Hochschild cohomology and the fundamental group(s) of a finite
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.