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Classical simple Lie 2-algebras of odd toral rank and a contragredient Lie 2-algebra of toral rank 4
Volume 62, no. 1
(2021),
pp. 123–139
https://doi.org/10.33044/revuma.1555
Abstract
After the classification of simple Lie algebras over a field of characteristic
$p > 3$, the main problem not yet solved in the theory of finite dimensional
Lie algebras is the classification of simple Lie algebras over a field of
characteristic 2. The first result for this classification problem ensures
that all finite dimensional Lie algebras of absolute toral rank 1 over an
algebraically closed field of characteristic 2 are soluble. Describing simple
Lie algebras (respectively, Lie $2$-algebras) of finite dimension of absolute
toral rank (respectively, toral rank) 3 over an algebraically closed field
of characteristic 2 is still an open problem.
In this paper we show that there are no classical type simple Lie $2$-algebras with
toral rank odd and furthermore that the simple contragredient Lie $2$-algebra
$G(F_{4, a})$ of dimension 34 has toral rank 4. Additionally, we give
the Cartan decomposition of $G(F_{4, a})$.
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Published by the Unión Matemática Argentina |
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