On the structure of graded Poisson color algebras

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.