On derivations of some classes of Leibniz algebras

TL;DR

This paper characterizes derivations of certain complex Leibniz algebras, providing explicit dimensions of their derivation algebras and classifying derivations for various classes of filiform Leibniz algebras.

Contribution

It offers a detailed description of derivations for naturally graded and non-Lie filiform Leibniz algebras, including dimension bounds and classifications.

Findings

Derivation algebra dimensions for NGF_1, NGF_2, NGF_3 are n+1, n+2, and 2n-1 respectively.

For L in FLb_n, derivation dimension ranges from n-1 to n+1.

For L in SLb_n, derivation dimension ranges from n-1 to n+2.

Abstract

In the paper we describe the derivations of complex $n$-dimensional naturally graded filiform Leibniz algebras $NGF_1, NGF_2\ \text{and} \ \ NGF_3.$ We show that the dimension of the derivation algebras of $NGF_1$ and $NGF_2$ equals $n+1$ and $n+2,$ respectively, while the dimension of the derivation algebra of $NGF_3$ is equal to $2n-1.$ The second part of the paper deals with the description of the derivations of complex $n$-dimensional filiform non Lie Leibniz algebras, obtained from naturally graded non Lie filiform Leibniz algebras. It is well known that this class is split into two classes denoted by $FLb_n$ and $SLb_n.$ Here we found that for $L\in FLb_n$ we have $n-1 \leq dim\ Der(L)\leq n+1$ and for algebras $L$ from $SLb_n$ the inequality $n-1\leq dim\ Der(L) \leq n+2$ holds true.