The action of various operations on the cohomology of nilpotent Lie algebras is studied. In the cohomology of any Lie algebra, we show that the existence of certain nontrivial compositions of higher cohomology operations implies the existence of hypercube-like structures in cohomology, which in turn establishes the Toral Rank Conjecture for that Lie algebra. We provide examples in low dimensions and exhibit an infinite family of nilpotent Lie algebras of arbitrary dimension for which such structures exist. A new proof of the Toral Rank Conjecture is also given for free two-step nilpotent Lie algebras.

The action of various operations on the cohomology of nilpotent Lie algebras is studied. In the cohomology of any Lie algebra, we show that the existence of certain nontrivial compositions of higher cohomology operations implies the existence of hypercube-like structures in cohomology, which in turn establishes the Toral Rank Conjecture for that Lie algebra. We provide examples in low dimensions and exhibit an infinite family of nilpotent Lie algebras of arbitrary dimension for which such structures exist. A new proof of the Toral Rank Conjecture is also given for free two-step nilpotent Lie algebras.

The action of various operations on the cohomology of nilpotent Lie algebras is studied. In the cohomology of any Lie algebra, we show that the existence of certain nontrivial compositions of higher cohomology operations implies the existence of hypercube-like structures in cohomology, which in turn establishes the Toral Rank Conjecture for that Lie algebra. We provide examples in low dimensions and exhibit an infinite family of nilpotent Lie algebras of arbitrary dimension for which such structures exist. A new proof of the Toral Rank Conjecture is also given for free two-step nilpotent Lie algebras.

Neste trabalho estudamos as 2-álgebras de Lie simples, de dimensão finita e de posto toral 3, sobre um corpo algebricamente fechado de característica 2. Nós conjecturamos que a única 2-álgebra de Lie simples de este tipo é W(1, 3). Assim, nosso principal objetivo é verificar a veracidade desta conjectura para estas álgebras de pequenas dimensões. Como resultados, provamos que esta conjectura é certa para todas estes álgebras de dimensão menor ou igual a 16, e também em alguns casos especiais quando a dimensão é 17. / In this work we study the simple Lie 2-algebras of finite dimension, and toral rank 3 over an algebraically closed field characteristic 2. We surmise that the only simple Lie 2-algebra of this type is W(1, 3). So, our main objective is to study the truthful of this conjecture for these algebras of small dimensions. As a result, we prove that this conjecture is true for all these algebras less than or equal to 16 dimension, and also in some special cases when the dimension is 17.

S.Skryabin demonstrou que qualquer álgebra de Lie simples de dimensão finita sobre um corpo de característica 2 possui posto toroidal 2. Duas 2- álgebras de Lie de dimensão 31 foram estudadas. Neste trabalho, mostramos que a primeira delas contem uma base toroidal absoluta de dimensão três, assim como a segunda, que foi estudada por Grishkov e Guerreiro anteriormente. Utilizando uma decomposicão de Cartan, exibimos um isomorfismo entre as duas 2- álgebras de Lie de dimensão 31. Este resultado foi sugerido depois de encontrar uma sub álgebra de dimensão 12 n ao solúvel e 7 isomorfas 2-sub álgebras de Lie de dimensão 7 nas duas álgebras. Finalmente, exploramos uma 2- álgebra de Lie de dimensão 34 como o fim de encontrar base toroidal absoluta de dimensão 4. Apoiamos os cálculos com algumas códigos no linguajem de MATLAB que permitiram optimizar e acelerar a pesquisa. / S.Skryabin showed that any finite dimensional simple Lie algebra over a field of characteristic 2 has absolute toral rank 2. Two 31-dimensional 2-algebras were known. In this work, we show that the first of these algebras, contains a 3-dimensional maximal toral subalgebra, as the second one, which was studied by Grishkov e Guerreiro previously. Using a Cartan decomposition we establish an isomorphism between the two 31-dimensional 2-algebras. This result was suggested after finding a 12-dimensional not soluble subalgebra and seven 7-dimensional isomorphic 2-subalgebras in both algebras. Finally, a 34-dimensional 2-Lie algebra was studied in order to find 4-dimensional maximal toral subalgebras. Some computations in this work were performed with help of MATLAB.

Neste trabalho estudamos as 2-álgebras de Lie simples, de dimensão finita e de posto toral 3, sobre um corpo algebricamente fechado de característica 2. Nós conjecturamos que a única 2-álgebra de Lie simples de este tipo é W(1, 3). Assim, nosso principal objetivo é verificar a veracidade desta conjectura para estas álgebras de pequenas dimensões. Como resultados, provamos que esta conjectura é certa para todas estes álgebras de dimensão menor ou igual a 16, e também em alguns casos especiais quando a dimensão é 17. / In this work we study the simple Lie 2-algebras of finite dimension, and toral rank 3 over an algebraically closed field characteristic 2. We surmise that the only simple Lie 2-algebra of this type is W(1, 3). So, our main objective is to study the truthful of this conjecture for these algebras of small dimensions. As a result, we prove that this conjecture is true for all these algebras less than or equal to 16 dimension, and also in some special cases when the dimension is 17.