Solvable Extensions of Nilpotent Complex Lie Algebras of Type {2n,1,1}

Abstract

We investigate derivations of nilpotent complex Lie algebras of type {2n, 1, 1} with the aim to classify solvable complex Lie algebras the commutator ideals of which have codimension 1 and are nilpotent Lie algebras of type {2n, 1, 1}.

A class of nilpotent Lie algebras admitting a compact subgroup of automorphisms

“…with J (0) the Jordan block of dimension and eigenvalue λ = 0. By a conjugation, we put them in the shape −Ā 1 A 2 A 3 A 1 , and when we take A h to be scalar complex matrices of the form A 1 = ixI , A 2 = −Ā 3 = zI , with x ∈ R and z ∈ C, we get the compact subalgebra k of the elements of the shape −ixI zI −zI ixI , which is manifestly isomorphic to sp (1).…”

“…. , d c } of a nilpotent Lie algebra g with descending central series g (i) = [g, g (i−1) ] is defined, according to the literature beginning with Vergne [20], by the integers d i = dim g (i− 1) g (i) . Nilpotent (real or complex) Lie algebras of type {n, 2} have been classified firstly by Gauger [11], applying the canonical reduction of the pair F 1 , F 2 , but, according to results of Belitskii, Lipyanski, and Sergeichuk [3], it is not possible to carry this argument further.…”

“…We mention that also nilpotent Lie algebras of type {n, 1, 1} can be explicitly described (cf. [2]), and derivations of a nilpotent Lie algebra of type {2n, 1, 1} are determined in [1].…”

Non-nilpotent Leibniz algebras with one-dimensional derived subalgebra

“…One of them is the Levi decomposition, which states that any Leibniz algebra over a field F of characteristic zero is the semidirect sum of its radical and a semisimple Lie algebra. This makes clear the importance of the problem of classification of solvable and nilpotent Lie / Leibniz algebras, which has been dealt with since the early 20th century (see [2], [3], [4], [9], [10], [11], [13] and [14], just for giving a few examples).…”