3-filiform Leibniz algebras of maximum length

Abstract

This work completes the study of the solvable Leibniz algebras, more precisely, it completes the classification of the 3-filiform Leibniz algebras of maximum length [4]. Moreover, due to the good structure of the algebras of maximum length, we also tackle some of their cohomological properties. Our main tools are the previous result of Cabezas and Pastor [3], the construction of appropriate homogeneous basis in the considered connected gradation and the computational support provided by the two programs implemented in the software Mathematica.We will say that a Z-graded Leibniz algebra L admits a connected gradationLet us define the naturally graded algebras as follows:⊆ L i+j and we obtain the graded algebra grL. If grL and L are isomorphic, in notation grL ∼ = L, we say that L is a naturally graded algebra.The above constructed gradation is called natural gradation.Definition 1.2. The number l(⊕L) = l(V k1 ⊕ V k1+1 ⊕ · · · ⊕ V k1+t ) = t + 1 is called the length of the gradation, where ⊕L is a connected gradation. The gradation ⊕L has maximum length if l(⊕L) = dim(L).We define the length of an algebra L by: l(L) = max{l(⊕L) such that ⊕ L = V k1 ⊕ · · · ⊕ V kt is a connected gradation}. An algebra L is called of maximum length if l(L) = dim(L). The set R(L) = {x ∈ L : [y, x] = 0, ∀y ∈ L} is called the right annihilator of L. R x denotes the operator R x : L → L such that R x (y) = [y, x], ∀y ∈ L and it is called the right operator. The set Cent(L) = {z ∈ L : [x, z] = [z, x] = 0, ∀x ∈ L} is called the center of L.Let x be a nilpotent element of the set L \ L 2 . For the nilpotent operator R x we define a descending sequence C(x) = (n 1 , n 2 , . . . , n k ), which consists of the dimensions of the Jordan blocks of the operator R x . In the set of such sequences we consider the lexicographic order, that is, C(x) = (n 1 , n 2 , . . . , n k ) < C(y) = (m 1 , m 2 , . . . , m s ) if and only if there exists i ∈ N such that n j = m j for any j < i and n i < m i .Definition 1.3. The sequence C(L) = max C(x) x∈L\L 2 is called the characteristic sequence of the algebra L.

p -Filiform Leibniz algebras of maximum length

“…In [1], [5] - [7] the classification of p-filiform Leibniz algebras of maximum length for 0 ≤ p ≤ 3 is already closed.…”