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 \(H^1(A)\).