Quantum Drinfeld Hecke Algebras
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.
Cite
@article{arxiv.1111.4975,
title = {Quantum Drinfeld Hecke Algebras},
author = {Viktor Levandovskyy and Anne V. Shepler},
journal= {arXiv preprint arXiv:1111.4975},
year = {2019}
}
Comments
29 pages. Last example corrected; some indices in the last theorem were accidentally transposed and now appear in correct order