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 $L={⨁}_{i\in I}{I}_i$ and $A={⨁}_{j\in J}{A}_j$, 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 $A_j{I}_i\ne 0$. 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.

arXiv admin note: substantial text overlap with arXiv:1706.07084

References in corpus (3)

• On Graded classical 2-absorbing second submodules of graded modules over graded commutative rings
• Gradings of Lie algebras, magical spin geometries and matrix factorizations
• On graded $I_e$-prime submodules of graded modules over graded commutative rings

Cited by in corpus (1)

• Foundations of Noncommutative Carrollian Geometry via Lie-Rinehart Pairs