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Counterexamples to the Zassenhaus conjecture on simple modular Lie algebras.
- Source :
-
Journal of Algebra . Sep2023, Vol. 629, p21-37. 17p. - Publication Year :
- 2023
-
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 ≥ 1. For n = 1 this recovers the known counterexample of psl 3 (F). We show that the outer derivation algebra of H (2 ; (1 , n)) (2) is isomorphic to (sl 2 (F) ⋉ V (2)) ⊕ F n − 1 , where V (2) is the natural representation of sl 2 (F). We also study other known simple Lie algebras in characteristic three, but they do not yield a new counterexample. [ABSTRACT FROM AUTHOR]
- Subjects :
- *LOGICAL prediction
*ALGEBRA
*LIE algebras
Subjects
Details
- Language :
- English
- ISSN :
- 00218693
- Volume :
- 629
- Database :
- Academic Search Index
- Journal :
- Journal of Algebra
- Publication Type :
- Academic Journal
- Accession number :
- 163638163
- Full Text :
- https://doi.org/10.1016/j.jalgebra.2023.04.005