Graded Poisson Algebras

1.1. Graded vector spaces. By a Z-graded vector space (or simply, graded vector space) we mean a direct sum A = ⊕i∈ZAi of vector spaces over a field k of characteristic zero. The Ai are called the components of A of degree i and the degree of a homogeneous element a ∈ A is denoted by |a|. We also denote by A[n] the graded vector space with degree shifted by n, namely, A[n] = ⊕i∈Z(A[n])i with (A[n])i = Ai+n. The tensor product of two graded vector spaces A and B is again a graded vector space whose degree r component is given by (A⊗B)r = ⊕p+q=rAp ⊗Bq. The symmetric and exterior algebra of a graded vector space A are defined respectively as S(A) = T (A)/IS and ∧ (A) = T (A)/I∧, where T (A) = ⊕n≥0A is the tensor algebra of A and IS (resp. I∧) is the two-sided ideal generated by elements of the form a ⊗ b − (−1)|a| |b|b ⊗ a (resp. a ⊗ b + (−1)|a| |b|b ⊗ a), with a and b homogeneous elements of A. The images of A⊗n in S(A) and ∧ (A) are denoted by S(A) and ∧n(A) respectively. Notice that there is a canonical decalage isomorphism S(A[1]) ' ∧n(A)[n].