Non-filiform characteristically nilpotent and complete Lie algebras

Abstract

One of the main achievements of the paper under review is the construction of new classes of characteristically nilpotent Lie algebras that are not filiform. In fact, in Theorem 4.5 one describes, for arbitrary m 4, a characteristically nilpotent Lie algebra of dimension 2m+ 2 whose characteristic sequence is (2m− 1, 2, 1). The starting point of that construction is given by the Lie algebras denoted by g4(m,m−1). The latter algebra is characterized in Theorem 3.7 as the only naturally graded central extension of L2m−1 by C with nilindex 2m − 1, where L2m−1 is the Lie algebra having a basis {X1, . . . ,X2m} such that [X1,Xi] = Xi+1 for 2 i 2m − 1, and [Xi,Xj ] = 0 for the other pairs of basis vectors. The main idea of the aforementioned construction of non-filifor characteristically nilpotent Lie algebras is to consider deformations of the algebras g4(m,m−1).