Leibniz algebras and graphs

Abstract

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