Quantum Drinfeld Hecke Algebras

TL;DR

This paper studies deformations of quantum polynomial rings under finite group actions, extending classical algebraic structures to the quantum setting and providing criteria for their PBW property using noncommutative Groebner bases.

Contribution

It introduces necessary and sufficient conditions for quantum Drinfeld Hecke algebras to satisfy PBW properties, extending symplectic reflection and graded Hecke algebras to quantum cases.

Findings

Established PBW criteria via noncommutative Groebner bases.

Classified graded automorphisms of quantum 3-space.

Applied results to abelian groups and quantum planes.

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.