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

TL;DR

This paper characterizes the projective indecomposable modules for restricted Zassenhaus algebras in characteristic 2, showing they all have maximal dimension and are induced from maximal tori, contrasting with higher characteristic cases.

Contribution

It establishes that all projective indecomposable modules for these algebras are of maximal dimension and induced from maximal tori, revealing a unique structural property in characteristic 2.

Findings

All projective indecomposable modules have maximal dimension 2^{2^n-1}.

These modules are isomorphic to induced modules from maximal tori.

This behavior contrasts with finite-dimensional simple restricted Lie algebras in characteristic p>3.

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 $\mathrm{ind}^{\mathfrak{W}}_{\mathfrak{t}}(\mathbb{F}(\mu))$ 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 $p>3$.