A nonlinear analogue of additive commutators

TL;DR

This paper introduces and explores polynomial commutators, a nonlinear analogue of additive commutators, demonstrating their algebraic generation properties, matrix representations, and trace behaviors in various algebraic structures.

Contribution

It extends the concept of polynomial commutators to broader algebraic contexts and analyzes their structural and functional properties in division rings and matrix algebras.

Findings

Polynomial commutators can generate maximal subfields and entire rings in division rings.

Matrices similar to zero-diagonal matrices are polynomial commutators.

Every matrix can be expressed as a product of at most three polynomial commutators under mild conditions.

Abstract

We study a nonlinear analogue of additive commutators, known as \textit{polynomial commutators}, defined by $p(ab) - p(ba)$ for a polynomial $p \in F[x]$ and elements $a, b$ in an algebra $R$ over a field $F$. Originally introduced by Laffey and West for matrices over fields, this notion is here extended to broader algebraic settings. We first show that in division rings, polynomial commutators can generate maximal subfields and even the entire ring as an algebra. In the matrix setting, we prove that matrices similar to ones with zero diagonal are polynomial commutators, and under mild assumptions, every matrix can be written as a product of at most three such commutators. Furthermore, we demonstrate that the matrix algebra can be decomposed as the sum of its center and the linear span of all polynomial commutators. Using the theory of rational identities in division rings, we also exhibit that the trace of a polynomial commutator in the matrix ring can be nonzero in noncommutative cases. Lastly, we explore the size of polynomial commutators via matrix norms.