Quantum group actions on quantum planes

Abstract:

A finite subgroup G of GL_(V) acts naturally on the polynomial ring C[V], where V is a C-vector space.

Classical invariant theory is concerned with questions about the invariant ring S = C[V]G. For example, for which finite subgroups G does S have nice homological properties? Can we classify all possible invariant rings S for finite subgroups G of GL_(V)? For a fixed G, can we describe S explicitly in terms of generators and relations?

We present a noncommutative version of the above setup and discuss the analogous questions.