Universal constructions for Poisson algebras. Applications

Abstract

We introduce the universal algebra of two Poisson algebras P and Q as a commutative algebra A:=P(P,Q) satisfying a certain universal property. The universal algebra is shown to exist for any finite dimensional Poisson algebra P and several of its applications are highlighted. For any Poisson P-module U, we construct a functor U⊗−:AM→QPM from the category of A-modules to the category of Poisson Q-modules which has a left adjoint whenever U is finite dimensional. Similarly, if V is an A-module, then there exists another functor −⊗V:PPM→QPM connecting the categories of Poisson representations of P and Q and the latter functor also admits a left adjoint if V is finite dimensional. If P is n-dimensional, then P(P):=P(P,P) is the initial object in the category of all commutative bialgebras coacting on P. As an algebra, P(P) can be described as the quotient of the polynomial algebra k[Xij|i,j=1,⋯,n] through an ideal generated by 2n3 non-homogeneous polynomials of degree ≤2. Two applications are provided. The first one describes the automorphisms group AutPoiss(P) as the group of all invertible group-like elements of the finite dual P(P)o. Secondly, we show that for an abelian group G, all G-gradings on P can be explicitly described and classified in terms of the universal coacting bialgebra P(P).