On the Lie Algebra Structure on Hochschild Cohomology of Koszul Quiver Algebras

Abstract

We present the Gerstenhaber algebra structure on Hochschild cohomology of Koszul algebras defined by quivers and relations using the idea of homotopy liftings. There is a canonical way of constructing a minimal (graded) projective resolution K of a Koszul quiver algebra over its enveloping algebra. This resolution was shown to have a comultiplicative structure. Our presentation involves the use of the resolution K and the comultiplicative structure on it. We present general forms of homotopy lifting maps for cocycles defined on Hochschild cohomology using K. To demonstrate the theory, we study the Hochschild cohomology ring of a family of quiver algebras and present explicit examples of homotopy lifting maps for cocycles of degrees 1 and 2. As an application to the theory of deformation of algebras, we specify Hochschild 2-cocycles satisfying the Maurer-Cartan equation.