Leibniz algebras: a brief review of current results

Abstract

Let $L$ be an algebra over a field $F$ with the binary operations $+$ and $[\cdot,\cdot]$. Then $L$ is called a left Leibniz algebra if it satisfies the left Leibniz identity $[[a,b],c]=[a,[b,c]]-[b,[a, c]]$ for all $a,b,c\in L$. This paper is a brief review of some current results, which related to finite-dimensional and infinite-dimensional Leibniz algebras.