On the structure of graded $3$-Lie-Rinehart algebras

Abstract

We study the structure of a graded 3-Lie-Rinehart algebra L over an associative and commutative graded algebra A. For G an abelian group, we show that if (L, A) is a tight G-graded 3-Lie-Rinehart algebra, then L and A decompose as L = i∈I L i and A = j∈J A j , where any L i is a non-zero graded ideal of L satisfying [L i1 , L i2 , L i3 ] = 0 for any i 1 , i 2 , i 3 ∈ I different from each other, and any A j is a non-zero graded ideal of A satisfying A j A l = 0 for any l, j ∈ J such that j = l, and both decompositions satisfy that for any i ∈ I there exists a unique j ∈ J such that A j L i = 0. Furthermore, any (L i , A j ) is a graded 3-Lie-Rinehart algebra. Also, under certain conditions, it is shown that the above decompositions of L and A are by means of the family of their, respectively, graded simple ideals.