Leibniz algebras and graphs

Abstract

We consider a Leibniz algebra ${\mathfrak L} = {\mathfrak I} \oplus {\mathfrak V}$ over an arbitrary base field $\mathbb{F}$, being ${\mathfrak I}$ the ideal generated by the products $[x,x], x \in {\mathfrak L}$. This ideal has a fundamental role in the study presented in our paper. A basis $\B=\{v_i\}_{i \in I}$ of ${\mathfrak L}$ is called multiplicative if for any $i,j \in I$ we have that $[v_i,v_j] \in {\mathbb F}v_k$ for some $k \in I$. We associate an adequate graph $\Gamma({\mathfrak L},\B)$ to ${\mathfrak L}$ relative to $\B$. By arguing on this graph we show that ${\mathfrak L}$ decomposes as a direct sum of ideals, each one being associated to one connected component of $\Gamma({\mathfrak L},\B)$. Also the minimality of ${\mathfrak L}$ and the division property of ${\mathfrak L}$ are characterized in terms of the weak symmetry of the defined subgraphs $\Gamma({\mathfrak L},\B_{\mathfrak I})$ and $\Gamma({\mathfrak L},\B_{\mathfrak V})$.