The Invariant Theory of Artin-Schelter Regular Algebras

Classical invariant theory studies the ring of invariants k[x1, . . . , xn] G under the linear action of a group G on a commutative polynomial ring k[x1, . . . , xn]. To extend this theory to a noncommutative context, we replace the polynomial ring with an Artin-Schelter regular algebra A (that when commutative is isomorphic to a commutative polynomial ring), and study the invariants AG under the action of a finite group, or, more generally, a finite dimensional Hopf algebra. In this talk we will present a survey of some techiques that have been used to generalize classical results: (1) homological concepts related to a theorem of Auslander, (2) homological regularities and bounds on the degrees of generators of AH, and (3) the representations of H to construct algebras A on which H acts inner faithfully.

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