Graded Lie-Rinehart algebras

Abstract

We introduce the class of graded Lie-Rinehart algebras as a natural generalization of the one of graded Lie algebras. For $G$ an abelian group, we show that if $L$ is a tight $G$-graded Lie-Rinehart algebra over an associative and commutative $G$-graded algebra $A$ then $L$ and $A$ decompose as the orthogonal direct sums $L = \bigoplus_{i \in I}I_i$ and $A = \bigoplus_{j \in J}A_j$, where any $I_i$ is a non-zero ideal of $L$, any $A_j$ is a non-zero ideal of $A$, and both decompositions satisfy that for any $i \in I$ there exists a unique $j \in J$ such that $A_jI_i \neq 0$. Furthermore, any $I_i$ is a graded Lie-Rinehart algebra over $A_j$. Also, under mild conditions, it is shown that the above decompositions of $L$ and $A$ are by means of the family of their, respective, gr-simple ideals.