On the classification of simple Lie algebras of dimension seven over fields of characteristic 2
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On the classification of simple Lie algebras of dimension seven over fields of characteristic 2 Alexander Grishkov1,2 · Marinês Guerreiro3 · Wilian Francisco de Araujo4
© Instituto de Matemática e Estatística da Universidade de São Paulo 2020
Abstract This paper is the second part of paper (Grishkov and Guerreiro in São Paulo J Math Sci v4(1):93–107, 2010) about simple 7-dimensional Lie algebras over an algebraically closed field k of characteristic two. In this paper we prove that all simple 7-dimensional Lie algebras over k of absolute toral rank three are isomorphic to the Cartan algebra W1 or the Hamilton algebra H2 . We hope to prove that those algebras are the unique simple 7-dimensional Lie algebras over the field k. Observe that in the case of absolute toral rank 2 this fact was proved in [2]. Keywords Simple Lie algebra · Toral subalgebra · Absolute toral rank
Communicated by Vyacheslav Futorny. A. Grishkov: Supported by FAPESP and CNPq Processo 307824/2016-0, Brazil and RFBR, Grant 16-01-00577a, Russian. M. Guerreiro: Supported by FAPESP Processo N. 04/07774-2, Brazil. * Alexander Grishkov [email protected] Marinês Guerreiro [email protected] Wilian Francisco de Araujo [email protected] 1
Instituto de Matemática e Estatística, Universidade de São Paulo Rua do Matão 1010, São Paulo CEP 05508‑090, Brazil
2
Omsk State University, n.a. F.M.Dostoevskii pr. Mira 55‑A, Omsk, Russia 644077
3
Departamento de Matemática, Centro de Ciências Exatas e Tecnológicas, Universidade Federal de Viçosa, Viçosa, M.G., Brazil
4
Universidade Tecnológica Federal do Paraná, R. Cristo Rei, Vila Becker, Toledo, PR, Brazil
13
Vol.:(0123456789)
São Paulo Journal of Mathematical Sciences
1 Introduction The classification of simple finite dimensional Lie algebras over an algebraically closed field k of characteristic two is not finished until now. The first case when we have not yet the classification of the simple n-dimensional Lie algebras over k is the case n = 7 . We know two simple 7-dimensional Lie algebras over k, the Cartan algebra W1 and the Hamilton algebra H2 . Recall the definition of those algebras from [2]. In the two following tables we give the multiplication of the 2-closure W of the Witt–Zassenhaus algebra W1 and the 2-closure H of the Hamilton algebra H2 . 𝜂
𝜅
𝜅
y−1
y0
y1
y2
y3
y4
y5
𝜂
0
y4
y2
y5
0
0
0
0
0
0
𝜅
y4
𝜅
0
0
0
y−1
y0
y1
y2
y3
𝜅 [2]
y2
0
0
0
0
0
0
y−1
y0
y1
y−1
y5
0
0
𝜅
y−1
y0
y1
y2
y3
y4
y0
0
0
0
y−1
y0
y1
0
y3
0
y5
y1
0
y−1
0
y0
y1
y2
0
y4
y5
0
y2
0
y0
0
y1
0
0
0
y5
0
0
y3
0
y1
y−1
y2
y3
y4
y5
𝜂
0
0
y4
0
y2
y0
y3
0
y5
0
0
0
0
y5
0
y3
y1
y4
y5
0
0
0
0
0
Table of multiplication of the 2-closure W of W1. In this table the last seven elements form a basis of the simple Lie algebra W1 . t
m
n
V0
V1
E1
E0
F0
G
t
t
0
0
V0
V1
0
0
m
0
0
E0
0
0
0
0
F1
F0
0
V1
V0
n
0
E0
0
0
F1
G
0
E1
0
0
0
V0
V0
0
0
0
0
V1
V1
V1
0
F1
0
m
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