Counterexamples to the Zassenhaus conjecture on simple modular Lie algebras

Abstract

We provide an infinite family of counterexamples to the conjecture of Zassenhaus on the solvability of the outer derivation algebra of a simple modular Lie algebra. In fact, we show that the simple modular Lie algebras $H(2;(1,n){)}^{(2)}$ of dimension $3^{n+1}-2$ in characteristic $p=3$ do not have a solvable outer derivation algebra for all $n\ge 1$. For $n=1$ this recovers the known counterexample of ${\mathfrak{psl}}_3(F)$. We show that the outer derivation algebra of $H(2;(1,n){)}^{(2)}$ is isomorphic to $({\mathfrak{sl}}_2(F)⋉V(2))\oplus F^{n-1}$, where $V(2)$ is the natural representation of ${\mathfrak{sl}}_2(F)$. We also study other known simple Lie algebras in characteristic three, but they do not yield a new counterexample.