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 = L i∈I Li and A = L j∈J Aj , where any Li is a non-zero graded ideal of L satisfying [Li1 ,Li2 ,Li3 ] = 0 for any i1, i2, i3 ∈ I different from each other, and any Aj is a non-zero graded ideal of A satisfying AjAl = 0 for any l, j ∈ J such that j 6= l, and both decompositions satisfy that for any i ∈ I there exists a unique j ∈ J such that AjLi 6= 0. Furthermore, any (Li , Aj ) is a graded 3-Lie-Rinehart algebra. Also, under certain conditions, it is shown that the above decompositions