Commuting maps on the Heisenberg algebra

TL;DR

This paper investigates linear maps that commute with all elements in the Heisenberg algebra, revealing that such maps can deviate from the standard form and providing a characterization of these maps.

Contribution

It characterizes linear commuting maps on the Heisenberg algebra, showing they can differ from the standard form and providing new insights into their structure.

Findings

Not all commuting maps are of the standard form.

A complete characterization of commuting maps on the Heisenberg algebra.

Identification of conditions under which commuting maps deviate from standard form.

Abstract

Given a ring $R$ with center $Z(R)$, we say a linear map $f:R\rightarrow R$ is commuting if $[f(x),x]=0$ for all $x\in R$. Such a map has a standard form if there exists $\lambda\in R$ and additive $\mu:R\rightarrow Z(R)$ such that $f(x)=\lambda x+\mu(x)$ for all $x\in R$. We characterize the linear commuting maps over the $n\times n$ Heisenberg algebra, showing that such maps need not be of the standard form.