Categorical Constructions for 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.

Key Words:

• Bialgebra
• (co)Product
• (co)Limit
• (co)Complete
• (co)Reflective
• Hopf algebra

2000 Mathematics Subject Classification:

• 18A30, 18A40

ACKNOWLEDGMENTS

The author wishes to thank Professor Gigel Militaru, who suggested the problem studied here, for his great support and for the useful comments from which this manuscript has benefitted, as well as the referee for valuable suggestions and for indicating the article [9] and [10].

The author acknowledges partial support from CNCSIS grant 24/28.09.07 of PN II “Groups, quantum groups, corings and representation theory”.

Dedicated to the memory of Professor Liliana Pavel.

Notes

Communicated by E. Puczylowski.