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Using the notion of compatibility between Poisson brackets and cluster structures in the coordinate rings of simple Lie groups, Gekhtman Shapiro and Vainshtein conjectured a correspondence between the two. Poisson Lie groups are classified…
Given a fusion category $\mathcal{C}$ and an indecomposable $\mathcal{C}$-module category $\mathcal{M}$, the fusion category $\mathcal{C}^*_\mathcal{M}$ of $\mathcal{C}$-module endofunctors of $\mathcal{M}$ is called the (Morita) dual…
A ring R shall be called F-noetherian if every finite subset of R is contained in a (left and right) noetherian subring of R . For example, every commutative ring is tightly F-noetherian in the sense that every finite subset of R generates…
We reformurate a central extension of Felder's elliptic quantum group in the FRST formulation as a topological algebra E_{q,p}(gl_N) over the ring of formal power series in p. We then discuss the isomorphism between E_{q,p}(gl_N) and the…
A categorical formalism for directed graphs is introduced, featuring natural notions of morphisms and subgraphs, and leading to two elementary descriptions of the free-properad monad, first in terms of presheaves on elementary graphs,…
We construct a long exact sequence computing the obstruction space, pi_1(BrPic(C_0)), to G-graded extensions of a fusion category C_0. The other terms in the sequence can be computed directly from the fusion ring of C_0. We apply our result…
Beginning with the data of a quiver Q, and its dimension vector d, we construct an algebra D_q=D_q(Mat_d(Q)), which is a flat q-deformation of the algebra of differential operators on the affine space Mat_d(Q). The algebra D_q is…
We build representations of the affine and double affine braid groups and Hecke algebras of type $C^\vee C_n$, based upon the theory of quantum symmetric pairs $(U,B)$. In the case $U=U_q(gl_N)$, our constructions provide a quantization of…
We modify and give complete proofs for the results of Etingof-Schiffmann-Varchenko on traces of intertwiners of untwisted quantum affine algebras in the opposite coproduct and the standard grading. More precisely, we show that certain…
We describe basic motivations behind quantum or noncommutative probability, introduce quantum L\'evy processes on compact quantum groups, and discuss several aspects of the study of the latter in the example of quantum permutation groups.…
This note is a development of our two previous papers, arXiv:1212.3392v1 and 1306.3660v1. The fundamental question is whether there exists a Galois theory, in which the Galois group is a quantum group. For a linear equations with respect to…
We found a necessary and sufficient condition for the existence of the tensor product of modules over a vertex algebra. We defined the notion of vertex bilinear map and we provide two algebraic construction of the tensor product, where one…
For a braided finite tensor category $\mathcal{C}$ with unit object $1 \in \mathcal{C}$, Lyubashenko considered a certain Hopf algebra $\mathbb{F} \in \mathcal{C}$ endowed with a Hopf pairing $\omega: \mathbb{F} \otimes \mathbb{F} \to 1$ to…
We show that the Weyl-Kac type character formula holds for the integrable highest weight modules over the quantized enveloping algebra of any symmetrizable Kac-Moody Lie algebra, when the parameter $q$ is not a root of unity.
We determine convex PBW-type Lyndon bases for two-parameter quantum groups $U_{r,s}(F_4)$ with detailed commutation relations. We construct a finite-dimensional Hopf algebra $\mathfrak u_{r,s}(F_4)$, as a quotient of $U_{r,s}(F_4)$ by a…
This is a continuation of our previous paper 1502.01744. We examine a class of non-commutative algebras A that depend on an elliptic curve and a translation automorphism of it. They may be defined in terms of the 4-dimensional Sklyanin…
In this paper, we establish the connection between the quantized W-algebra of ${\frak sl}(2,1)$ and quantum parafermions of $U_q(\hat {\frak sl}(2))$ that a shifted product of the two quantum parafermions of $U_q(\hat {\frak sl}(2))$…
In this article, we describe a construction of a holomorphic vertex operator algebras of central charge $24$ whose weight one Lie algebra has type $A_{6,7}$.
Let $U_q(\mathfrak{g})$ be the quantum supergroup of $\mathfrak{gl}_{m|n}$ or the modified quantum supergroup of $osp_{m|2n}$ over the field of rational functions in $q$, and let $V_q$ be the natural module for $U_q(\mathfrak{g})$. There…
We establish Drinfeld realization for the two-parameter twisted quantum affine algebras using a new method. The Hopf algebra structure for Drinfeld generators is given for both untwisted and twisted two-parameter quantum affine algebras,…