Commuting maps on the Heisenberg algebra
Jordan Bounds, Ellis Edinkrah
TL;DR
This work examines linear commuting maps on the Heisenberg algebra $\mathcal{H}_n(F)$, showing that such maps need not conform to the standard form familiar from matrix algebras. The authors develop a coordinate-decomposition approach and establish a precise classification: every linear commuting map $f$ with $[f(X),X]=0$ admits a representation $f(X)=\{X,A\}+X^\tau B+CX^\tau+\zeta(X)$ with $A\in\mathcal{C}_n$, $B,C\in\mathcal{P}_n$, and $\zeta$ additive into the center $Z(\mathcal{H}_n)$; this demonstrates a richer structure than in the standard-form paradigm. By constructing explicit nonstandard examples and a detailed proof, the paper shows that commuting maps on $\mathcal{H}_n$ can deviate from the conventional form, contributing to the broader theory of commuting maps on nilpotent algebras and potentially informing related physics contexts involving the Heisenberg structure.
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 $λ\in R$ and additive $μ:R\rightarrow Z(R)$ such that $f(x)=λx+μ(x)$ 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.