Solvable Leibniz Algebras with Filiform Nilradical

Abstract

In this paper we continue the description of solvable Leibniz algebras whose nilradical is a filiform algebra. In fact, solvable Leibniz algebras whose nilradical is a naturally graded filiform Leibniz algebra are described in [6] and [8]. Here we extend the description to solvable Leibniz algebras whose nilradical is a filiform algebra. We establish that solvable Leibniz algebras with filiform Lie nilradical are Lie algebras.Mathematics Subject Classification 2010: 17A32, 17A65, 17B30.

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

“…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.…”

“…(3) If (n ≥ 5), then we apply the identities given in (6) We apply the same identities as in case (2) for (n = 4). (7) We apply the same identities as in case (1) for (n = 4).…”

“…If a 4,5 = 0, then we scale it to 1 and obtain a continuous family of Leibniz algebras: [e 5 , e 4 ] = 2(a − 1)e 2 , (a = 0, a = 1). (7) We apply the transformation e ′ 1 = e 1 + a 2,1 a e 2 , e ′ i = e i , (2 ≤ i ≤ 5) and assign d := a·b 2,3 +c·a 2,1 a , f := a·a 2,3 −c·a 2,1 a . Then this case becomes the same as case (7)…”

“…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].…”

“…re iφ e 4 , e ′ 5 = e 5 and obtain a Leibniz algebra given below: [e 5 , e 3 ] = −e 1 + ae 3 , (a = 0, a = 1). (7) We apply the transformation e ′…”