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In vertex algebra theory, fusion rules are described as the dimension of the vector space of intertwining operators between three irreducible modules. We describe fusion rules in the category of weight modules for the Weyl vertex algebra.…

We extend the Dong-Mason theorem on the irreducibility of modules for orbifold vertex algebras from [C. Dong, G. Mason, Duke Math. J. 86 (1997)] 305-321] for the category of weak modules. Let $V$ be a vertex operator algebra, $g$ an…

We introduce shift algebras as certain crossed product algebras based on general function spaces and study properties, as well as the classification, of a particular class of modules depending on a set of matrix parameters. It turns out…

We provide a parameterization of all fusion subcategories of the equivariantization by a group action on a fusion category. As applications, we classify the Hopf subalgebras of a family of semisimple Hopf algebras of Kac-Paljutkin type and…

We study unitary pseudonatural transformations (UPTs) between fibre functors Rep(G) -> Hilb, where G is a compact quantum group. For fibre functors F_1, F_2 we show that the category of UPTs F_1 -> F_2 and modifications is isomorphic to the…

We introduce a Frobenius-like structure for the $\frak{sl}_2$ Gaudin model. Namely, we introduce potential functions of the first and second kind. We describe the Shapovalov form in terms of derivatives of the potential of the first kind…

We provide a method to construct new associators out of Drinfel'd's KZ associator. We obtain two analytic families of associators whose coefficients we can describe explicitly by a generalization of multiple zeta values. The two families…

Modular functors, i.e. consistent systems of projective representations of mapping class groups of surfaces, have been constructed for non-semisimple modular categories already decades ago. Concepts from homological algebra have not been…

We present two classes of examples of Hopf algebroids associated with noncommutative principal bundles. The first comes from deforming the principal bundle while leaving unchanged the structure Hopf algebra. The second is related to…

The most standard description of symmetries of a mathematical structure produces a group. However, when the definition of this structure is motivated by physics, or information theory, etc., the respective symmetry objects might become more…

We notice that the famous pentagon identity for quantum dilogarithm functions and the five-term relation for certain operators related to Macdonald polynomials discovered by Garsia and Mellit can both be understood as specific cases of a…

In this note we show a close relation between the following objects: Classical Yang -- Baxter equation (CYBE), conformal algebras (also known as vertex Lie algebras), and averaging operators on Lie algebras. It turns out that the singular…

This paper aims to deepen the theory of bijective non-degenerate set-theoretic solutions of the Yang-Baxter equation, not necessarily involutive, by means of q-cycle sets. We entirely focus on the finite indecomposable ones among which we…

We construct reflection functors for Yetter-Drinfeld modules over Nichols systems and discuss their fundamental properties. We will obtain properties about the geometry of the support of Nichols systems and their Yetter-Drinfeld modules, by…

For $n,m\in \mathbb{N}$, let $H_{n,m}$ be the dual of the Radford algebra of dimension $n^{2}m$. We present new finite-dimensional Nichols algebras arising from the study of simple Yetter-Drinfeld modules over $H_{n,m}$. Along the way, we…

We describe how to find quantum determinants and antipode formulas from braided vector spaces using the FRT-construction and finite-dimensional Nichols algebras. It generalizes the construction of quantum function algebras using quantum…

Let p be an odd prime and k an algebraically closed field of characteristic zero. We classify nonsemisimple Hopf algebras over k of dimension 8p with the Chevalley property, and give partial results on nonsemisimple Hopf algebras of…

We classify all finite-dimensional Hopf algebras over an algebraically closed field of characteristic zero such that its coradical is isomorphic to the algebra of functions over a dihedral group D_m, with m=4a> 11. We obtain this…

Using the standard filtration associated with a generalized lifting method, we determine all finite-dimensional Hopf algebras over an algebraically closed field of characteristic zero whose coradical generates a Hopf subalgebra isomorphic…

Let $G$ be a connected, simply connected simple complex algebraic group and let $\epsilon$ be a primitive $\ell$th root of unity with $\ell$ odd and coprime with $3$ if $G$ is of type $G_{2}$. We determine all Hopf algebra quotients of the…