PhD Public Lecture (Math) - Maye Cardenas Montoya

It is known that satisfying a (non-trivial) polynomial identity has a significant impact on the structure of an associative algebra A over a field. In this talk, we present a series of conditions that ensure that A is a PI-algebra. Suppose R is a unitary associative algebra. Our focus will be on associative algebras A equipped with an R-module action with the property that the algebra of endomorphisms on A defined by the R-action is of finite dimension. Using the added structure of an R-action, we extend the classical notion of polynomial identity to so-called R-identities and ask: is the existence of an R-identity sufficient to ensure that A is a PI-algebra? We prove that if the R-action is `compatible' with the multiplicative structure of A, a suitable condition on the R-identity yields a positive result. Next, we introduce the notion of R-rewritable algebras and prove a more general result: if the R-action is `compatible' with the multiplicative structure of an R-rewritable algebra A, then A is a PI-algebra. A key ingredient in this result is a numerical sequence, denoted pi_n(A), that shares some important properties of the codimension sequence of A and turns out to be quite interesting on its own. In particular, we prove that A is a PI-algebra if and only if pi_n(A)<n!, for some positive integer n. Time permitting, we show that our main results come with natural Lie-theoretic analogues. This work extends a collection of results in associative and Lie PI-theory.