Polynomial identities in associative algebras

Abstract: The main goal of this talk is to introduce the basic definitions and to present some of the most important results of the theory of polynomial identities (PI-theory) for associative algebras. When an algebra satisfies a non-trivial polynomial identity we call it a PI-algebra.

Let $A$ be an associative algebra over a field $F$ of characteristic zero and let $c_n(A)$ be its sequence of codimensions. Such a sequence was introduced by Regev in '72 and it provide an effective way of measuring the growth of the polynomial identities satisfied by a given algebra. He proved that any PI-algebra has codimension sequence exponentially bounded. From that moment, the sequence of codimensions became a powerful tool in PI-theory and it has been extensively studied by several authors. In this direction, I shall present two celebrated results. The first one, proved by Kemer, characterizes those algebras having a polynomial growth of the codimension sequence. The second one is a theorem of Giambruno and Zaicev, solving in the affirmative a conjecture of Amitsur.

group theoryrings and algebras

Audience: researchers in the topic

Al@Bicocca take-away