Advances in the Theory of Compact Groups and Pro-Lie Groups in the Last Quarter Century

Karl H Hofmann,Sidney A Morris

doi:10.3390/axioms10030190

Karl H Hofmann, Sidney A Morris

Open Access

https://doi.org/10.3390/axioms10030190

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Abstract

This article surveys the development of the theory of compact groups and pro-Lie groups, contextualizing the major achievements over 125 years and focusing on some progress in the last quarter century. It begins with developments in the 18th and 19th centuries. Next is from Hilbert’s Fifth Problem in 1900 to its solution in 1952 by Montgomery, Zippin, and Gleason and Yamabe’s important structure theorem on almost connected locally compact groups. This half century included profound contributions by Weyl and Peter, Haar, Pontryagin, van Kampen, Weil, and Iwasawa. The focus in the last quarter century has been structure theory, largely resulting from extending Lie Theory to compact groups and then to pro-Lie groups, which are projective limits of finite-dimensional Lie groups. The category of pro-Lie groups is the smallest complete category containing Lie groups and includes all compact groups, locally compact abelian groups, and connected locally compact groups. Amongst the structure theorems is that each almost connected pro-Lie group G is homeomorphic to RI×C for a suitable set I and some compact subgroup C. Finally, there is a perfect generalization to compact groups G of the age-old natural duality of the group algebra R[G] of a finite group G to its representation algebra R(G,R), via the natural duality of the topological vector space RI to the vector space R(I), for any set I, thus opening a new approach to the Hochschild-Tannaka duality of compact groups.

Highlights

Certain areas of mathematical research draw their particular fascination from the fact that they are based between two principal domains of mathematics, such as algebra and topology
The complete solution of Hilbert’s Fifth Problem arrived in 1952 (9 years after the death of Hilbert), when Andrew Mattei Gleason (1921–2008), Deane Montgomery (1909–1992), and Leo Zippin (1905–1995) settled it with a positive answer. This effort was crowned by the fundamental discovery in 1953 by Hidehiko Yamabe (1923–1960) that: in a topological group G whose component factor group Gt is compact, any compact identity neighborhood of G contains a closed normal subgroup N, such that the factor group G/N is a Lie group, precisely one of those Lie groups, which had so fascinated Hilbert in 1900
We have had some success in getting the basics of a structure theory of abelian pro-Lie groups formulated. (See Reference [6], 5.20).) we proved the following result

Summary

Introduction

Certain areas of mathematical research draw their particular fascination from the fact that they are based between two principal domains of mathematics, such as algebra and topology. Between these two, we find algebraic topology and topological algebra. Groups in their abstract form can to be traced back to Augustin-Louis. Cauchy (1789–1857), Niels Henrik Abel (1802–1829), and Évariste Galois (1811–1832), when groups were formative in the development of abstract algebra. 1847.) Topology as an independent area had not yet crystallized, though Geometry was quite present, when Felix Klein (1849–1925) and Sophus Lie (1842–1899) (and followers, such as Friedrich Engel (1861–1941) and Wilhelm Karl Joseph Killing (1847–1923)) founded the area of what later became named Lie groups. Geometry, and analysis were thoroughly mixed into the genesis of Lie group theory

Hilbert’s Fifth Problem and Locally Compact Groups

Pro-Lie Groups

Linear Algebra Meets Pro-Lie Group Theory

Postscript