Free Poisson Hopf algebras generated by coalgebras

Abstract

We construct the analogue of Takeuchi's free Hopf algebra in the setting of Poisson Hopf algebras. More precisely, we prove that there exists a free Poisson Hopf algebra on any coalgebra or, equivalently that the forgetful functor from the category of Poisson Hopf algebras to the category of coalgebras has a left adjoint. In particular, we also prove that the category of Poisson Hopf algebras is a reflective subcategory of the category of Poisson bialgebras. Along the way, we describe coproducts and coequalizers in the category of Poisson Hopf algebras, therefore showing that the latter category is cocomplete.Comment: to appear in J. Math. Phy

Universal coacting Poisson Hopf algebras

“…where š iPI B `Pi , U ˘denotes the coproduct in Poiss F of the universal coacting Poisson algebras B `Pi , U ˘, i P I. We refer the reader to [2] for more detail on the (co)completeness of the categories Poiss F , PoissBialg F , PoissHopf F and explicit constructions of certain (co)limits including coproducts. Theorem 3.8.…”

“…With this Poisson bialgebra structure, the pair `BpP q, ψ BpP q ˘is proved to be the universal Poisson bialgebra of P (see Definition 3.9). Furthermore, the existence of a free Poisson Hopf algebra on any Poisson bialgebra ( [2]) allows us to construct a universal coacting Poisson Hopf algebra `HpP q, ψ HpP q ˘on any finite dimensional Poisson algebra P , i.e. ψ HpP q is a Poisson comodule algebra structure on P such that for any other Poisson Hopf algebra H and any Poisson comodule algebra structure ρ H : P Ñ P b H there exists a unique Poisson Hopf algebra homomorphism g : HpP q Ñ H for which the following diagram commutes:…”