Universal coacting Poisson Hopf algebras

Abstract

We introduce the analogue of Manin’s universal coacting (bialgebra) Hopf algebra for Poisson algebras. First, for two given Poisson algebras P and U, where U is finite dimensional, we construct a Poisson algebra $$\mathcal {B}(P,\, U)$$ together with a Poisson algebra homomorphism $$\psi _{\mathcal {B}(P,\,U)} :P \rightarrow U \otimes \mathcal {B}(P,\, U)$$ satisfying a suitable universal property. $$\mathcal {B}(P,\, U)$$ is shown to admit a Poisson bialgebra structure for any pair of Poisson algebra homomorphisms subject to certain compatibility conditions. If $$P=U$$ is a finite dimensional Poisson algebra then $$\mathcal {B}(P) = \mathcal {B}(P,\, P)$$ admits a unique Poisson bialgebra structure such that $$\psi _{\mathcal {B}(P)}$$ becomes a Poisson comodule algebra and, moreover, the pair $$\bigl (\mathcal {B}(P),\, \psi _{\mathcal {B}(P)}\bigl )$$ is the universal coacting bialgebra of P. The universal coacting Poisson Hopf algebra $$\mathcal {H}(P)$$ on P is constructed as the initial object in the category of Poisson comodule algebra structures on P by using the free Poisson Hopf algebra on a Poisson bialgebra (Agore in J Math Phys 10:083502, 2014).