2004

It is an open question whether the smash product of a semisimple Hopf algebra and a semiprime module algebra is semiprime. In this talk we show that the smash product of a commutative semiprime module algebra over a semisimple cosemisimple Hopf algebra is semiprime. In particular we show that the central H-invariant elements of the Martindale ring of quotients of a module algebra form a von Neumann regular and self-injective ring whenever A is semiprime. For a semiprime Goldie PI H-module algebra A with central invariants we show that A#H is semiprime if and only if the H-action can be extended to the classical ring of quotients of A if and only if every non-trivial H-stable ideal of A contains a non-zero H-invariant element.

El grupo de automorfismos algebraicos del espacio afín es un grupo algebraico de dimensión infinita relativamente poco estudiado. En esta charla presentaremos los resultados clásicos y recientes sobre la estructura de este grupo. Expondremos igualmente las versiones no conmutativas y cuánticas.

In this talk I will first recall the definition of Hochschild cohomology and its main properties. Then I will define triangular algebras and give some examples. Finally I will explain how to compute the cohomology of these algebras.

Dado un cubrimiento de Galois de una categoria lineal, resulta una graduacion de la categoria base por el grupo de Galois. En un trabajo reciente con Eduardo Marcos mostramos que el cubrimiento es isomorfo al smash producto categorico.

Generalized Weyl algebras, as defined by V. V. Bavula [St. Petersburgh Math. J., 1993], are a family of algebras containing both some classical objects (enveloping algebras and their prime quotients, Weyl algebras, invariant sub-algebras, ...) and their quantum analogues. These algebras are generated by two generators over a k-algebra R, with relations given thanks to an automorphism and a central element of R. We are interested in problems of classification for such algebras. In this talk based on a joint work with A. Solotar, we will consider isomorphisms between generalized Weyl algebras, giving a complete answer to this problem in the quantum case for R = k[h]. We will give separation results too up to rational equivalence and Morita-equivalence for these algebras.

We will describe a certain class of infinite-dimensional Lie algebras (elliptic, also called toroidal or double-loop algebras), and explain their appearance in some fundamental classification problems for systems of linear differential equations on a complex projective line (the so-called Deligne-Simpson problem). We will then give a geometric approach to these Lie algebras and their quantum deformations.

The involutions on a central simple algebra were described by A. Albert. It is known that if R is the algebra of all n⨉n matrices over an algebraically closed field then there exist, up to isomorphism, only two involutions for R, namely the usual transpose involution, and if n is even, the usual symplectic involution. The talk will be concerned with an analogous question about the algebra of upper triangular matrices over an infinite field of characteristic different from two. The main result to be discussed is that the only involutions on this algebra are the ones induced by involutions on the corresponding full matrix algebra. In particular one has that studying polynomial identities with involution for this algebra one may work with the analogs of the transpose and the symplectic involutions only, as it is the case with the full matrix algebra.

In order to obtain the main result mentioned above one needs detailed information about the existence of certain “good” central idempotents that are fixed by the involution. Furthermore one uses ideas from structure theory of associative algebras as well as combinatorics and linear algebra.