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 ).
Agradeço, primeiramente à Deus, por sua infinita misericórdia e fidelidade. Aos meus pais, Armando Payares H., e Marlene Guevara H., razão maior da minha existência. À minha esposa e companheira incondicional Mariela Perez A., pela paciência, amor e compreensão nas minhas ausências. Agradeço a CAPES pelo importante apoio financeiro. À minha família, pelo apoio, incentivo, compreensão e paciência. Quero agradecer ao Prof. Dr. Alexandre Grichkov, pela excelente orientação. Aos membros da banca pelas correções, sugestões e orientações, para a versão final da tese. Sou grato aos meus amigos de doutorado, pela força e amizade.
In this paper we first study some properties of the finite-dimensional simple restricted Lie algebras. In the literature is found a one-to-one correspondence between them and finite-dimensional simple Lie algebras over a field of positive characteristic. Motivated by this fact, we give a one-to-one correspondence between their morphisms, which allow us to conclude that such categories are equivalent.
Simple Lie algebras of finite dimension over an algebraically closed field of characteristic 0 or p > 3 were recently classified. However, the problem over an algebraically closed field of characteristics 2 or 3 there exist only partial results. The first result on the problem of classification of simple Lie algebra of finite dimension over an algebraically closed field of characteristic 2 is that these algebras have absolute toral rank greater than or equal to 2. In this paper we show that there are not simple Lie 2-algebras with toral rank 3 over an algebraically closed field of characteristic 2 and dimension less or equal to 16.
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