On the structure of graded Poisson color algebras

Abstract: In this paper we introduce the class of graded Poisson color algebras as the natural generalization of graded Poisson algebras and graded Poisson superalgebras. For $\Lambda$ an arbitrary abelian group, we show that any of such $\Lambda$-graed Poisson color algebra $\mathcal{P}$, with a symmetric $\Lambda$-support is of the form $\mathcal{P} = \mathcal{U}\oplus\sum_j I_j$, with $\mathcal{U}$ a subspace of $\mathcal{P}_1$ and any $I_j$ a well described graded ideal of $\mathcal{P},$ satisfying ${I_j, I_k}+I_j I_k=0$ if $j\neq i.$ Furthermore, under certain conditions, the gr-simplicity of $\mathcal{P}$ is characterized and it is shown that $\mathcal{P}$ is the direct sum of the family of its graded simple ideals.