On split regular Hom-Leibniz-Rinehart algebras

Abstract

In this paper, we introduce the notion of the Hom-Leibniz-Rinehart algebra as an algebraic analogue of Hom-Leibniz algebroid, and prove that such an arbitrary split regular Hom-Leibniz-Rinehart algebra \(L\) is of the form \(L=U+\sum_\gamma I_\gamma\) with \(U\) a subspace of a maximal abelian subalgebra \(H\) and any \(I_\gamma\), a well described ideal of \(L\), satisfying \([I_\gamma, I_\delta]= 0\) if \([\gamma]\neq [\delta]\). In the sequel, we develop techniques of connections of roots and weights for split Hom-Leibniz-Rinehart algebras respectively. Finally, we study the structures of tight split regular Hom-Leibniz-Rinehart algebras.