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arXiv · 2209.14822

Counterexamples to the Zassenhaus conjecture on simple modular Lie algebras

Dietrich Burde·Wolfgang Moens·Pilar Páez-Guillán
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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.

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Dietrich Burde, Wolfgang Moens, Pilar Páez-Guillán. 2022-09-29. Counterexamples to the Zassenhaus conjecture on simple modular Lie algebras. https://arxiv.org/abs/2209.14822

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