Publication
Universal coacting Poisson Hopf algebras
Journal Contribution - Journal Article
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 B(P,U) together with a Poisson algebra homomorphism ψB(P,U):P→U⊗B(P,U) satisfying a suitable universal property. 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 B(P)=B(P,P) admits a unique Poisson bialgebra structure such that ψB(P) becomes a Poisson comodule algebra and, moreover, the pair (B(P),ψB(P)) is the universal coacting bialgebra of P. The universal coacting Poisson Hopf algebra 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).