Maps Preserving Zero Lie Brackets on a Maximal Nilpotent Subalgebra of the Symplectic Algebra

Abstract:

      Let $F$ be a field with char $F\neq2$, $l$ a maximal nilpotent subalgebra of the symplectic algebra $\sp(2m,F)$. In this paper, we characterize linear maps of $l$ which preserve zero Lie brackets in both directions. It is shown that for $m\geq 4$, a map $\varphi$ of $l$ preserves zero Lie brackets in both directions if and only if $\varphi=\psi_{c}\sigma_{T_0}\lambda_{\alpha}\phi_{d}\eta_{f}$, where $\psi_{c}, \sigma_{T_0}, \lambda_{\alpha}, \phi_{d}, \eta_{f}$ are the standard maps preserving zero Lie brackets in both directions.