Graded Lie-Rinehart algebras
Elisabete Barreiro, Antonio J. Calder\'on, Rosa M. Navarro, Jos\'e, M. S\'anchez

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 an abelian group, we show that if is a tight -graded Lie-Rinehart algebra over an associative and commutative -graded algebra then and decompose as the orthogonal direct sums and , where any is a non-zero ideal of , any is a non-zero ideal of , and both decompositions satisfy that for any there exists a unique such that . Furthermore, any is a graded Lie-Rinehart algebra over . Also, under mild conditions, it is shown that the above decompositions of and are by means of the family of their, respective, gr-simple ideals.
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Taxonomy
TopicsAdvanced Topics in Algebra · Algebraic structures and combinatorial models · Advanced Algebra and Logic
