The projective indecomposable modules for the restricted Zassenhaus algebras in characteristic 2

Abstract

It is shown that for the restricted Zassenhaus algebra $\mathfrak{W}=\mathfrak{W}(1,n)$, $n>1$, defined over an algebraically closed field $\mathbb{F}$ of characteristic 2 any projective indecomposable restricted $\mathfrak{W}$-module has maximal possible dimension $2^{2^n-1}$, and thus is isomorphic to some induced module ${\mathrm{ind}}_{\mathfrak{t}}^{\mathfrak{W}}(\mathbb{F}(\mu ))$ for some torus of maximal dimension $\mathfrak{t}$. This phenomenon is in contrast to the behavior of finite-dimensional simple restricted Lie algebras in characteristic $p>3$.