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Hochschild cohomology governs deformations of algebras, and its graded Liestructure plays a vital role. We study this structure for the Hochschildcohomology of the skew group algebra formed by a finite group acting on analgebra by automorphisms. We examine the Gerstenhaber bracket with a viewtoward deformations and developing bracket formulas. We then focus on thelinear group actions and polynomial algebras that arise in orbifold theory andrepresentation theory; deformations in this context include graded Heckealgebras and symplectic reflection algebras. We give some general resultsdescribing when brackets are zero for polynomial skew group algebras, whichallow us in particular to find noncommutative Poisson structures. For abeliangroups, we express the bracket using inner products of group characters.Lastly, we interpret results for graded Hecke algebras.

Group actions on algebras and the graded Lie structure of Hochschild cohomology