Counterexamples to the Zassenhaus conjecture on simple modular Lie algebras

2022/09/29 by Dietrich Burde, Burde, Dietrich, Wolfgang Alexander Moens +3
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Advanced Algebra and Geometry

paper · pdf · doi:10.48550/arxiv.2209.14822

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 3ⁿ⁺¹-2 in characteristic p=3 do not have a solvable outer derivation algebra for all n≥ 1. For n=1 this recovers the known counterexample of \mathfrakpsl₃(F). We show that the outer derivation algebra of H(2;(1,n))⁽²⁾ is isomorphic to (\mathfraksl₂(F)\ltimes V(2))⊕ F^n-1, where V(2) is the natural representation of \mathfraksl₂(F). We also study other known simple Lie algebras in characteristic three, but they do not yield a new counterexample.

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