Complex Nilpotent Structures in Soluble Lie Algebras

Abstract

Considering a Lie Algebra (g, [.,.]) With complex J structure, it is possible to define a new Lie bracket in g Considering a Lie Algebra g, [.,.] With complex structure J, it is possible to define in g a new Lie bracket, so that it can be shown that the subspaces g1,0 and g0,1 are sub-algebras of Lie isomorphic ag, [*]. For that, we only consider complex integrable structures. We will show that if these sub-algebras are nilpotent, then (g, [.,.]) It will be soluble. In this sense, a characterization of the Lie Algebras (g, [*] J) with a nilpotent s-step complex structure will be made, in order to study the behavior of the original Lie bracket [.,.], Thus allowing the construction of examples of Lie algebras of dim = 6.