Classical simple Lie 2-algebras of odd toral rank and a contragredient Lie 2-algebra of toral rank 4

Carlos R Payares Guevara,Fabián A Arias Amaya

doi: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(F4,a) of dimension 34 has toral rank 4. Additionally, we give the Cartan decomposition of G(F4,a).

Highlights

The classification of the simple Lie algebras over an algebraically closed field of characteristic p with p ∈ {2, 3} is still an open problem
Grishkov classified Lie algebras of absolute toral rank 2. They announced in [2] the following result: All finite dimensional simple Lie algebras over an algebraically closed field of characteristic 2 of absolute toral rank 2 are classical of dimension 3, 8, 14 or 26
Every finite dimensional simple Lie 2-algebra over a field of characteristic 2 of toral rank 2 is isomorphic to A2, G2 or D4

Summary

Introduction

The classification of the simple Lie algebras over an algebraically closed field of characteristic p with p ∈ {2, 3} is still an open problem. Grishkov classified Lie algebras of absolute toral rank 2 They announced in [2] (work in progress) the following result: All finite dimensional simple Lie algebras over an algebraically closed field of characteristic 2 of absolute toral rank 2 are classical of dimension 3, 8, 14 or 26. The only classical type simple Lie 2-algebras of toral rank 4 over an algebraically closed field of characteristic 2 are the following: sl5(K), psl6(K), sp10(K)(2), and sp12(K)(2)/z(gl12(K)) (see Corollary 5.6). Theorem 2 gives us an example of a non-classical simple Lie 2-algebra, which should be taken into account in future investigations related to the problem of classifying the simple Lie 2-algebras of toral rank 4. Veısfeıler in [13] has toral rank 4, and we give the Cartan decomposition of this algebra

Preliminaries

Classical type simple Lie 2-algebras and their toral rank