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We describe the Gerstenhaber bracket structure on Hochschild cohomology ofKoszul quiver algebras in terms of homotopy lifting maps. There is a projectivebimodule resolution of Koszul quiver algebras that admits a comultiplicativestructure. Introducing new scalars, we describe homotopy lifting mapsassociated to Hochschild cocycles using the comultiplicative structure. We showthat the scalars can be described by some recurrence relations and we giveseveral examples where these scalars appear in the literature. In particular,for a member of a family of quiver algebras, we describe Hochschild 2-cocyclesand their associated homotopy lifting maps and determine the Maurer-Cartanelements of the quiver algebra in two ways: (i) by the use of homotopy liftingmaps and (ii) by the use of a combinatorial star product that arises from thedeformation of algebras using reduction systems.

Homotopy lifting maps on Hochschild cohomology and connections to deformation of algebras using reduction systems