Graded Lie-Rinehart algebras

TL;DR

This paper introduces graded Lie-Rinehart algebras, generalizing graded Lie algebras, and studies their decomposition into orthogonal sums of ideals, revealing their structure and relationships with graded associative algebras.

Contribution

It extends the concept of graded Lie algebras to Lie-Rinehart algebras and characterizes their decompositions into simple ideals, providing new structural insights.

Findings

Decomposition of graded Lie-Rinehart algebras into orthogonal sums of ideals.

Each ideal forms a graded Lie-Rinehart algebra over a corresponding ideal of the associative algebra.

Decompositions are achieved via families of gr-simple ideals.

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.