Commuting maps on the Heisenberg algebra

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.