Character groups of Hopf algebras as infinite-dimensional 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.

Overview of (pro-)Lie Group Structures on Hopf Algebra Character Groups

“…1.9 Example (Character groups of Hopf algebras) 1. The universal enveloping algebra U (g) of a Lie algebra g over K is a Hopf algebra (see [11, 3.6]) 4 . Every character φ ∈ G (U (g), K) corresponds to a Lie algebra homomorphism φ | g : g → K which in turn factors naturally through a linear map φ ∼ : g/([g, g]) → K, yielding a group isomorphism…”

“…We deduce that exp makes sense as a mapping to ∏ n∈N 0 Hom K (H n , B) whence it makes sense as a mapping into Hom K (H , B). To see that exp is K-analytic one uses functional calculus for densely graded algebras (see [4,Lemma B.6] for details).…”

“…That exp restricts to a bijection g(H , B) → G (H , B) is well known in the literature. We refer to [4,Lemma B.11] for a proof.…”

“…In [5] we have shown that the Butcher group can be turned into an infinitedimensional Lie group. All of these results can be interpreted in the general framework of (infinite-dimensional) Lie group structures on character groups of Hopf algebras developed in [4].…”

“…The present article provides an introduction to the theory developed in [4] with a view towards applications and further research. To be as accessible as possible, we consider the results from the perspective of examples arising in numerical analysis and mathematical physics.…”