Benjamin Briggs, MSRI

Title: Hochschild cohomology and the fundamental group(s) of a finite dimensional algebra

Abstract: This talk is about the Lie algebra structure of the first Hochschild cohomology of a finite dimensional algebra. I will explain how to classify the maximal tori in this Lie algebra using the homotopy theory of quivers. More precisely: every maximal torus in HH^1(A) arises as the dual of the fundamental group of associated to a presentation of A. This extends work of Farkas, Green, Marcos, de la Peña, Saorín and Le Meur.  Since Hochschild cohomology is well-known to be a derived (and stable) invariant, and since the fundamental group contains information about the Gabriel quiver of A, we can use this to build some useful derived (and stable) invariants. As one consequence of this idea, we prove that there are only finitely many monomial algebras in any derived equivalence class of finite dimensional algebras. This is all joint work with Lleonard Rubio y Degrassi.