Universal constructions for Poisson algebras. Applications

We introduce the \emph{universal algebra} of two Poisson algebras $P$ and $Q$ as a commutative algebra $A:={\mathcal P} (P, \, Q )$ 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 $U \otimes - \colon {}_{A} {\mathcal M} \to {}_Q{\mathcal P}{\mathcal M}$ 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 $- \otimes V \colon {}_P{\mathcal P}{\mathcal M} \to {}_Q{\mathcal P}{\mathcal M}$ 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 ${\mathcal P} (P) := {\mathcal P} (P, \, P)$ is the initial object in the category of all commutative bialgebras coacting on $P$. As an algebra, ${\mathcal P} (P)$ can be deescribed as the quotient of the polynomial algebra $k[X_{ij} \, | \, i, j = 1, \cdots, n]$ through an ideal generated by $2 n^3$ non-homogeneous polynomials of degree $\leq 2$. Two applications are provided. The first one describes the automorphisms group ${\rm Aut}_{\rm Poiss} (P)$ as the group of all invertible group-like elements of the finite dual ${\mathcal P} (P)^{\rm o}$. 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 ${\mathcal P} (P)$.