Smoothness of commutative Hopf algebras

TL;DR

This paper investigates the smoothness properties of commutative Hopf algebras in various categories, establishing equivalences and conditions for smoothness over different fields and categorical contexts.

Contribution

It proves the equivalence of several smoothness conditions for Hopf algebras in symmetric monoidal categories and characterizes these properties over fields of characteristic zero and positive characteristic.

Findings

All ordinary Hopf algebras over a field of characteristic zero are smooth.

Hopf algebras in super-vector spaces have stronger smoothness properties.

Weaker smoothness properties are identified for positive characteristic cases.

Abstract

Hopf algebras, most generally in a semisimple abelian symmetric monoidal category, are here supposed to be commutative but not to be of finite-type, and their (equivariant) smoothness are discussed. Given a Hopf algebra $H$ in a category such as above, it is proved that the following are equivalent: (i) $H$ is smooth as an algebra; (ii) $H$ is smooth as an $H$-comodule algebra; (iii) the product morphism $S_H^2(H^+) \to H^+$ defined on the 2nd symmetric power is monic. Working over a field $k$ of characteristic zero, we prove: (1) every ordinary Hopf algebra, i.e., such in the category $\mathsf{Vec}$ of vector spaces, satisfies the equivalent conditions (i)--(iii) and some others; (2) every Hopf algebra in the category $\mathsf{sVec}$ of super-vector spaces has a certain property that is stronger than (i). In the case where $\operatorname{char}k=p>0$, there are shown weaker properties of ordinary Hopf algebras and of Hopf algebras in $\mathsf{sVec}$ or in the ind-completion $\mathsf{Ver}_p^{\mathrm{ind}}$ of the Verlinde category.