Some Nilpotent Lie Algebras and Its Applications

Abstract

Abstract. We examine the problem of describing filiform Lie algebras over the field C admitting a semisimple derivation. As applications of obtained results we give a description of Lie algebras with a given filiform nilradical.

Chern-flat and Ricci-flat invariant almost Hermitian structures

“…16 , e26 , e 56 , e 35 , e 45 t e 6 ∧ f 1 − e 5 ∧ f 2 + f 34 : de6 ∧ f 1 = de 5 ∧ f 6 ∧ f 1 − e 5 ∧ f 2 + f 34 : e 13 ∧ f 1 = e 12 ∧ f2 0, 0, 0, 0, 0, 12 e 36 , e 46 , e 56 t 4.3, we have the following result which generalizes the cases of the h 3 (R)⊕ R and h 3 (C) = (0, 0, 0, 0, 13 + 42, 14 + 23).…”

“…It follows from Corollary 1.3 that studying Chern-flat structures on nilmanifolds is equivalent to studying anti-abelian almost complex structures on nilpotent Lie algebras (nilalgebras). According to published classifications, there are just 3 isomorphism classes of 4-dimensional nilalgebras and there are 34 isomorphism classes of 6-dimensional nilpotent Lie algebras [20,16]. These isomorphism classes can be described using an adequate choice of the basis as in [22] (see Table 1).…”

Determination of 7-Dimensional Indecomposable Nilpotent Complex Lie Algebras by Adjoining a Derivation to 6-Dimensional Lie Algebras

“…From now on, we speak only of adjoint cohomology. 9 As in [8] the value ξ = 1 has to be excluded as it gives an algebra isomorphic to 2357 D . 10 There are 3 missing values for λ : λ = 1 (v = ∞) which refer to 1357 Q , λ = 0 (v = −1) which refer to 1357 P and λ = −1 (v = 0) which is the value dismissed for g 7,1.2(iλ) .…”

“…Several classifications appeared later based on different approaches [1,9,24,26]. However, due to the lack of a good invariant and the complexity of the calculations involved, the completeness issue of the lists was a recurrent problem.…”

“…We get forg δ 15 nonisomorphic algebras (Tables 7,8,9). Table The cross table (Table 10) presents the strain of each of the 30 complex NLA's of dimension 6, including the 10 direct products g 5, j × C, (1 j 6), g 4 × C 2 , n × n, n × C 3 , C 6 .…”