Classification of solvable Leibniz algebras with null-filiform nilradical

Abstract

In this paper we classify solvable Leibniz algebras whose nilradical is a null-filiform algebra. We extend the obtained classification to the case when the solvable Leibniz algebra is decomposed as a direct sum of its nilradical, which is a direct sum of null-filiform ideals, and a onedimensional complementary subspace. Moreover, in this case we establish that these ideals are ideals of the algebra, as well.2010 Mathematics Subject Classification. 17A32, 17A65, 17B30.

Infinitesimal deformations of naturally graded filiform Leibniz algebras

“…, a n−1 ) ∈ X . (2) Note that the algebra ν 2 is the unique (up to isomorphism) solvable Leibniz algebra with null-filiform nilradical [15].…”

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

“…Since N is the nilradical of L, no nontrivial linear combination of the matrices A a , (n+ 1 ≤ a ≤ p) is nilpotent i.e. the matrices A a must be "nil-independent" [8,18].…”

“…This method for Lie algebras is based on what was shown by Mubarakzyanov in [18]: the dimension of the complimentary vector space to the nilradical does not exceed the number of nil-independent derivations of the nilradical. This result was extended to Leibniz algebras by Casas, Ladra, Omirov and Karimjanov [8] with the help of [2]. 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].…”

“…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].…”

An irreducible component of the variety of Leibniz algebras having trivial intersection with the variety of Lie algebras

“…To a certain extent, this actually holds, [4,5] albeit strong differences are soon encountered. [6] Possibly the best studied case is that of nilpotent Leibniz algebras, [7,8] as well as the classification problem in low dimensions. The analysis of the variety of Leibniz algebras LE n and its irreducible components enlarges that of the case of Lie algebras, and refers in particular to the rigidity problem.…”