Split 3-Lie-Rinehart color algebras

Abstract

In this paper we introduce a class of 3−color algebras which are called split 3−Lie-Rinehart color algebras as the natural generalization of the one of split Lie-Rinehart algebras. We characterize their inner structures by developing techniques of connections of root systems and weight systems associated to a splitting Cartan subalgebra. We show that such a tight split 3−Lie-Rinehart color algebras (L, A) decompose as the orthogonal direct sums L = ⊕ i∈I L i and A = ⊕ j∈J A j , where any L i is a non-zero graded ideal of L stisfying [Li 1 , Li 2 , Li 3 ] = 0 if i 1 , i 2 , i 3 ∈ I be different from each other and any A j is a non-zero graded ideal of A stisfying A j1 A j2 = 0 if J 1 = j 2 . 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 split 3−Lie-Rinehart color algebra. Also, under certain conditions, it is shown that the above decompositions of L and A are by means of the family of their, respective, simple ideals.

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

“…split 3-Lie-Rinehart algebra is example of graded 3-Lie-Rinehart algebra. So this paper extends the results obtained in [23].…”