We consider finite groups acting on quantum (or skew) polynomial rings.
Deformations of the semidirect product of the quantum polynomial ring with the
acting group extend symplectic reflection algebras and graded Hecke algebras to
the quantum setting over a field of arbitrary characteristic. We give necessary
and sufficient conditions for such algebras to satisfy a Poincare-Birkhoff-Witt
property using the theory of noncommutative Groebner bases. We include
applications to the case of abelian groups and the case of groups acting on
coordinate rings of quantum planes. In addition, we classify graded
automorphisms of the coordinate ring of quantum 3-space. In characteristic
zero, Hochschild cohomology gives an elegant description of the
Poincare-Birkhoff-Witt conditions.Comment: 29 pages. Last example corrected; some indices in the last theorem
were accidentally transposed and now appear in correct orde
