Functional identities of one variable

TL;DR

This paper characterizes the form of trace functions of multilinear maps on prime algebras when they commute with their argument, extending previous results to finite-dimensional cases and applying this to functional identities.

Contribution

It extends the classification of commuting trace functions to finite-dimensional prime algebras of dimension at most d^2, and applies this to general functional identities.

Findings

Characterization of trace functions commuting with their argument.

Extension of previous results to finite-dimensional algebras.

Solution of general functional identities involving multilinear maps.

Abstract

Let $A$ be a centrally closed prime algebra over a characteristic 0 field $k$, and let $q:A\to A$ be the trace of a $d$-linear map (i.e., $q(x)=M(x,...,x)$ where $M:A^d\to A$ is a $d$-linear map). If $[q(x),x]=0$ for every $x\in A$, then $q$ is of the form $q(x) =\sum_{i=0}^{d} \mu_i(x)x^i$ where each $\mu_i$ is the trace of a $(d-i)$-linear map from $A$ into $k$. For infinite dimensional algebras and algebras of dimension $>d^2$ this was proved by Lee, Lin, Wang, and Wong in 1997. In this paper we cover the remaining case where the dimension is $ \le d^2$. Using this result we are able to handle general functional identities of one variable on $A$; more specifically, we describe the traces of $d$-linear maps $q_i:A\to A$ that satisfy $\sum_{i=0}^m x^i q_i(x)x^{m-i}\in k$ for every $x\in A$.