Weak polynomial identities and their applications

TL;DR

This paper surveys the theory of weak polynomial identities in associative algebras, exploring their applications to polynomial identities, central polynomials, and the finite basis problem, with specific results on degree three identities.

Contribution

It provides a comprehensive overview of weak polynomial identities and introduces new results on identities of degree three in associative and related algebras.

Findings

Weak polynomial identities can be used to analyze polynomial identities and central polynomials.

Results on weak polynomial identities of degree three are presented.

Applications to the finite basis problem are discussed.

Abstract

Let $R$ be an associative algebra over a field $K$ generated by a vector subspace $V$. The polynomial $f(x_1,\ldots,x_n)$ of the free associative algebra $K\langle x_1,x_2,\ldots\rangle$ is a weak polynomial identity for the pair $(R,V)$ if it vanishes in $R$ when evaluated on $V$. We survey results on weak polynomial identities and on their applications to polynomial identities and central polynomials of associative and close to them nonassociative algebras and on the finite basis problem. We also present results on weak polynomial identities of degree three.