On extended graded Poisson algebras

We introduce the class of extended graded Poisson algebras as a generalization of the one of graded Poisson algebras and study its structure. If P is an extended graded Poisson algebra, we show that P is of the form P=U+∑iIi with U a linear subspace of P0 and any Ii a well described ideal of P, satisfying {Ii,Ij}+IiIj=0 if i≠qj. It is also shown that, under certain conditions, P is the direct sum of the family of its simple ideals.