A geometric classification of Lie groups
type: Item
instance of: scholarly article
title: A geometric classification of Lie groups
zbMATH Open document ID: 0971.22007
DOI: 10.4171/RMI/271
published in: Revista Matemática Iberoamericana
publication date: 2001-11-07T00:00:00Z
full work available at URL: https://eudml.org/doc/39586
review text: This paper is part of a program aimed at the study of Lie groups by way of heat diffusion on the group. To describe the flavor of the results, let \(G\) be a connected Lie group with Lie algebra \({\mathbf g}\), and consider the left Haar measure, denoted \(dx\), on \(G\). Let \(\Omega\) be a neighborhood of the identity \(e\) such that \(\Omega= \Omega^{-1}\), and define \(\gamma (n)=\) Haar measure \((\Omega^n)\) for \(n\geq 1\). Let \(\varphi\in C_0(G)\), and consider the measure \(d\mu(x)= \varphi(x)dx\), where \(\varphi\) is chosen such that \(\mu\) is a probability measure and the inverse map preserves \(\mu\). Let \(\varphi_n\) be the function such that \(d\mu^{*n}(x)= \varphi_n(x)dx\), where \(\mu^{*n}\) denotes the convolution of \(\mu\) with itself \(n\) times. Fix \(g\in G\), and define \(\varphi(n)=\varphi_n(g)\). Then the unimodular Lie groups can be separated into two disjoint classes. This classification can be achieved (i) geometrically, essentially by the condition that \(\gamma(n)\) grows exponentially or not, (ii) algebraically, by a condition on the Lie algebra known as the \(R\)-condition, and (iii) analytically, essentially by the condition that \(\varphi(n) =O(e^{-cn^{1 \over 3}})\) for some \(c>0\) or not. However, these conditions are insufficient to provide a dichotomy for general connected (non-unimodular) Lie groups, for which the situation is more subtle. Previous work of the author and collaborators has identified the appropriate algebraic and analytical properties to extend this classification to arbitrary connected Lie groups. The present paper completes this classification scheme by identifying the geometric characterization of the dichotomy. The author begins the paper with a very well written overview of the field and a guide to the reader, both of which make this paper much easier to digest and appreciate.
Mathematics Subject Classification ID: 22E30
Mathematics Subject Classification ID: 43A80
zbMATH DE Number: 1485606
zbMATH Keywords: heat diffusion
zbMATH Keywords: unimodular Lie groups
zbMATH Keywords: connected Lie groups
zbMATH Keywords: dichotomy
author: N. Th. Varopoulos
MaRDI profile type: Publication
Wikidata QID: Q115211906
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description: scientific article; zbMATH DE number 1485606