Noncommutative Invariant Theory



Luigi Ferraro
Wake Forest University

Thursday, September 19th, 2019
Carswell Hall, Room 101

Abstract: Group actions are ubiquitous in mathematics. To study an algebraic object it is often useful to understand what groups act on it. A major role in the development of commutative algebra has been played by the study of the invariants of the action of a finite group on a commutative polynomial ring.

In recent years there has been a growing interest in studying group actions on noncommutative rings. Of main interest are actions on Artin-Schelter (AS) regular rings, which are rings that share many of the homological properties of commutative polynomial rings.

Analogous to group actions, rings can be studied by understanding what Lie algebras act on them as derivations. Unifying groups and Lie actions are Hopf actions. A Hopf algebra is not only an algebra, but also a coalgebra, and the notion of Hopf action uses this extra structure. Noncommutative rings usually admit few group actions, which is why Hopf actions are of more interest when studying them.

In this talk, we will study group/Hopf actions on “noncommutative” polynomial rings. In particular we will be interested in properties of the invariant subring.

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