Character groups of Hopf algebras as infinite-dimensional Lie groups

TL;DR

This paper develops an infinite-dimensional Lie group framework for character groups of graded and connected Hopf algebras, establishing their structure as regular Lie groups and exploring subgroup properties.

Contribution

It introduces a Lie group structure on character groups of Hopf algebras valued in locally convex algebras, extending the theory to non-graded cases as pro-Lie groups.

Findings

Character groups of graded, connected Hopf algebras form regular infinite-dimensional Lie groups.

Subgroups associated with Hopf ideals are closed Lie subgroups.

Non-graded Hopf algebra character groups are pro-Lie groups.

Abstract

In this article character groups of Hopf algebras are studied from the perspective of infinite-dimensional Lie theory. For a graded and connected Hopf algebra we construct an infinite-dimensional Lie group structure on the character group with values in a locally convex algebra. This structure turns the character group into a Baker--Campbell--Hausdorff--Lie group which is regular in the sense of Milnor. Furthermore, we show that certain subgroups associated to Hopf ideals become closed Lie subgroups of the character group. If the Hopf algebra is not graded, its character group will in general not be a Lie group. However, we show that for any Hopf algebra the character group with values in a weakly complete algebra is a pro-Lie group in the sense of Hofmann and Morris.