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

• Main Authors: Karl H. Hofmann, Sidney A. Morris
• Format: Article
• Language: English
• Published: MDPI AG 2021-08-01
• Series: Axioms
• Subjects: topological group; Lie group; compact group; pro-Lie group; Lie algebra; duality
• Online Access: https://www.mdpi.com/2075-1680/10/3/190

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 <i>G</i> is homeomorphic to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi mathvariant="double-struck">R</mi><mi>I</mi></msup><mo>×</mo><mi>C</mi></mrow></semantics></math></inline-formula> for a suitable set <i>I</i> and some compact subgroup <i>C</i>. Finally, there is a perfect generalization to compact groups <i>G</i> of the age-old natural duality of the group algebra <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="double-struck">R</mi><mo>[</mo><mi>G</mi><mo>]</mo></mrow></semantics></math></inline-formula> of a finite group <i>G</i> to its representation algebra <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi mathvariant="double-struck">R</mi><mo>)</mo></mrow></semantics></math></inline-formula>, via the natural duality of the topological vector space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mi>I</mi></msup></semantics></math></inline-formula> to the vector space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mrow><mo>(</mo><mi>I</mi><mo>)</mo></mrow></msup></semantics></math></inline-formula>, for <i>any</i> set <i>I</i>, thus opening a new approach to the Hochschild-Tannaka duality of compact groups.