On epimorphisms and monomorphisms of Hopf algebras

TL;DR

This paper demonstrates that in the category of Hopf algebras, epimorphisms and monomorphisms can be non-surjective or non-injective respectively, providing counterexamples and exploring their properties.

Contribution

It provides explicit examples of non-surjective epimorphisms and non-injective monomorphisms in Hopf algebras, challenging common assumptions about these morphisms.

Findings

Universal map from a Hopf algebra to its enveloping Hopf algebra is an epimorphism but not necessarily surjective.

Examples of non-faithfully flat and non-faithfully coflat maps of Hopf algebras.

Dual results for monomorphisms and injectivity in Hopf algebra category.

Abstract

We provide examples of non-surjective epimorphisms $H\to K$ in the category of Hopf algebras over a field, even with the additional requirement that $K$ have bijective antipode, by showing that the universal map from a Hopf algebra to its enveloping Hopf algebra with bijective antipode is an epimorphism in $\halg$, although it is known that it need not be surjective. Dual results are obtained for the problem of whether monomorphisms in the category of Hopf algebras are necessarily injective. We also notice that these are automatically examples of non-faithfully flat and respectively non-faithfully coflat maps of Hopf algebras.