Two-Term Polynomial Identities

TL;DR

This paper investigates algebras satisfying specific two-term multilinear identities, revealing conditions under which they become eventually commutative or nilpotent, and confirming the Specht conjecture for certain cases.

Contribution

It establishes bounds on the degree of eventual commutativity for such algebras and proves the Specht conjecture in cases of arbitrary characteristic.

Findings

Algebras with $q=1$ and non-fixed permutations become eventually commutative with a bounded degree.

The degree of eventual commutativity is at most $2n-3$, and this bound is sharp.

Algebras with $q e 1$ are necessarily nilpotent.

Abstract

We study algebras satisfying a two-term multilinear identity, namely one of the form $x_1 \cdots x_n= q x_{\sigma(1)} \cdots x_{\sigma(n)}$, where $q$ is a parameter from the base field. We show that such algebras with $q=1$ and $\sigma$ not fixing 1 or $n$ are eventually commutative in the sense that the equality $x_1\cdots x_k = x_{\tau(1)} \cdots x_{\tau(k)}$ holds for $k$ large enough and all permutations $\tau \in S_k$. Calling the minimal such $k$ the degree of eventual commutativity, we prove that $k$ is never more than $2n-3$, and that this bound is sharp. For various natural examples, we prove that $k$ can be taken to be $n+1$ or $n+2$. In the case when $q \ne 1$, we establish that the algebra must be nilpotent. We, moreover, demonstrate that if an algebra is eventually commutative of arbitrary characteristic, then it has a finite basis of its polynomial identities, thus confirming the Specht conjecture in this particular case.