Pro-Lie Groups: A survey with Open Problems
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paper_title: Pro-Lie groups which are infinite-dimensional Lie groups

paper_content:

A pro-Lie group is a projective limit of a family of finite-dimensional Lie groups. In this note we show that a pro-Lie group $G$ is a Lie group in the sense that its topology is compatible with a smooth manifold structure for which the group operations are smooth if and only if $G$ is locally contractible. We also characterize the corresponding pro-Lie algebras in various ways. Furthermore, we characterize those pro-Lie groups which are locally exponential, that is, they are Lie groups with a smooth exponential function which maps a zero neighborhood in the Lie algebra diffeomorphically onto an open identity neighborhood of the group.
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paper_title: Character groups of Hopf algebras as infinite-dimensional Lie groups

paper_content:

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.
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paper_title: Local splitting of locally compact groups and pro-Lie groups

paper_content:

Abstract In the book “The Lie Theory of Connected Pro-Lie Groups” the authors proved the local splitting theorem for connected pro-Lie groups. George A. A. Michael subsequently proved this theorem for almost connected pro-Lie groups. Here his result is proved more directly using the machinery of the aforementioned book.
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paper_title: Local splitting of locally compact groups and pro-Lie groups

paper_content:

Abstract In the book “The Lie Theory of Connected Pro-Lie Groups” the authors proved the local splitting theorem for connected pro-Lie groups. George A. A. Michael subsequently proved this theorem for almost connected pro-Lie groups. Here his result is proved more directly using the machinery of the aforementioned book.
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paper_title: Simplified Proofs for the Pro-Lie Group Theorem and the One-Parameter Subgroup Lifting Lemma

paper_content:

This note is devoted to the theory of projective limits of finite-dimensional Lie groups, as developed in the recent monograph ``The Lie Theory of Connected Pro-Lie Groups'' by K.H. Hofmann and S.A. Morris. We replace the original, highly non-trivial proof of the One-Parameter Subgroup Lifting Lemma given in the monograph by a shorter and more elementary argument. Furthermore, we shorten (and correct) the proof of the so-called Pro-Lie Group Theorem, which asserts that pro-Lie groups and projective limits of Lie groups coincide.
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paper_title: On Inverse Limits of Finite Dimensional Lie Groups

paper_content:

We give a short proof of the Hofmann–Morris Theorem characterizing inverse limits of finite dimensional Lie groups [Hofmann, K. H., and S. A. Morris: Projective limits of finite dimensional Lie groups, Proc. Lond. Math. Soc. 87 (2003), 647–676, Theorem 4.7]. The proof depends on the Gleason–Palais characterization of finite dimensional Lie groups [Gleason, A., and R. Palais: On a class of transformation groups, Amer. J. Math. 79 (1957), 631–648, Theorem 7.2]. Mathematics Subject Classification: 22A05
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paper_title: Topological vector spaces

paper_content:

This chapter presents the most basic results on topological vector spaces. With the exception of the last section, the scalar field over which vector spaces are defined can be an arbitrary, non-discrete valuated field K; K is endowed with the uniformity derived from its absolute value. The purpose of this generality is to clearly identify those properties of the commonly used real and complex number field that are essential for these basic results. Section 1 discusses the description of vector space topologies in terms of neighborhood bases of 0, and the uniformity associated with such a topology. Section 2 gives some means for constructing new topological vector spaces from given ones. The standard tools used in working with spaces of finite dimension are collected in Section 3, which is followed by a brief discussion of affine subspaces and hyperplanes (Section 4). Section 5 studies the extremely important notion of boundedness. Metrizability is treated in Section 6. This notion, although not overly important for the general theory, deserves special attention for several reasons; among them are its connection with category, its role in applications in analysis, and its role in the history of the subject (cf. Banach [1]). Restricting K to subfields of the complex numbers, Section 7 discusses the transition from real to complex fields and vice versa.
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paper_title: On Inverse Limits of Finite Dimensional Lie Groups

paper_content:

We give a short proof of the Hofmann–Morris Theorem characterizing inverse limits of finite dimensional Lie groups [Hofmann, K. H., and S. A. Morris: Projective limits of finite dimensional Lie groups, Proc. Lond. Math. Soc. 87 (2003), 647–676, Theorem 4.7]. The proof depends on the Gleason–Palais characterization of finite dimensional Lie groups [Gleason, A., and R. Palais: On a class of transformation groups, Amer. J. Math. 79 (1957), 631–648, Theorem 7.2]. Mathematics Subject Classification: 22A05
---
paper_title: Pro-Lie groups which are infinite-dimensional Lie groups

paper_content:

A pro-Lie group is a projective limit of a family of finite-dimensional Lie groups. In this note we show that a pro-Lie group $G$ is a Lie group in the sense that its topology is compatible with a smooth manifold structure for which the group operations are smooth if and only if $G$ is locally contractible. We also characterize the corresponding pro-Lie algebras in various ways. Furthermore, we characterize those pro-Lie groups which are locally exponential, that is, they are Lie groups with a smooth exponential function which maps a zero neighborhood in the Lie algebra diffeomorphically onto an open identity neighborhood of the group.
---
paper_title: Local splitting of locally compact groups and pro-Lie groups

paper_content:

Abstract In the book “The Lie Theory of Connected Pro-Lie Groups” the authors proved the local splitting theorem for connected pro-Lie groups. George A. A. Michael subsequently proved this theorem for almost connected pro-Lie groups. Here his result is proved more directly using the machinery of the aforementioned book.
---
paper_title: The Generalized Gluskov-Iwasawa Local Splitting Theorem

paper_content:

In this paper we obtain an extension of Gluskov- Iwasawa local splitting theorem for locally compact groups as the prod- uct of a compact group and a local Lie group, to inverse limits of finite dimensional Lie groups. Fo ra ny topological group G let G0 be its identity component and call G almost connected if G/G0 is compact. If G is a locally compact group which is either connected in (8, Theorem 11) or almost connected in (4, The- orem A) it is proved there that for every 1 ∈ U open ⊆ G, there exists a compact normal N ≤ G, N ⊆ U such that G/N Lie group and N × G/N is locally isomorphic to G.W ecall this result the Gluskov-Iwasawa local split- ting theorem for locally compact groups . We wish to generalize this result to almost connected projective Lie groups, i.e. inverse limits of finite di- mensional Lie groups or equivalently complete Hausdorff topological groups such that every neighborhood of the identity element of which contains a closed normal subgroup that gives rise to a finite dimensional Lie quotient group (9). An almost connected locally compact group is a projective Lie group (10, p.175), so our result generalizes the Gluskov-Iwasawa local split- ting theorem for locally compact groups and we call this generalization the generalized Gluskov-Iwasawa local splitting theorem. We obtain a sufficient criterion for the validity of the generalized Gluskov-Iwasawa local splitting theorem which extends the condition that is automatically satisfied in the locally compact case. This generalization has also been considered in (7). We show that our sufficient condition is equivalent to the one obtained in (7, Theorem 3.2) but the proofs we give are different and provide a vast simplifi- cation of the ones in (7). We remark that the proof of (7, Theorem 3.2) gives the stated assertion only for connected groups so our Theorem 2.2 actually
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paper_title: Character groups of Hopf algebras as infinite-dimensional Lie groups

paper_content:

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.
---
paper_title: Character groups of Hopf algebras as infinite-dimensional Lie groups

paper_content:

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.
---

Title: Pro-Lie Groups: A survey with Open Problems
Section 1: Introduction
Description 1: Provide background information on the motivation, historical context, and foundational work leading to the study of pro-Lie groups.

Section 2: The topology of pro-Lie groups
Description 2: Discuss the topological structure of connected and almost connected pro-Lie groups and their equivalence to other topological constructs.

Section 3: Pro-Lie groups as projective limits of Lie groups
Description 3: Explain the definition and importance of pro-Lie groups as projective limits of finite-dimensional Lie groups, including key theorems and open questions.

Section 4: Weakly complete vector groups
Description 4: Address the characteristics and significance of weakly complete vector groups in the context of pro-Lie groups, including related propositions and definitions.

Section 5: The open mapping theorem
Description 5: Investigate conditions under which a surjective morphism of pro-Lie groups is a quotient morphism and the implications of the open mapping theorem for pro-Lie groups.

Section 6: Lie theory
Description 6: Introduce the Lie theory as it applies to pro-Lie groups, focusing on the relationship between pro-Lie groups and their Lie algebras, as well as fundamental related theorems.

Section 7: Later developments
Description 7: Summarize significant advances and additional results in the theory of pro-Lie groups that have emerged since earlier foundational work, highlighting key theorems and their implications.

Section 8: A natural occurrence of pro-Lie groups and pro-Lie algebras
Description 8: Discuss how weakly complete topological vector spaces and pro-Lie groups naturally arise in mathematical environments, providing illustrative examples and the underlying theory.

Section 9: Further open questions
Description 9: List and elaborate on additional open questions concerning pro-Lie groups that are significant for ongoing research in this field.