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 $J_1\ne j_2.$ 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.

References in corpus (5)

• Gauge Symmetry and Supersymmetry of Multiple M2-Branes
• Algebraic structures on parallel M2-branes
• Modeling Multiple M2's
• 3-Lie-Rinehart Algebras
• Hom 3-Lie-Rinehart Algebras