Shephard–Todd–Chevalley Theorem for Skew Polynomial Rings

Abstract

Abstract. We prove the following generalization of the classical ShephardTodd-Chevalley Theorem. Let G be a finite group of graded algebra automorphisms of a skew polynomial ring A := kp ij [x 1 , · · · , xn]. Then the fixed subring A G has finite global dimension if and only if G is generated by quasireflections. In this case the fixed subring A G is isomorphic a skew polynomial ring with possibly different p ij 's. A version of the theorem is proved also for abelian groups acting on general quantum polynomial rings.

Quantum binary polyhedral groups and their actions on quantum planes

“…The latter algebras are called Artin-Schelter (AS) regular algebras of global dimension 2. Previous work [12], [14], [15], [16] demonstrates that there is a rich invariant theory in this context. The goal of this paper is to classify noncommutative analogues of linear actions of finite subgroups of SL 2 (k) on AS regular algebras of global dimension 2 and study the resulting rings of invariants.…”

“…More generally for the first question in part (b), we ask if there is a version of the Shephard-Todd-Chevalley theorem for finite dimensional Hopf actions on AS regular algebras. This task has been addressed for finite group actions on skew polynomial rings [16].…”

Gorenstein subrings of invariants under Hopf algebra actions

“…One such setting is the action of a finite group on an Artin-Schelter regular algebra, which when commutative, is a polynomial ring. Here successful generalizations of the theory include a noncommutative analogue of Watanabe's Theorem on the Gorenstein property of invariant subrings following the introduction of the homological determinant [JoZ,JiZ2], and a noncommutative analogue of the Shephard-ToddChevalley Theorem for skew polynomial rings after a reasonable notion of a quasi-reflection was established [KKZ1,KKZ2]. It is natural to consider extending the invariant theory from a group action to a Hopf algebra action.…”

“…It is natural to consider extending the invariant theory from a group action to a Hopf algebra action. Replacing finite group actions by finite dimensional Hopf algebra actions creates an additional dimension of noncommutativity (or rather noncocommutativity from the point of view of the Hopf algebra) that can produce more invariant subrings, as shown in [KKZ2,Proposition 0.5(ii)]. …”

Noncommutative complete intersections

“…In this paper we use our work in noncommutative invariant theory to propose several notions of a noncommutative graded complete intersection. Moreover, the existence of noncommutative analogues of commutative complete intersection invariant subalgebras broadens our continuing project of establishing an invariant theory for finite groups acting on Artin-Schelter regular algebras that is parallel to classical invariant theory (see [28][29][30][31][32]). …”