Hopf algebra is said to be jointly inner faithful if it does not factor simultaneously through a proper Hopf quotient of $H$. We show that tensor and free products of jointly inner faithful ... More

Noncommutative Geometry of Quantized CoveringsApr 30 2019This research is devoted to the noncommutative generalization of topological coverings. Otherwise since topological coverings are related to the set of geometric constructions one can obtain noncommutative generalizations of these constructions. Here ... More

Grothendieck-Teichmueller group, operads and graph complexes: a surveyApr 30 2019This paper attempts to provide a more or less self-contained introduction into theory of the Grothendieck-Teichmueller group and Drinfeld associators using the theory of operads and graph complexes.

Polynomial functors and two-parameter quantum symmetric pairsApr 29 2019Apr 30 2019We develop a theory of two-parameter quantum polynomial functors. Similar to how (strict) polynomial functors give a new interpretation of polynomial representations of the general linear groups $GL_n$, the two-parameter polynomial functors give a new ... More

Polynomial functors and two-parameter quantum symmetric pairsApr 29 2019We develop a theory of two-parameter quantum polynomial functors. Similar to how (strict) polynomial functors give a new interpretation of polynomial representations of the general linear groups $GL_n$, the two-parameter polynomial functors give a new ... More

Polynomial functors and two-parameter quantum symmetric pairsApr 29 2019May 07 2019We develop a theory of two-parameter quantum polynomial functors. Similar to how (strict) polynomial functors give a new interpretation of polynomial representations of the general linear groups $GL_n$, the two-parameter polynomial functors give a new ... More

Unitary quantum groups vs quantum reflection groupsApr 29 2019We study the intermediate liberation problem for the real and complex unitary and reflection groups, namely $O_N,U_N,H_N,K_N$. For any of these groups $G_N$, the problem is that of understanding the structure of the intermediate quantum groups $G_N\subset ... More

BGG category for the quantum Schrödinger algebraApr 29 2019In this paper, we study the BGG category $\mathcal{O}$ for the quantum Schr{\"o}dinger algebra $U_q(\mathfrak{s})$, where $q$ is a nonzero complex number which is not a root of unity. If the central charge $\dot z\neq 0$, using the module $B_{\dot z}$ ... More

On the spectrum of the local $\mathbb{P}^2$ mirror curveApr 28 2019We address the spectral problem of the normal quantum mechanical operator associated to the quantized mirror curve of the toric (almost) del Pezzo Calabi--Yau threefold called local $\mathbb{P}^2$ in the case of complex values of Planck's constant.

On the Representation theory of the Infinite Temperley-Lieb algebraApr 28 2019We begin the study of the representation theory of the infinite Temperley-Lieb algebra. We fully classify its finite dimensional representations, then introduce infinite link state representations and classify when they are irreducible or indecomposable. ... More

Deformation quantization and Kähler geometry with moment mapApr 26 2019In the first part of this paper we outline the constructions and properties of Fedosov star product and Berezin-Toeplitz star product. In the second part we outline the basic ideas and recent developments on Yau-Tian-Donaldson conjecture on the existence ... More

Retractability of solutions to the Yang-Baxter equation and $p$-nilpotency of skew bracesApr 26 2019Using Bieberbach groups we study multipermutation involutive solutions to the Yang-Baxter equation. We find a linear representation of structure groups of involutive solutions; this representation is then used to study the unique product property in such ... More

Cut-and-join structure and integrability for spin Hurwitz numbersApr 25 2019Spin Hurwitz numbers are related to characters of the Sergeev group, which are the expansion coefficients of the Q Schur functions, depending on odd times and on a subset of all Young diagrams. These characters involve two dual subsets: the odd partitions ... More

Singular crossings and Ozsváth-Szabó's Kauffman-states functorApr 24 2019Recently, Ozsv\'ath and Szab\'o introduced some algebraic constructions computing knot Floer homology in the spirit of bordered Floer homology, including a family of algebras B(n) and, for a generator of the braid group on n strands, a certain type of ... More

Adapted Sequence for Polyhedral Realization of Crystal BasesApr 24 2019The polyhedral realization of crystal base has been introduced by A.Zelevinsky and the second author([T.Nakashima, A.Zelevinsky, Adv. Math. 131, no. 1 (1997)]), which describe the crystal base $B(\infty)$ as a polyhedral convex cone in the infinite $\mathbb{Z}$-lattice ... More

Adapted Sequence for Polyhedral Realization of Crystal BasesApr 24 2019Apr 25 2019The polyhedral realization of crystal base has been introduced by A.Zelevinsky and the second author([T.Nakashima, A.Zelevinsky, Adv. Math. 131, no. 1 (1997)]), which describe the crystal base $B(\infty)$ as a polyhedral convex cone in the infinite $\mathbb{Z}$-lattice ... More

Hopfological algebra for infinite dimensional Hopf algebrasApr 23 2019We consider "Hopfological" techniques as in \cite{Ko} but for infinite dimensional Hopf algebras, under the assumption of being co-Frobenius. In particular, $H=k[{\mathbb Z}]\#k[x]/x^2$ is the first example, whose corepresentations category is d.g. vector ... More

Hopfological algebra for infinite dimensional Hopf algebrasApr 23 2019May 17 2019We consider "Hopfological" techniques as in \cite{Ko} but for infinite dimensional Hopf algebras, under the assumption of being co-Frobenius. In particular, $H=k[{\mathbb Z}]\#k[x]/x^2$ is the first example, whose corepresentations category is d.g. vector ... More

Quantum Boson Algebra and Poisson Geometry of the Flag VarietyApr 23 2019In his work on crystal bases \cite{Kas}, Kashiwara introduced a certain degeneration of the quantized universal enveloping algebra of a semi-simple Lie algebra $\mathfrak g$, which he called a quantum boson algebra. In this paper, we construct Kashiwara ... More

The Elliptic Drinfeld Center of a Premodular CategoryApr 20 2019Given a tensor category C, one constructs its Drinfeld center Z(C) which is a braided tensor category, having as objects pairs (X, lambda), where X in Obj(C) and lambda is a half-braiding. For a premodular category C, we construct a new category Zel(C) ... More

Quantizations of local surfaces and rebel instantonsApr 20 2019We construct explicit deformation quantizations of the noncompact complex surfaces $Z_k := \operatorname{Tot} (\mathcal O_{\mathbb P^1} (-k))$ and describe their effect on instanton moduli spaces. We introduce the concept of rebel instantons, as being ... More

Quadratic $d$-numbersApr 20 2019Here we constructively classify quadratic $d$-numbers: algebraic integers in quadratic number fields generating Galois-invariant ideals. We prove the subset thereof maximal among their Galois conjugates in absolute value is discrete in $\mathbb{R}$. Our ... More

Shuffle algebras and perverse sheavesApr 19 2019We relate shuffle algebras, as defined by Nichols, Feigin-Odesskii and Rosso, to perverse sheaves on symmetric products of the complex line (i.e., on the spaces of monic polynomials stratified by multiplicities of roots). More precisely, we construct ... More

On the Cohomological Hall Algebra of the Kronecker QuiverApr 19 2019We give a short introduction to Cohomological Hall algebras of quivers and describe the semistable Cohomological Hall algebra of central slope of the Kronecker quiver in terms of generators and relations.

Schur sector of Argyres-Douglas theory and $W$-algebraApr 19 2019We study the Schur index, the Zhu's $C_2$ algebra, and the Macdonald index of a four dimensional $\mathcal{N}=2$ Argyres-Douglas (AD) theories from the structure of the associated two dimensional $W$-algebra. The Schur index is derived from the vacuum ... More

Khovanov-Rozansky homology for infinite multi-colored braidsApr 19 2019We define a limiting $\mathfrak{sl}_N$ Khovanov-Rozansky homology for semi-infinite positive multi-colored braids, and we show that this limiting homology categorifies a highest-weight projector for a large class of such braids. This effectively completes ... More

A complete classification of pivotal fusion categories $\otimes$-generated by an object of dimension $\frac{1 + \sqrt{5}}{2}$Apr 18 2019In this paper we give a complete classification of pivotal fusion categories $\otimes$-generated by an object of dimension $\frac{1 + \sqrt{5}}{2}$. We show that all such categories arise as certain wreath products of either the Fibonacci category, or ... More

Classification of simple weight modules for the $N=2$ superconformal algebraApr 18 2019In this paper, we classify all simple weight modules with finite dimensional weight spaces over the $N=2$ superconformal algebra. As an application, we give a new proof of the classification of such modules for the $N=1$ superconformal algebra, which ... More

Categorification and the quantum GrassmannianApr 16 2019In \cite{JKS} we gave an (additive) categorification of Grassmannian cluster algebras, using the category $\CM(A)$ of Cohen-Macaulay modules for a certain Gorenstein order $A$. In this paper, using a cluster tilting object in the same category $\CM(A)$, ... More

About a question of Gateva-Ivanova and Cameron on square-free set-theoretic solutions of the Yang-Baxter equationApr 16 2019In this paper, we introduce a new sequence $\bar{N}_m$ to find a new estimation of the cardinality $N_m$ of the minimal involutive square-free solution of level $m$. As an application, using the first values of $\bar{N}_m$, we improve the estimations ... More

Classifying (Weak) Coideal Subalgebras of Weak Hopf C*-AlgebrasApr 16 2019We develop a general approach to the problem of classification of weak coideal C*-subalgebras of weak Hopf C*-algebras. As an example, we consider weak Hopf C*-algebras and their weak coideal C*-subalgebras associated with Tambara Yamagami categories. ... More

Finite symmetric tensor categories with the Chevalley property in characteristic $2$Apr 16 2019We prove an analog of Deligne's theorem for finite symmetric tensor categories $\mathcal{C}$ with the Chevalley property over an algebraically closed field $k$ of characteristic $2$. Namely, we prove that every such category $\mathcal{C}$ admits a symmetric ... More

The matched product of the solutions to the Yang-Baxter equation of finite orderApr 16 2019In this work, we focus on the set-theoretical solutions of the Yang-Baxter equation which are of finite order and not necessarily bijective. We use the matched product of solutions as a unifying tool for treating these solutions of finite order, that ... More

Duality for Knizhnik-Zamolodchikov and Dynamical OperatorsApr 15 2019We consider the Knizhnik-Zamolodchikov and Dynamical operators, both differential and difference, in the context of the (gl(k), gl(n))-duality for the space of polynomials in kn anticommuting variables. We show that the Knizhnik-Zamolodchikov and Dynamical ... More

Quantum toroidal algebra associated with $\mathfrak{gl}_{m|n}$Apr 15 2019We introduce and study the quantum toroidal algebra $\mathcal{E}_{m|n}(q_1,q_2,q_3)$ associated with the superalgebra $\mathfrak{gl}_{m|n}$ with $m\neq n$, where the parameters satisfy $q_1q_2q_3=1$. We give an evaluation map. The evaluation map is a ... More

Quantum increasing sequences generate quantum permutation groupsApr 15 2019We answer a question of A. Skalski and P.M. So{\l}tan (2016) about inner faithfulness of the S.~Curran's map of extending a quantum increasing sequence to a quantum permutation in full generality. To do so, we exploit some novel techniques introduced ... More

Polynomial-valued constant hexagon cohomologyApr 15 2019Hexagon relations are algebraic realizations of four-dimensional Pachner moves. `Constant' -- not depending on a 4-simplex in a triangulation of a 4-manifold -- hexagon relations are proposed, and their polynomial-valued cohomology is constructed. This ... More

Rank $n$ swapping algebra for GrassmannianApr 15 2019The rank $n$ swapping algebra is the Poisson algebra defined on the ordered pairs of points on a circle using the linking numbers, where a subspace of $(\mathbb{K}^n \times \mathbb{K}^{n*})^r/\operatorname{GL}(n,\mathbb{K})$ is its geometric mode. In ... More

A two-variable series for knot complementsApr 12 2019The physical 3d $\mathcal{N}=2$ theory T[Y] was previously used to predict the existence of some 3-manifold invariants $\hat{Z}_{a}(q)$ that take the form of power series with integer coefficients, converging in the unit disk. Their radial limits at the ... More

Kuperberg invariants for balanced sutured 3-manifoldsApr 11 2019We construct quantum invariants of balanced sutured 3-manifolds with a $Spin^{c}$ structure out of an involutive (possibly non-unimodular) Hopf superalgebra $H$. If $H$ is the Borel subalgebra of $U_{q}(\mathfrak{gl}(1|1))$, we show that our invariant ... More

Reflection $K$ matrices associated with an Onsager coideal of $U_p(A^{(1)}_{n-1})$, $U_p(B^{(1)}_n)$, $U_p(D^{(1)}_n)$ and $U_p(D^{(2)}_{n+1})$Apr 11 2019We determine the intertwiners of a family of Onsager coideal subalgebras of the quantum affine algebra $U_p(A^{(1)}_{n-1})$ in the fundamental representations and $U_p(B^{(1)}_{n}), U_p(D^{(1)}_{n}), U_p(D^{(2)}_{n+1})$ in the spin representations. They ... More

Reflection $K$ matrices associated with an Onsager coideal of $U_p(A^{(1)}_{n-1})$, $U_p(B^{(1)}_n)$, $U_p(D^{(1)}_n)$ and $U_p(D^{(2)}_{n+1})$Apr 11 2019Apr 20 2019We determine the intertwiners of a family of Onsager coideal subalgebras of the quantum affine algebra $U_p(A^{(1)}_{n-1})$ in the fundamental representations and $U_p(B^{(1)}_{n}), U_p(D^{(1)}_{n}), U_p(D^{(2)}_{n+1})$ in the spin representations. They ... More

Gelfand-Tsetlin modules in the Coulomb contextApr 10 2019This paper gives a new perspective on the theory of principal Galois orders as developed by Futorny, Ovsienko, Hartwig and others. Every principal Galois order can be written as $eFe$ for any idempotent $e$ in an algebra $F$, which we call a flag Galois ... More

Wreath Macdonald polynomials as eigenstatesApr 10 2019We show that the wreath Macdonald polynomials for $\mathbb{Z}/\ell\mathbb{Z}\wr\Sigma_n$, when naturally viewed as elements in the vertex representation of the quantum toroidal algebra $U_{\mathfrak{q},\mathfrak{d}}(\ddot{\mathfrak{sl}}_\ell)$, diagonalize ... More

A note on S-dual basis in free fermion systemApr 09 2019Free fermion system is the simplest quantum field theory which has the symmetry of Ding-Iohara-Miki algebra (DIM). DIM has S-duality symmetry, known as Miki automorphism which defines the transformation of generators. In this note, we introduce the second ... More

F-algebra--Rinehart Pairs and Super F-algebroidsApr 09 2019In this note we define F-algebra--Rinehart pairs and super F-algebroids and study the connection between them.

Non-commutative Rényi Entropic Uncertainty PrinciplesApr 08 2019In this paper, we calculate the norm of the string Fourier transform on subfactor planar algebras and characterize the extremizers of the inequalities for parameters $0<p,q\leq \infty$. Furthermore, we establish R\'{e}nyi entropic uncertainty principles ... More

Categorified Chern character and cyclic cohomologyApr 08 2019We examine Hopf cyclic cohomology in the same context as the analysis of the geometry of loop spaces $LX$ in derived algebraic geometry and the resulting close relationship between $S^1$-equivariant quasi-coherent sheaves on $LX$ and $D_X$-modules. Furthermore, ... More

One parameter family of Jordanian twistsApr 08 2019We propose an explicit generalization of the Jordanian twist proposed in $r$-symmetrized form by Giaquinto and Zhang. We present explicit formulas for the corresponding star product and twisted coproduct. Finally, we show that our generalization coincides ... More

Topological generation results for free unitary and orthogonal groupsApr 08 2019We show that for every $N\ge 3$ the free unitary group $U^+_N$ is topologically generated by its classical counterpart $U_N$ and the lower-rank $U^+_{N-1}$. This allows for a uniform inductive proof that a number of finiteness properties, known to hold ... More

A twisted local index formula for curved noncommutative two toriApr 08 2019We consider the Dirac operator of a general metric in the canonical conformal class on the noncommutative two torus, twisted by an idempotent (representing the $K$-theory class of a general noncommutative vector bundle), and derive a local formula for ... More

$h$-adic quantum vertex algebras associated with rational $R$-matrix in types $B$, $C$ and $D$Apr 07 2019We introduce the $h$-adic quantum vertex algebras associated with the rational $R$-matrix in types $B$, $C$ and $D$, thus generalizing the Etingof--Kazhdan's construction in type $A$. Next, we construct the algebraically independent generators of the ... More

Lie, associative and commutative quasi-isomorphismApr 07 2019Over a field of characteristic zero, we show that two commutative differential graded (dg) algebras are quasi-isomorphic if and only if they are quasi-isomorphic as associative dg algebras. We also show the Koszul dual statement that two dg Lie algebras ... More

Toroidal prefactorization algebras associated to holomorphic fibrations and a relationship to vertex algebrasApr 05 2019Let $X$ be a complex manifold, $\pi: E \rightarrow X$ a locally trivial holomorphic fibration with fiber $F$, and $\mathfrak{g}$ a Lie algebra with an invariant symmetric form. We associate to this data a holomorphic prefactorization algebra $\mathcal{F}_{\mathfrak{g}, ... More

Affine group dg-schemes and linear representations 1 - Basic theory and Tannakian reconstructionsApr 05 2019We develop a basic theory of affine group dg-schemes, their Lie algebraic counterparts and linear representations. We prove Tannaka type reconstruction theorems that an affine group dg-scheme can be recovered from the dg-tensor category of its linear ... More

The graph of a Weyl algebra endomorphismApr 05 2019Endomorphisms of Weyl algebras are studied using bimodules. Initially, for a Weyl algebra over a field of characteristic zero, Bernstein's inequality implies that holonomic bimodules finitely generated from the right or left form a monoidal category. ... More

The duality of $\mathfrak{gl}_{m|n}$ and $\mathfrak{gl}_k$ Gaudin modelsApr 04 2019We establish a duality of the non-periodic Gaudin model associated with superalgebra $\mathfrak{gl}_{m|n}$ and the non-periodic Gaudin model associated with algebra $\mathfrak{gl}_k$. The Hamiltonians of the Gaudin models are given by expansions of a ... More

Chiral Algebra, Localization, Modularity, Surface defects, And All ThatApr 04 2019We study the 2D vertex operator algebra (VOA) construction in 4D $\mathcal{N}=2$ superconformal field theories (SCFT) on $S^3 \times S^1$, focusing both on old puzzles as well as new observations. The VOA lives on a two-torus $\mathbb{T}^2\subset S^3\times ... More

Quantum differentials by super bosonisation and super bicrossproductApr 04 2019We introduce general methods to construct well-behaved quantum differential calculi or DGAs on biproduct Hopf algebras $A{\buildrel\hookrightarrow\over \twoheadleftarrow}A\ltimes B$ (where $B$ is a braided Hopf algebra in the category of $A$-crossed modules) ... More

Double Yangian and the universal R-matrixApr 04 2019We describe the double Yangian of the general linear Lie algebra $\mathfrak{gl}_N$ by following a general scheme of Drinfeld. This description is based on the construction of the universal $R$-matrix for the Yangian. To make the exposition self contained, ... More

Hall algebras associated to complexes of fixed sizeApr 04 2019Let $\A$ be a finitary hereditary abelian category with enough projectives. We study the Hall algebra of complexes of fixed size over projectives. Explicitly, we first give a relation between Hall algebras of complexes of fixed size and cyclic complexes. ... More

Differential calculus of Hochschild pairs for infinity-categoriesApr 04 2019In this paper, we provide a conceptual new construction of the algebraic structure on the pair of the Hochschild cohomology spectrum (cochain complex) and Hochschild homology spectrum, which is analogous to the structure of calculus on a manifold. This ... More

Conjugacy classes and centralizers for pivotal fusion categoriesApr 03 2019A criterion for M\"uger centralizer of a fusion subcategory of a braided non-degenerate fusion category is given. Along the way we extend some identities on the space of class functions of a fusion category introduced by Shimizu in \cite{scalg}. We also ... More

The structure of connected (graded) Hopf algebrasApr 03 2019In this paper, we establish a structure theorem for connected graded Hopf algebras over a field of characteristic $0$ by claiming the existence of a family of homogeneous generators and a total order on the index set that satisfy some excellent conditions. ... More

Fusion rules for $\mathbb{Z}_{2}$-orbifolds of affine and parafermion vertex operator algebrasApr 03 2019This paper is about the orbifold theory of affine and parafermion vertex operator algebras. It is known that the parafermion vertex operator algebra $K(sl_2,k)$ associated to the integrable highest weight modules for the affine Kac-Moody algebra $A_1^{(1)}$ ... More

Cluster algebra structures on module categories over quantum affine algebrasApr 02 2019We study monoidal categorifications of certain monoidal subcategories $\mathcal{C}_J$ of finite-dimensional modules over quantum affine algebras, whose cluster algebra structures coincide and arise from the category of finite-dimensional modules over ... More

Quantum $SL_2$, Infinite curvature and Pitman's 2M-X theoremApr 01 2019It is understood that Pitman's theorem in probability theory is intimately related to the representation theory of $\mathcal{U}_{q}(\mathfrak{sl}_2)$, in the so-called crystal regime $q \rightarrow 0$. This relationship has been explored by Biane and ... More

Relative Serre functor for comodule algebrasMar 31 2019Let $\mathcal{C}$ be a finite tensor category, and let $\mathcal{M}$ be an exact left $\mathcal{C}$-module category. A relative Serre functor of $\mathcal{M}$, introduced by Fuchs, Schaumann and Schweigert, is an endofunctor $\mathbb{S}$ on $\mathcal{M}$ ... More