Split 3-Lie-Rinehart color algebras

TL;DR

This paper introduces split 3-Lie-Rinehart color algebras, characterizes their structure via root and weight systems, and demonstrates their decomposition into orthogonal sums of simple ideals.

Contribution

It generalizes split Lie-Rinehart algebras to 3-Lie-Rinehart color algebras and provides a detailed structural decomposition framework.

Findings

Decomposition of algebras into orthogonal sums of graded ideals.

Identification of conditions for simple ideal decomposition.

Establishment of the structure of 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 LieRinehart 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 $(\LL, A)$ decompose as the orthogonal direct sums $\LL =\oplus_{i\in I}\LL_i$ and $A =\oplus_{j\in J}A_j,$ where any $\LL_i$ is a non-zero graded ideal of $\LL$ satisfying $[\LL{i_1}, \LL{i_2}, \LL{i_3}]=0$ if $i_1, i_2, i_3\in I$ be different from each other and any $A_j$ is a non-zero graded ideal of A satisfying $A_{j_1}A_{j_2}=0$ if $J_1\neq j_2.$ Both decompositions satisfy that for any $i\in I$ there exists a unique $j\in J$ such that $A_j\LL_i = 0$. Furthermore, any $(\LL_i , A_j )$ is a split $3-$LieRinehart color algebra. Also, under certain conditions, it is shown that the above decompositions of $\LL$ and $A$ are by means of the family of their, respective, simple ideals.