Let H be an involutory Hopf algebra over a field of characteristic zero, M and N two finite dimensional left H-modules such that M ⊗ N is a semisimple H-module. Then M and N are semisimple H-modules. This is a generalization of a theorem proved by J.-P. Serre for group algebras. A version of the theorem above for monoidal categories is also given.

Let G be a connected reductive algebraic group defined over an algebraically closed field F of characteristic not 2. Denote the Lie algebra of G by 9. In this paper we shall classify the isomorphism classes of ordered pairs of commuting involutorial automorphisms of G. This is shown to be independent of the characteristic of F and can be applied to describe all semisimple locally symmetric spac...

In this paper an algebraic star product and differential one defined on a regular coadjoint orbit of a compact semisimple group are compared. It is proven that there is an injective algebra homomorphism between the algebra of polynomials with the algebraic star product and the algebra of differential functions with the differential star product structure. Investigation supported by the Universi...

We show that the locally free class group of an order in a semisimple algebra over a number field is isomorphic to a certain ray class group. This description is then used to present an algorithm that computes the locally free class group. The algorithm is implemented in MAGMA for the case where the algebra is a group ring over the rational numbers.

In this paper, we prove that the variety of modal FLew-algebras is generated by its finite simple members. The result is obtained by showing that every free modal FLew-algebra is semisimple and then showing that every variety generated by a simple modal FLew-algebra is generated by a set of finite simple modal FLew-algebras.

We study projective objects in the category OHc(0) of the Cherednik algebra introduced recently by Berest, Etingof and Ginzburg. We prove that it has enough projectives and that it is a highest weight category in the sense of Cline, Parshall and Scott, and therefore satisfies an analog of the BGG-reciprocity formula for a semisimple Lie algebra.

We give a uniform geometric realization for the cluster algebra of an arbitrary finite type with principal coefficients at an arbitrary acyclic seed. This algebra is realized as the coordinate ring of a certain reduced double Bruhat cell in the simply connected semisimple algebraic group of the same Cartan-Killing type. In this realization, the cluster variables appear as certain (generalized) ...

Introduction For this course you need a very good understanding of linear algebra; a good knowledge of group theory and the representation theory of finite groups will also help. The main sources for these notes are the books [6] and [8]. We give complete proofs of all statements with the exception of the conjugacy of Cartan subgroups, the uniqueness theorem for semisimple Lie algebras, and the...

We investigate an analogue of the Wedderburn principal theorem for associative conformal algebras with finite faithful representations. It is shown that the radical splitting property for an algebra of this kind holds if the maximal semisimple factor of this algebra is unital, but does not hold in general.

Maschke functors, semisimple functors and separable functors of the second kind. Abstract We introduce separable functors of the second kind (or H-separable functors) and H-Maschke functors. H-separable functors are generalizations of separable functors. Various necessary and sufficient conditions for a functor to be H-separable or H-Maschke, in terms of generalized (co)Casimir elements (integr...