We determine the structure of Hopf algebras that admit an extension of a group algebra by the cyclic group of order 2. We study the corepresentation theory of such Hopf algebras, which provide a generalization, at the Hopf algebra level, of the so called Tambara-Yamagami fusion categories. As a byproduct, we show that every semisimple Hopf algebra of dimension < 36 is necessarily group-theoreti...

We reprove the theorem of Feigin and Frenkel relating the center of the critical level enveloping algebra of the Kac-Moody algebra for a semisimple Lie algebra to opers (which are certain de Rham local systems with extra structure) for the Langlands dual group. Our proof incorporates a construction of Beilinson and Drinfeld relating the Feigin-Frenkel isomorphism to (more classical) Langlands d...

We construct quasi-Hopf algebras associated with a semisimple Lie algebra, a complex curve and a rational differential. This generalizes our previous joint work with V. Rubtsov (Israel J. Math. (1999) and q-alg/9608005).

Alexander Premet has stated the following problem: what is a relation between subregular nilpotent representations of a classical semisimple restricted Lie algebra and non-commutative deformations of the corresponding singularities? We solve this problem for type A.

We start by briefly explaining key results on the representation theory of semisimple Lie algebras in large enough positive characteristic. After that, we start a new topic: the complex representation theory of finite groups of Lie type and Hecke algebras. Today we consider the most basic group: G := GLn(Fq). We introduce the Hecke algebra as the endomorphism algebra of the G-module C[B \ G]. W...

We study the Leibniz $n$-algebra $\textbf{U}_n(\mathfrak{L})$, whose multiplication is defined via bracket of a algebra $\mathfrak{L}$ as $[x_1,\dots,x_n]=[x_1,[\dots, [x_{n-2},[x_{n-1},x_n]]\dots]]$. show that $\textbf{U}_n(\mathfrak{L})$ simple if and only Lie algebra. An analogue Levi's theorem for algebras in $\textbf{U}_n(\textbf{Lb})$ established it proven $n$-kernel any semisimple $\text...

We study some non-semisimple representations of affine Temperley–Lieb algebras and related cellular algebras. In particular, we classify extensions between simple standard modules. Moreover, we construct a completion which is an infinite dimensional cellular algebra.