On the structure of split regular Hom-Lie Rinehart algebras

Abstract: The aim of this paper is to study the structures of split regular Hom-Lie Rinehart algebras. Let $(L,A)$ be a split regular Hom-Lie Rinehart algebra. We first show that $L$ is of the form $L=U+\sum_{[\gamma]\in\Gamma/\thicksim}I_{[\gamma]}$ with $U$ a vector space complement in $H$ and $I_{[\gamma]}$ are well described ideals of $L $ satisfying $I_{[\gamma]},I_{[\delta]}=0$ if $I_{[\gamma]}\neq I_{[\delta]}$. Also, we discuss the weight spaces and decompositions of $A$ and present the relation between the decompositions of $L$ and $A$. Finally, we consider the structures of tight split regular Hom-Lie Rinehart algebras.