On the structure of graded Poisson color algebras
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In this paper we introduce a class of graded Poisson color algebras as a natural generalization of graded Poisson algebras and graded Poisson superalgebras. For $\Lam$ an arbitrary abelian group, we show that any of such $\Lam$-graded Poisson color algebra $\pp,$ with a symmetric $\Lam$-support is of the form $\pp=\uu\oplus\sum_{j}I_{j},$ with $\uu$ a subspace of $\pp_1$ and any $I_j$ a well described graded ideal of $\pp,$ satisfying $\{I_j, I_k\}+I_j I_k=0$ if $j\neq i.$ Furthermore, under certain conditions, the gr-simplicity of $\pp$ is characterized and it is shown that $\pp$ is the direct sum of the family of its graded simple ideals.
Abstract
In this paper we introduce a class of graded Poisson color algebras as a natural generalization of graded Poisson algebras and graded Poisson superalgebras. For $\Lam$ an arbitrary abelian group, we show that any of such $\Lam$-graded Poisson color algebra $\pp,$ with a symmetric $\Lam$-support is of the form $\pp=\uu\oplus\sum_{j}I_{j},$ with $\uu$ a subspace of $\pp_1$ and any $I_j$ a well described graded ideal of $\pp,$ satisfying $\{I_j, I_k\}+I_j I_k=0$ if $j\neq i.$ Furthermore, under certain conditions, the gr-simplicity of $\pp$ is characterized and it is shown that $\pp$ is the direct sum of the family of its graded simple ideals.