Power commuting and centralizing maps on the ring of strictly upper triangular matrices

TL;DR

This paper addresses the classification of $2$-power commuting maps on the ring $N_n(F)$ of strictly upper triangular matrices over a field $F$ of characteristic zero, i.e., maps $f$ with $[f(X),X^2]=0$ for all $X$. It proves that for $n\ge 4$, every linear such map decomposes as $f(X)=\lambda X+\mu(X)$ with $\mu(N_n)\subset\Psi$, where $\Psi$ is the 4-dimensional subspace spanned by $e_{1,n-1},e_{1,n},e_{2,n-1},e_{2,n}$. As a corollary, centralizing maps (those with $[g(X),X]\in Z(N_n)$) also admit a standard decomposition $g(X)=\lambda X+\mu(X)$ with $\mu(N_n)\subset\Omega$, where $\Omega$ is the 3-dimensional subspace spanned by $e_{1,n-1},e_{1,n},e_{2,n}$. The results show that $2$-power commuting maps need not be commuting and provide a complete description of centralizing maps, thereby extending the commuting-map theory to a 2-power setting for strictly upper triangular matrix rings.

Abstract

Let $N_n(F)$ denote the ring of strictly upper triangular matrices with entries in a field $F$ of characteristic zero and center $Z(N_n(F))$. We characterize the $2$-power commuting maps over $N_n(F)$, maps satisfying the identity $[f(X),X^2]=0$ for all $X\in N_n(F)$. As a consequence, we also obtain a characterization of the maps centralizing maps over $N_n(F)$, maps satisfying $[f(X),X]\in Z(N_n(F))$ for all $X\in N_n(F)$.

Theorems & Definitions (20)

• Theorem 1.1: Theorem 1 in bounds16
• Theorem 1.2
• Corollary 1.3
• Example 2.1
• Example 2.2
• Example 2.3
• Lemma 2.4
• proof
• Lemma 2.5
• proof