Categorical Constructions for Hopf Algebras

TL;DR

This paper establishes the existence of right adjoints for the embedding of Hopf algebras into bialgebras and the forgetful functor to algebras, solving a longstanding problem and explicitly describing categorical constructions.

Contribution

It proves that every bialgebra has a Hopf coreflection and each algebra admits a cofree Hopf algebra, providing new categorical insights into Hopf algebra theory.

Findings

Every bialgebra has a Hopf coreflection.

Existence of cofree Hopf algebras on algebras.

Explicit descriptions of coequalizers and coproducts in Hopf algebras.

Abstract

We prove that both, the embedding of the category of Hopf algebras into that of bialgebras and the forgetful functor from the category of Hopf algebras to the category of algebras, have right adjoints; in other words: every bialgebra has a Hopf coreflection and on every algebra there exists a cofree Hopf algebra. In this way we give an affirmative answer to a forty years old problem posed by Sweedler. On the route the coequalizers and the coproducts in the category of Hopf algebras are explicitly described.