Quantum Drinfeld Hecke Algebras, with Viktor Levandovskky. Submitted. 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 PBW conditions.
Finite groups actings
linearly: Hochschild cohomology and the cup product, with Sarah
Witherspoon.
Advances in Mathematics, 226 (4), 2011, 2884--2910. Abstract:
When a finite group acts linearly on a complex vector space, the
natural semi-direct product of the group and the polynomial ring over
the space forms a skew group algebra. This algebra plays the role of
the coordinate ring of the resulting orbifold and serves as a
substitute for the ring of invariant polynomials from the viewpoint of
geometry and physics. Its Hochschild cohomology predicts various
Hecke algebras and deformations of the orbifold. In this article, we
investigate the ring structure of the Hochschild cohomology of the skew
group algebra. We show that the cup product coincides with a natural
smash product, transferring the cohomology of a group action into a
group action on cohomology. We express the algebraic structure of
Hochschild cohomology in terms of a partial order on the group (modulo
the kernel of the action). This partial order arises after
assigning to each group element the codimension of its fixed point
space. We describe the algebraic structure for Coxeter groups,
where this partial order is given by the reflection length function; a
similar combinatorial description holds for an infinite family of
complex reflection groups.
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