Bracket structure on Hochschild cohomology of Koszul quiver algebras using homotopy liftings

Abstract: We present the Gerstenhaber algebra structure on the Hochschild cohomology of Koszul algebras defined by quivers and relations using the idea of homotopy liftings. E.L. Green, G. Hartman, E.N. Marcos and O. Solberg provided a canonical way of constructing a minimal projective bimodule resolution of a Koszul quiver algebra. The resolution has a comultiplicative structure which we use to define homotopy lifting maps. We first present a short example that demonstrates the theory. We then study the Gerstenhaber algebra structure on Hochschild cohomology of a family of bound quiver algebras, some members of which are counterexamples to the Snashall-Solberg finite generation conjecture. We give examples of homotopy lifting maps for degree $2$ and degree $1$ cocycles and draw connections to derivation operators. As an application, we describe Hochschild 2-cocycles satisfying the Maurer-Cartan equation.