Solvable Leibniz algebras with naturally graded non-Liep-filiform nilradicals

Abstract

In this paper solvable Leibniz algebras with naturally graded non-Lie p-filiform (n − p ≥ 4) nilradical and with one-dimensional complemented space of nilradical are described. Moreover, solvable Leibniz algebras with abelian nilradical and extremal (minimal, maximal) dimensions of complemented space nilradical are studied. The rigidity of solvable Leibniz algebras with abelian nilradical and maximal dimension of its complemented space is proved.

Local and 2-local derivations of solvable Leibniz algebras

“…, f n } is a basis of a n . In [1] such solvable algebras in case of k = n are classified and it is proved that any 2n-dimensional solvable Leibniz algebra with nilradical a n is isomorphic the direct sum of two dimensional algebras, i.e., isomorphic to the algebra…”

On some solvable Leibniz algebras and their completeness

“…Since N/N 2 is abelian, the solvable quotient Leibniz algebra algebra R/N 2 has abelian nilradical. Due to a result of the paper [2] the algebra R/N 2 has to have the table of multiplications (3.1).…”

Solvable Leibniz algebras with naturally graded non-Lie p-filiform nilradicals whose maximal complemented space of its nilradical

“…In this section we recall some results of the paper [1], which will be used below. We denote by a k the k-dimensional abelian algebras and by R(a k , s) the solvable Leibniz algebra with a k as nilradical and s as the dimension of complemented space to a k .…”

“…In this section we give a description of solvable Leibniz algebras whose nilradical is a naturally graded p-filiform Leibniz algebra and the dimension of Q is maximal. Firstly, we recall the derivations of the algebras µ i , i = 1, 2, 3 given in [1].…”

“…First we shall prove that HL 2 (R m , R m ) = 0 for any 1 ≤ m ≤ n − k + 1, then using this result we shall prove that HL 2 (R n−k+m , R n−k+m ) = 0 for any 2 ≤ m ≤ k. In both cases we shall use the induction method. The first step follows from the results of paper [1], where it is proved the triviality of the second cohomology group of the quotient algebra R 1 = {e 1 , f i , y i , y k+1 }, with 1 ≤ i ≤ k.…”