Classification of solvable Leibniz algebras with naturally graded filiform nilradical

Abstract

In this paper we show that the method for describing solvable Lie algebras with given nilradical by means of non-nilpotent outer derivations of the nilradical is also applicable to the case of Leibniz algebras. Using this method we extend the classification of solvable Lie algebras with naturally graded filiform Lie algebra to the case of Leibniz algebras. Namely, the classification of solvable Leibniz algebras whose nilradical is a naturally graded filiform Leibniz algebra is obtained.

Infinitesimal deformations of naturally graded filiform Leibniz algebras

“…In order to achieve the purpose of the subsection we need the matrix form of a derivation of the filiform Leibniz algebra F 1 n [14]:…”

“…Further we shall use the result in [14] which describes the derivations of the filiform Leibniz algebra F 2 n . Namely, any derivation of F 2 n has the following matrix form:…”

Solvable extensions of the naturally graded quasi-filiform Leibniz algebra of second type L4

“…Leibniz algebras inherit an important property of Lie algebras which is that the right (left) multiplication operator of a right (left) Leibniz algebra is a derivation [9]. Besides the algebra of right (left) multiplication operators is endowed with a structure of a Lie algebra by means of the commutator [9].…”

“…Leibniz algebras inherit an important property of Lie algebras which is that the right (left) multiplication operator of a right (left) Leibniz algebra is a derivation [9]. Besides the algebra of right (left) multiplication operators is endowed with a structure of a Lie algebra by means of the commutator [9]. Also the quotient algebra by the two-sided ideal generated by the square elements of a Leibniz algebra is a Lie algebra [19], where such ideal is the minimal, abelian and in the case of non-Lie Leibniz algebras it is always non trivial.…”

“…Besides, similarly to the case of Lie algebras, for a solvable Leibniz algebra L we also have the inequality dim nil(L) ≥ 1 2 dim L [18]. There is the following work performed over the field of characteristic zero using this method: Casas, Ladra, Omirov and Karimjanov classified solvable Leibniz algebras with null-filiform nilradical [8]; Omirov and his colleagues Casas, Khudoyberdiyev, Ladra, Karimjanov, Camacho and Masutova classified solvable Leibniz algebras whose nilradicals are a direct sum of null-filiform algebras [13], naturally graded filiform [9,14], triangular [12] and finally filiform [7]. Bosko-Dunbar, Dunbar, Hird and Stagg attempted to classify left solvable Leibniz algebras with Heisenberg nilradical [5].…”