Monograph

# Polynomial Identities of Algebras with Actions: A Unified Combinatorial Approach

## Degree

Doctor of Philosophy

Mathematics

Riley, David M.

## Abstract

Fix a field k. In this thesis, we establish different criteria for when a given (associative or Lie) k-algebra is a PI-algebra.

Fix a unitary associative k-algebra R. Our focus is on associative k-algebras A endowed with an R-module action ρ:R\to End_k(A) with the property that ρ(R) is finite-dimensional. Our main result asserts that if such an algebra is endowed with an R-module action that is `compatible’ with the multiplicative structure of A, then A is a PI-algebra if (and only if), for some positive integer d and all a_1,...,a_d in A, the product a_1···a_d is a linear combination of elements of the form (R·a_σ(1))···(R·a_σ(d)), where σ is a non-identity permutation. An analogous result is given for Lie algebras.

To prove these results, we first establish a new combinatorial characterization of PI-algebras which allows us to recover an explicit polynomial identity for the algebra in question.

## Summary for Lay Audience

An algebra is a set of objects together with operations of multiplication, addition, and scalar multiplication by elements of a field (such as the real numbers) such that these operations behave nicely with one another. Suppose that we can find a non-zero polynomial $f(x_1,\dots, x_n)$ in non-commuting indeterminates $x_1,\dots, x_n$ that vanishes when evaluated at arbitrary elements of a given algebra $A$ over a field; in this case, we say that $A$ is a PI-algebra satisfying the polynomial identity $f\equiv 0$. Satisfying a polynomial identity has far reaching consequences on the structure of the algebra in question. Thus, it is interesting to provide criteria for when a given algebra is a PI-algebra. This is the general goal of this thesis.