Abstract

Nichols algebras are a family of associative algebras defined by generators and relations depending on an endomorphism of a vector space Votimes V satisfying the braid relation (braiding). They include exterior and symmetric algebras and quantized Borel subalgebras. The algebras introduced by Fomin and Zelevinsky to study Schubert calculus for flag varieties for GL(n) are also strictly related to Nichols algebras.

For a group G a special class of braidings can be associated with a G-module with a compatible G-grading. It has been conjectured that if G is finite simple non-abelian then all Nichols algebras obtained this way are infinite-dimensional. I will describe the state of the art of this conjecture and present results obtained with N. Andruskiewitsch and G. Garca and with Mauro Costantini for Suzuki and Ree for different families of finite simple groups.

Abstract

Let A be a finite dimensional algebra over an algebraically closed field. Hochschild cohomology records crucial information about A: its first degree component, denoted by HH^1(A), is a Lie algebra and it is invariant under Morita and derived equivalences. For symmetric algebras, it is also invariant under stable equivalences of Morita type. Although HH^1 is a powerful invariant, its Lie structure has been calculated only for few families of algebras.

In this talk I will show how the Lie structure of HH^1 is strongly related with the Ext-quiver of A. More precisely, if we assume that the Ext-quiver of A is a simple directed graph, then the Lie algebra of HH^1(A) is solvable. For quivers containing loops, I will determine sufficient conditions for the solvability of HH^1. Finally, I will apply these criteria to show the solvability of the first Hochschild cohomology of blocks with cyclic defect, all tame blocks of finite groups and some wild algebras.

This is part of two joint works with Markus Linckelmann, and with Andrea Solotar and Sibylle Schroll.