Anne V. Shepler,
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
Poincaré-Birkhoff-Witt property using the theory of noncommutative
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 PBW conditions.
skew polynomial rings, noncommutative Gröbner bases, graded Hecke algebras, symplectic reflection algebras, Hochschild cohomology
16S36 - Ordinary and skew polynomial rings and semigroup rings [See also 20M25]
16S35 - Twisted and skew group rings, crossed products
16S80 - Deformations of rings [See also 13D10, 14D15]
16W20 - Automorphisms and endomorphisms
16Z05 - Computational aspects of associative rings [See also 68W30]
16E40 - (Co)homology of rings and algebras (e.g. Hochschild, cyclic, dihedral, etc.)