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 $Γ({\mathfrak L},\B)$ 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 $Γ({\mathfrak L},\B)$. 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 $Γ({\mathfrak L},\B_{\mathfrak I})$ and $Γ({\mathfrak L},\B_{\mathfrak V})$.