Counterexamples to the Zassenhaus conjecture on simple modular Lie algebras

TL;DR

This paper constructs an infinite family of counterexamples to Zassenhaus's conjecture by showing certain simple modular Lie algebras have non-solvable outer derivation algebras, expanding understanding of their structure in characteristic three.

Contribution

It introduces an infinite family of counterexamples to Zassenhaus's conjecture, detailing the structure of their outer derivation algebras in characteristic three.

Findings

Counterexamples for all n ≥ 1 with non-solvable outer derivation algebras

Explicit isomorphism of outer derivation algebra to (sl_2(F)  V(2))  F^{n-1}

Other known simple Lie algebras in characteristic three do not provide new counterexamples

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 $(\mathfrak{sl}_2(F)\ltimes V(2))\oplus F^{n-1}$, 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.