A nonlinear analogue of additive commutators

TL;DR

The paper initiates a systematic study of nonlinear polynomial commutators $p(ab)-p(ba)$ in noncommutative algebra. It extends the additive-commutator framework to division rings and matrix algebras, establishing that $p[R,R]$ is nontrivial and can generate maximal subfields, and that $R$ can be generated by a polynomial commutator (up to conjugacy). In matrix settings, trace behavior of $p[A,B]$ can be nonzero, yet many traceless matrices arise as polynomial commutators, and matrices enjoy product-decomposition results into a bounded number of polynomial commutators, along with center–commutator decompositions. The authors also develop structural and size estimates, including span- and identity-type results, and three analytic tools (Frobenius-norm bound, numerical-radius bound, and an integral representation) to bound $||p[A,B]||$. Collectively, the work broadens the understanding of nonlinear commutators in noncommutative algebras and provides foundational tools for identities, representations, and operator-norm analyses.

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.

Theorems & Definitions (54)

• Theorem 1.2
• Theorem 2.1
• proof
• Theorem 2.2
• proof
• Theorem 2.3
• Lemma 2.4
• Lemma 2.5
• Lemma 2.6
• proof