Low-dimensional nilpotent Leibniz algebras and their automorphism groups

Abstract

Let \(L\) be an algebra over a field \(F\) with the binary operations \(+\) and \([,]\). Then \(L\) is called a Leibniz algebra if it satisfies the Leibniz identity: \([a,[b,c]]=[[a,b],c]+[b,[a,c]]\) for all \(a,b,c\in L\). A linear transformation \(f\) of \(L\) is called an endomorphism of \(L\), if \(f([a,b])=[f(a),f(b)]\) for all \(a,b\in L\). A bijective endomorphism of \(L\) is called an automorphism of \(L\). The set of all automorphisms of Leibniz algebra is a group with respect to the operation of multiplication of automorphisms. The main goal of this article is to describe the structure of the automorphism group of a certain type of nilpotent three-dimensional Leibniz algebras over arbitrary field \(F\).