arXiv math/06096842006-09-25math.GR

Pro-Lie groups which are infinite-dimensional Lie groups

K. H. Hofmann, K. -H. Neeb

英文摘要：

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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Infinite Approximate Subgroups of Soluble Lie Groups

We study infinite approximate subgroups of soluble Lie groups. Generalising a theorem of Fried and Goldman we show that approximate subgroups are close, in a sense to be defined, to genuine connected subgroups. Building up on this result we prove a structure theorem for approximate lattices in soluble Lie groups. This extends to soluble Lie groups a theorem about quasi-crystals due to Yves Meyer.

2019-09-26math.GR

无穷多个有限单李型群不能由两个素数阶元素不变生成

Infinitely many finite simple groups of Lie type are not generated invariably by two elements of prime order

University of Southern California（南加州大学）；Washington University（华盛顿大学）；Univerza na Primorskem（沿海大学）

本文证明了无穷多个辛群Sp_{2k}(2)不能由两个素数阶元素不变生成，扩展了有限单群生成性质的研究。

2026-09-29math.GR

On some pro-p groups from infinite-dimensional Lie theory

We initiate the study of some pro-p-groups arising from infinite-dimensional Lie theory. These groups are completions of some subgroups of incomplete Kac-Moody groups over finite fields, with respect to various completions of algebraic or geometric origin. We show topological finite generation for the pro-p Sylow subgroups in many complete Kac-Moody groups. This implies abstract simplicity of the latter groups. We also discuss with the question of (non-)linearity of these pro-p groups.

2013-02-18math.GR

Abelian extensions of infinite-dimensional Lie groups

In the present paper we study abelian extensions of connected Lie groups $G$ modeled on locally convex spaces by smooth $G$-modules $A$. We parametrize the extension classes by a suitable cohomology group $H^2_s(G,A)$ defined by locally smooth cochains and construct an exact sequence that describes the difference between $H^2_s(G,A)$ and the corresponding continuous Lie algebra cohomology space $H^2_c(\g,\a)$. The obstructions for the integrability of a Lie algebra extensions to a Lie group extension are described in terms of period and flux homomorphisms. We also characterize the extensions w…

2004-02-18math.GR

The Frobenius theorem for Banach distributions on infinite-dimensional manifolds and applications in infinite-dimensional Lie theory

We prove a Frobenius theorem for Banach distributions on manifolds that are modelled over locally convex spaces. Moreover, we recall how Frobenius theorems can be applied to infinite-dimensional Lie groups and obtain, that given a Lie subalgebra of the Lie algebra of a Lie group that is modelled over a locally convex space that admits an exponential map, the Lie subalgebra is integrable if it is complemented as a topological vector space and a Banach space with the induced topology.

2014-07-11math.DGmath.GR

Completeness of locally $k_ω$-groups and related infinite-dimensional Lie groups

Recall that a topological space is said to be a $k_ω$-space if it is the direct limit of an ascending sequence of compact Hausdorff topological spaces. If each point in a Hausdorff space $X$ has an open neighbourhood which is a $k_ω$-space, then $X$ is called locally $k_ω$. We show that a topological group is complete whenever the underlying topological space is locally $k_ω$. As a consequence, every infinite-dimensional Lie group modelled on a Silva space is complete.

2017-03-07math.GR

更多相关论文收起更多论文

Fundamental problems in the theory of infinite-dimensional Lie groups

In a preprint from 1982, John Milnor formulated various fundamental questions concerning infinite-dimensional Lie groups. In this note, we describe some of the answers (and partial answers) obtained in the preceding years.

2006-02-05math.FAmath.GR

Direct limits of infinite-dimensional Lie groups

Many infinite-dimensional Lie groups of interest can be expressed as a union of an ascending sequence of (finite- or infinite-dimensional) Lie groups. In this survey article, we compile general results concerning such ascending unions, describe the main classes of examples, and explain what the general theory tells us about these. In particular, we discuss: (1) Direct limit properties of ascending unions of Lie groups in the relevant categories; (2) Regularity in Milnor's sense; (3) Homotopy groups of direct limit groups and of Lie groups containing a dense union of Lie groups; (4) Subgroups o…

2008-04-01math.FAmath.GR

Hoelder continuous homomorphisms between infinite-dimensional Lie groups are smooth

It is an open problem whether every continuous homomorphism between infinite-dimensional Lie groups is smooth. In this article, we show that every Hoelder continuous homomorphism is smooth.

2004-04-30math.FAmath.GR

Measurable regularity of infinite-dimensional Lie groups based on Lusin measurability

We discuss Lebesgue spaces $\mathcal{L}^p([a,b],E)$ of Lusin measurable vector-valued functions and the corresponding vector spaces $AC_{L^p}([a,b],E)$ of absolutely continuous functions. These can be used to construct Lie groups $AC_{L^p}([a,b],G)$ of absolutely continuous functions with values in an infinite-dimensional Lie group $G$. We extend the notion of $L^p$-regularity of infinite-dimensional Lie groups introduced by Glöckner to this setting and adopt several results and tools.

2019-05-23math.FAmath.GR

Analytic Mappings Between LB-spaces and Applications in Infinite-Dimensional Lie Theory

We give a sufficient criterion for complex analyticity of nonlinear maps defined on direct limits of normed spaces. This tool is then used to construct new classes of (real and complex) infinite dimensional Lie groups: (a) groups of germs of analytic diffeomorphisms around a compact set in a Banach space and (b) unions of ascending sequences of Banach Lie groups.

2008-07-28math.FAmath.GR

An Infinite Dimensional Approach to the Third Fundamental Theorem of Lie

We revisit the third fundamental theorem of Lie (Lie III) for finite dimensional Lie algebras in the context of infinite dimensional matrices.

2008-02-16math.GRmath.RT

Linking Lie groupoid representations and representations of infinite-dimensional Lie groups

The present paper links the representation theory of Lie groupoids and infinite-dimensional Lie groups. We show that smooth representations of Lie groupoids give rise to smooth representations of associated Lie groups. The groups envisaged here are the bisection group and a group of groupoid self maps. Then representations of the Lie groupoids give rise to representations of the infinite-dimensional Lie groups on spaces of (compactly supported) bundle sections. Endowing the spaces of bundle sections with a fine Whitney type topology, the fine very strong topology, we even obtain continuous and…

2019-01-08math.GRmath.RT

Solutions to open problems in Neeb's recent survey on infinite-dimensional Lie groups

We solve three open problems concerning infinite-dimensional Lie groups posed in a recent survey article by K.-H. Neeb: (1) There exists a subgroup of some infinite-dimensional Lie group G which does not admit an initial Lie subgroup structure; (2) The pathology cannot occur if G is a direct limit of an ascending sequence of finite-dimensional Lie groups; (3) Every such direct limit group is a ``topological group with Lie algebra'' in the sense of Hofmann and Morris. Moreover, we prove a version of Borel's Theorem announced in the survey, ensuring the existence of compactly supported smooth di…

2008-01-12math.DGmath.GR

Direct limits of infinite-dimensional Lie groups compared to direct limits in related categories

Let G be a Lie group which is the union of an ascending sequence of Lie groups G_n (all of which may be infinite-dimensional). We study the question when G is the direct limit of the G_n's in the category of Lie groups, topological groups, smooth manifolds, resp., topological spaces. Full answers are obtained for G the group Diff_c(M) of compactly supported smooth diffeomorphisms of a sigma-compact smooth manifold M, and for test function groups C^infty_c(M,H) of compactly supported smooth maps with values in a finite-dimensional Lie group H. We also discuss the cases where G is a direct limit…

2006-06-03math.FAmath.GR

Character groups of Hopf algebras as infinite-dimensional Lie groups

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 …

2015-04-09math.GRmath.RA

Integrating central extensions of Lie algebras via Lie 2-groups

The purpose of this paper is to show how central extensions of (possibly infinite-dimensional) Lie algebras integrate to central extensions of étale Lie 2-groups. In finite dimensions, central extensions of Lie algebras integrate to central extensions of Lie groups, a fact which is due to the vanishing of π_2 for each finite-dimensional Lie group. This fact was used by Cartan (in a slightly other guise) to construct the simply connected Lie group associated to each finite-dimensional Lie algebra. In infinite dimensions, there is an obstruction for a central extension of Lie algebras to integra…

2015-01-28math.DGmath.GR

无穷维李群

Infinite-Dimensional Lie Groups

本文介绍了无穷维李群的基本理论，涵盖局部凸空间中的微积分、流形结构及各类李群的定义与性质。

2026-02-12math-phmath.DGmath.FA