Quantum Drinfeld Hecke Algebras
arXiv:1111.4975 · doi:10.4153/CJM-2013-012-2
Abstract
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.
29 pages. Last example corrected; some indices in the last theorem were accidentally transposed and now appear in correct order
Cited by in corpus (11)
- Poincare-Birkhoff-Witt deformations of smash product algebras from Hopf actions on Koszul algebras
- Poincare-Birkhoff-Witt Theorems
- Hochschild cohomology and quantum Drinfeld Hecke algebras
- Twisted quantum Drinfeld Hecke algebras
- Drinfeld Orbifold Algebras
- Quantum Drinfeld Orbifold Algebras
- Truncated quantum Drinfeld Hecke algebras and Hochschild cohomology
- PBW deformations of skew polynomial rings and their group extensions
- Cleft extensions of Koszul twisted Calabi-Yau algebras
- Stabilizing Automorphisms of Quantum Affine Space
- Deformations of Quantum Symmetric Algebras Extended by Groups