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.

to appear in J. Math. Phys

References in corpus (3)

• Poisson geometrical symmetries associated to non-commutative formal diffeomorphisms
• Poisson Hopf algebras associated to quantized enveloping algebras
• Universal enveloping algebras of Poisson Hopf algebras