# Cohomología de Hochschild y estructura de Gerstenhaber de las álgebras toupie - Dalia Artenstein (2016)

In this thesis we compute the Hochschild cohomology $H^∗(A)$ of a certain type of algebras called toupie algebras, and we describe the Gerstenhaber structure of $⊕^∞_{i=0} H^i(A)$. 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 $A^e$-module adapting to this case Bardzell’s resolution for monomial algebras. Using this resolution, we compute a k-basis for every cohomology space $H^i(A)$. The structure of $H^1(A)$ as a Lie algebra is described in detail as well as the module structure of $H^i(A)$ over $H^1(A)$.