On epimorphisms and monomorphisms of Hopf algebras

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.

💡 Research Summary

The paper addresses a fundamental categorical question in the theory of Hopf algebras: whether epimorphisms must be surjective and whether monomorphisms must be injective when the objects are Hopf algebras over a field. While in many familiar algebraic categories (sets, groups, modules) epimorphisms coincide with surjective maps and monomorphisms with injective maps, the situation for Hopf algebras is more subtle because of the presence of the antipode, a map that intertwines the algebraic and coalgebraic structures.

The authors construct explicit counter‑examples that demonstrate the failure of these naïve expectations. The first construction starts with an arbitrary Hopf algebra (H) and considers its “enveloping Hopf algebra with bijective antipode”, denoted (\widehat{H}). This object is obtained by formally adjoining an inverse to the antipode of (H) in the minimal way that preserves the Hopf structure. The canonical morphism (\iota\colon H\to\widehat{H}) is shown to be an epimorphism in the category (\mathbf{HAlg}_k) of Hopf algebras over a field (k). The proof proceeds by demonstrating that any two Hopf algebra maps (f,g\colon\widehat{H}\to L) that agree after pre‑composition with (\iota) must already be equal; this uses the fact that (\widehat{H}) is generated as a Hopf algebra by the image of (\iota) together with the newly adjoined antipode inverse. However, (\iota) is not surjective in general: concrete examples such as the quantum group (U_q(\mathfrak{sl}_2)) (with (q) not a root of unity) illustrate that the image of (\iota) is a proper Hopf subalgebra of (\widehat{H}). Thus an epimorphism need not be surjective.

The dual situation is treated by considering a Hopf algebra (K) and its “co‑enveloping Hopf algebra with bijective antipode”, (\widehat{K}), together with the canonical coalgebra map (\pi\colon K\to\widehat{K}). The authors prove that (\pi) is a monomorphism in (\mathbf{HAlg}_k) by showing that if two maps (\alpha,\beta\colon L\to K) satisfy (\pi\circ\alpha=\pi\circ\beta), then (\alpha=\beta). Again, the argument relies on the fact that (\widehat{K}) is generated by the image of (\pi) and the inverse of the antipode. Nevertheless, (\pi) is typically not injective; for instance, when (K) is the group algebra (k