  We consider a finite group acting on a vector space and the corresponding
skew group algebra generated by the group and the symmetric algebra of the
space. This skew group algebra illuminates the resulting orbifold and serves as
a replacement for the ring of invariant polynomials, especially in the eyes of
cohomology. One analyzes the Hochschild cohomology of the skew group algebra
using isomorphisms which convert between resolutions. We present an explicit
chain map from the bar resolution to the Koszul resolution of the symmetric
algebra which induces various isomorphisms on Hochschild homology and
cohomology, some of which have appeared in the literature before. This approach
unifies previous results on homology and cohomology of both the symmetric
algebra and skew group algebra. We determine induced combinatorial cochain maps
which invoke quantum differentiation (expressed by Demazure-BBG operators).
