可裂3-Lie-Rinehart色代数
Split 3-Lie-Rinehart color algebras
摘要：
本文引入了一类$3-$色代数，称之为分裂$3-$ Lie-Rinehart色代数，它是分裂Lie-Rinehart代数的自然推广.我们通过发展与分裂Cartan子代数相关的根系和权系的连接技术来刻画它们的内部结构。本文证明了这样的紧分裂$3-$ Lie-Rinehart色代数$(\LL, A)$分解为正交直和$\LL =\oplus_{i\in I}\LL_i$和$A =\oplus_{j\in J}A_j,$，其中任意$\LL_i$是满足$[\LL{i_1}, \LL{i_2}, \LL{i_3}]=0$的$\LL$的非零分次理想，如果$i_1, i_2, i_3\in I$互不相同，且任意$A_j$是非零分次理想，A的零分次理想满足$A_{j_1}A_{j_2}=0$如果$J_1\neq j_2.$两个分解都满足对任何$i\in I$存在唯一的$j\in J$使得$A_j\LL_i = 0$.此外，任何$(\LL_i , A_j )$都是分裂的$3-$ LieRinehart色代数。此外，在一定的条件下，它表明，上述分解的$\LL$和$A$是通过家庭的，各自的，简单的理想。
英文摘要：
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.
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