Dept Oral Exam

Speaker: Chris Plyley (Western)

" Polynomial Identities on Algebras with Action"

Time: 14:00

Room: MC 108

In order to study the properties of an algebra A, we sometimes endow A with some additional structure in the form of a prescribed action or a grading. Properties of A can often be determined based on the properties of the structure-induced subspaces of A. This tactic is especially effective in polynomial identity theory. For example, classic results state that an algebra with involution satisfies an identity as soon as the subspace of symmetric (or skew-symmetric) elements does, similarly, a group-graded associative algebra is a PI-algebra whenever the identity component of the grading is. Recent results incorporate combinatorial techniques to find quantitative versions of these classic theorems. After providing an introduction into the basics of polynomial identity theory, our first objective in this talk is to describe how to prove a similar result for associative algebras whose induced Lie or Jordan algebras are group-graded. Many types of additional structures on algebras are easily formalized as actions of Hopf algebras on A (for instance, group-gradings or actions by automorphisms). In these cases, a well-known duality between Hopf actions and group-gradings becomes a powerful tool. Other natural actions (for instance, anti-automorphisms) are of interest in PI-theory, but fall out of the range of this duality. Our second objective in this talk is to describe how we can extend this duality to incorporate more general Hopf actions. Following this, we describe a unified Hopf algebra approach to PI-theory that will encompass and improve all of the aforementioned results.