Lie semigroups and their applications
type: Item
instance of: scholarly article
title: Lie semigroups and their applications
zbMATH Open document ID: 0807.22001
published in: Lecture Notes in Mathematics
publication date: 1993-11-23T00:00:00Z
review text: Closed subsemigroups of Lie groups which are generated by their one- parameter subsemigroups are called Lie semigroups and have arisen in such varied contexts as geometric function theory, representation theory, the study of causality in mathematical relativity, and geometric control theory. In the last fifteen years there has occurred a rapid development of a full-blown theory of such semigroups. Major achievements of the theory up to 1989 appear in the monograph of \textit{J. Hilgert, K. H. Hofmann}, and \textit{J. D. Lawson} [Lie groups, convex cones, and semigroups (Oxford 1989; Zbl 0701.22001)]. It has become clear that certain classes of Lie semigroups play a vital role in representation theory and the harmonic analysis of symmetric spaces, and the main purpose of the monograph under review is to lead the reader through the structure theory of Lie semigroups up to these applications of the theory. There is some overlap with the earlier work, since the authors make some effort for the book to be relatively self-contained, but for the most part the book contains new developments, improved treatments and sharpened versions of previous theory, and topics not treated previously. The first three chapters treat the general Lie theory of semigroups, appropriately illustrated with a variety of examples. Chapters 4 and 5 develop the close ties between ordered homogeneous spaces and Lie semigroup theory, a theory which bears certain resemblance to causality theory in mathematical physics, but which is also currently the framework for new developments in harmonic analysis. Chapter 6 treats the theory of maximal semigroups, which are a useful tool for considering reachability and globality questions (questions that involve determining the semigroup associated with a wedge of tangent vectors in the Lie algebra). Chapter 7 introduces the centrally important class of semigroups called Ol'shanskij semigroups, and these are studied in some detail in Chapter 8 in the context of …
Mathematics Subject Classification ID: 22-02
Mathematics Subject Classification ID: 22E15
Mathematics Subject Classification ID: 22A15
Mathematics Subject Classification ID: 22A22
zbMATH DE Number: 463969
zbMATH Keywords: Lie groups
zbMATH Keywords: one-parameter subsemigroups
zbMATH Keywords: Lie semigroups
zbMATH Keywords: Lie theory of semigroups
zbMATH Keywords: ordered homogeneous spaces
zbMATH Keywords: maximal semigroups
zbMATH Keywords: Ol'shanskij semigroups
zbMATH Keywords: compression semigroups
author: Joachim Hilgert
author: Karl-Hermann Neeb
reviewed by: Jimmie D. Lawson
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 463969