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

TL;DR

This paper characterizes specific algebraic maps called 2-power commuting and centralizing maps on the ring of strictly upper triangular matrices over a field of characteristic zero, revealing their structural properties.

Contribution

It provides a detailed characterization of 2-power commuting and centralizing maps on $N_n(F)$, extending understanding of their algebraic structure and behavior.

Findings

Characterization of 2-power commuting maps satisfying $[f(X),X^2]=0$.

Description of centralizing maps with $[f(X),X] ext{ in } Z(N_n(F))$.

Results applicable to algebraic structures over fields of characteristic zero.

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 $X\in N_n(F)$.