Graded Lie-Rinehart algebras

Abstract

The present work introduces the class of graded Lie-Rinehart algebras as a natural generalization of graded Lie algebras. It is demonstrated that a tight G-graded Lie-Rinehart algebra L over a commutative and associative G-graded algebra A, where G is an abelian group, can be decomposed into the orthogonal direct sums L=⨁i∈IIi and A=⨁j∈JAj, where each Ii and Aj is a non-zero ideal of L and A, respectively. Additionally, both decompositions satisfy that for any i∈I, there exists a unique j∈J such that AjIi≠0 and that any Ii is a graded Lie-Rinehart algebra over Aj. In the case of maximal length, the aforementioned decompositions of L and A are through indecomposable (graded) ideals, and the (graded) simplicity of any Ii and any Aj are also characterized.