Itô’s theorem and metabelian Leibniz algebras

Abstract

We prove that the celebrated Itô's theorem for groups remains valid at the level of Leibniz algebras: if g is a Leibniz algebra such that g = A + B, for two abelian subalgebras A and B, then g is metabelian, i.e. [ [g, g], [g, g] ] = 0. A structure type theorem for metabelian Leibniz/Lie algebras is proved. All metabelian Leibniz algebras having the derived algebra of dimension 1 are described, classified and their automorphisms groups are explicitly determined as subgroups of a semidirect product of groups P * ⋉ k * × Aut k (P ) associated to any vector space P . 2010 Mathematics Subject Classification. 17A32, 17A60, 17B30. Key words and phrases. metabelian Leibniz (Lie) algebras. A.L. Agore is Postdoctoral Fellow of the Fund for Scientific Research Flanders (Belgium) (F.W.O. Vlaanderen).

Classification of Some Solvable Leibniz Algebras

“…gives α 1 , β 2 ≠ 0. If α 2 = 0 and β 1 = 0, then setting x = 1 β2 a − 1 α1β2 a 2 , y = b, z = a 2 we have A = span{x, y, z} with the nonzero products given in (2) where α = α1 β2 . If α 2 = 0 and β 1 ≠ 0 and α 1 = β 2 then choosing x = 1 2α1 a − 1 2α 2 1 a 2 , y = α1+β1 2α1 a 2 + 1 2 b, z = a 2 + b gives A = span{x, y, z} with the nonzero products given in (3).…”

“…Then A = span{x, y, z} with the nonzero products given in (2). Now suppose dim(Leib(A)) = 1 and dim(Z(A)) = 1.…”

“…In particular, we show that the only isomorphism class of non-Lie non-split non-nilpotent solvable Leibniz algebra with a one dimensional derived subalgebra is the two dimensional solvable cyclic Leibniz algebra. After completion of this paper it was brought to our attention that a general classification result for (right) Leibniz algebras with one dimensional derived subalgebra is given in ( [2], Theorem 3.4) . However, the explicit isomorphism classes are hard to determine using this general result.…”

“…If α 2 = 0 and β 1 ≠ 0 and α 1 = β 2 then choosing x = 1 2α1 a , y = α1+β1 2α1 a 2 + 1 2 b, z = a 2 + b gives A = span{x, y, z} with the nonzero products given in(3). On the other hand, if α 2 = 0 and β 1 ≠ 0 andα 1 ≠ β 2 then choosing x = 1 β2 a − 1 α1β2 a 2 , y = β1 β2−α1 a 2 + b, z = 1β gives A = span{x, y, z} with the nonzero products given in(2) with α = α1 β2 ∈ C {0, 1}. If α 2 ≠ 0 and α 1 +β 2 = 0 then choosing x = 1 (α2β1−β2α1) 1 2 a− α1 (α2β1−β2α1) 3 2 a 2 − α2 (α2β1−β2α1) 3 2 b, y = [(α 2 β 1 −β 2 α 1 ) 1 2 −α 1 ]a 2 −α 2 b, z = [(α 2 β 1 −β 2 α 1 ) 1 2 +α 1 ]a 2 +α 2 b gives A = span{x, y, z} with the nonzero products given in (2) with α = −1.…”

Metabelian Associative Algebras

“…Fite [8] and since then they became a very important topic of study within group theory [14]. At the level of Lie, or more general Leibniz algebras, the corresponding concept of metabelian Lie/Leibniz algebra, as 2-step solvable algebra, is also well known [2,6,7]. In this paper we introduce the associative algebra counterpart of a metabelian group by defining a metabelian algebra over a field k as an extension of an abelian algebra by an abelian algebra -the word 'abelian' is borrowed from Lie algebras, i.e.…”