Cohomología de Hochschild y estructura de Gerstenhaber de las álgebras toupie

Abstract

In this thesis we compute the Hochschild cohomology H∗(A) of a certain type of algebras calledtoupie algebras, and we describe the Gerstenhaber structure of ⊕1i=0 Hi(A). A quiver Q is called toupie if it has a unique source and a unique sink, and for any other vertex there is exactly one arrow starting at it and exactly one arrow ending at it. The algebra A is toupie if A = kQ/I with Q a toupie quiver and I any admissible ideal. We first construct a minimal projective resolution of A as Ae-module adapting to this case Bardzell’s resolution for monomial algebras. Using this resolution, we compute a k-basis for every cohomology space Hi(A). The structure of H1(A) as a Lie algebra is described in detail as well as the module structure of Hi(A) over H1(A).