Hochschild cohomology governs deformations of algebras, and its graded Lie
structure plays a vital role. We study this structure for the Hochschild
cohomology of the skew group algebra formed by a finite group acting on an
algebra by automorphisms. We examine the Gerstenhaber bracket with a view
toward deformations and developing bracket formulas. We then focus on the
linear group actions and polynomial algebras that arise in orbifold theory and
representation theory; deformations in this context include graded Hecke
algebras and symplectic reflection algebras.

When a finite group acts linearly on a complex vector space, the natural
semi-direct product of the group and the polynomial ring over the space forms a
skew group algebra. This algebra plays the role of the coordinate ring of the
resulting orbifold and serves as a substitute for the ring of invariant
polynomials from the viewpoint of geometry and physics. Its Hochschild
cohomology predicts various Hecke algebras and deformations of the orbifold. In
this article, we investigate the ring structure of the Hochschild cohomology of
the skew group algebra.

We give an explicit formula for the correspondence between simple
Yetter-Drinfeld modules for certain finite-dimensional pointed Hopf algebras
$H$ and those for cocycle twists $H^{\sigma}$ of $H$. This implies an
equivalence between modules for their Drinfeld doubles. To illustrate our
results, we consider the restricted two-parameter quantum groups
${\mathfrak{u}}_{r,s}({\mathfrak{sl}}_n)$ under conditions on the parameters
guaranteeing that ${\mathfrak{u}}_{r,s}({\mathfrak{sl}}_n)$ is a Drinfeld
double of its Borel subalgebra.