Solvable Leibniz Algebras with Heisenberg Nilradical

Abstract

All solvable Lie algebras with Heisenberg nilradical have already been classified. We extend this result to a classification of solvable Leibniz algebras with Heisenberg nilradical. As an example, we show the complete classification of all real or complex Leibniz algebras whose nilradical is the 3-dimensional Heisenberg algebra.

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

“…Right solvable extensions with a codimension one nilradical L 4 are found following the steps in Theorems 5.1.1, 5.1.2 and 5.1.3 with the main result summarized in Theorem 5.1.3, where it is shown there are eight such algebras: g n+1,i , (1 ≤ i ≤ 4, n ≥ 4), g 5,5 , g 5,6 , g 5,7 and g 5,8 . It is noticed that g n+1,1 is left as well when a = 0, g 5,5 is left when b = −1 and g n+1, 4 , g 5,7 , g 5,8 are right and left Leibniz algebras at the same time. There are four solvable indecomposable right Leibniz algebras with a codimension two nilradical: g n+2,1 , (n ≥ 5), g 6,2 , g 6,3 and g 6,4 stated in Theorem 5.1.4, where none of them is left.…”

“…We follow the steps in Theorems 5.2.1, 5.2.2 and 5.2.3 to find codimension one left solvable extensions. We notice in Theorem 5.2.3, that we have eight of them as well: l n+1,1 , l n+1,2 , l n+1, 3 , g n+1, 4 , l 5,5 , l 5,6 , g 5,7 and g 5,8 , such that l n+1,1 is right when a = 0 and l 5,5 is right when b = −1. We find four solvable indecomposable left Leibniz algebras with a codimension two nilradical as well: l n+2,1 , (n ≥ 5), l 6,2 , l 6,3 and l 6,4 stated in Theorem 5.2.4, where none of them is right.…”

“…We find four solvable indecomposable left Leibniz algebras with a codimension two nilradical as well: l n+2,1 , (n ≥ 5), l 6,2 , l 6,3 and l 6,4 stated in Theorem 5.2.4, where none of them is right. 5. Classification of solvable indecomposable Leibniz algebras with a nilradical L 4…”

“…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]. Left and right solvable extensions of R 18 [1], L 1 , L 2 and L 3 [6] over the field of real numbers were found by Shabanskaya in [22,23,24].…”

“…which is a limiting case of (5.1.3) with a = 0. Altogether (5.1.3) and all its limiting cases give us the algebra g 5,5 in full generality:…”