Holomorphy and convexity in Lie theory
type: Item
instance of: scholarly article
title: Holomorphy and convexity in Lie theory
zbMATH Open document ID: 0936.22001
published in: De Gruyter Expositions in Mathematics
publication date: 1999-12-14T00:00:00Z
review text: The main purpose of the author of this book is the introduction into the ``circle of ideas connecting holomorphic and unitary representations with invariant convexity in Lie algebras''. The author is a well-known German mathematician, his area of interests contains the Lie theory of semigroups and representation theory. This monograph is a remarkable collection of facts, theorems and theories concerning such subjects as holomorphy and convexity in Lie theory. The book is divided into five parts A-E. In Part A, consisting of Chapters I--IV, the abstract representation theory of involutive semigroups is given. In Chapter I the author develops the theory of positive kernels and reproducing kernel Hilbert spaces. Positive definite kernels are a tool to deal with Hilbert spaces of functions on a set \(X\) for which the evaluations in points of \(X\) are continuous. In Chapter II the representation theory of involutive semigroups, i.e. semigroups \(S\) endowed with an involutive antiautomorphism \(s\mapsto s^*\), is considered. In Chapter IV the author studies the relation between several continuity concepts for representations of topological involutive semigroups, whereas in Chapters I--III any topology is not taken into account. Part B consists of Chapters V and VI, where Chapter V provides the background on convex sets and convex functions in finite-dimensional vector spaces. Here the author develops results and facts on convex functions that will be needed in the remainder of the book. Chapter VI deals with the representation theory of cones and tubes. The representation theory of open cones \(W\) in real vector spaces \(V\) and the associated tube domains \(\Gamma_V(W):=V+iW\) is given. Since tubes are the simplest examples of Ol'shanskii semigroups, this chapter can be viewed as a discussion on holomorphic representation theory for the simple class of vector groups. Using the enveloping \(C^*\)-algebras of an open cone, the author obtains a generalization of Nussbau…
Mathematics Subject Classification ID: 22-02
Mathematics Subject Classification ID: 22E15
Mathematics Subject Classification ID: 22E45
Mathematics Subject Classification ID: 17-02
Mathematics Subject Classification ID: 17B05
Mathematics Subject Classification ID: 17B10
Mathematics Subject Classification ID: 32E10
Mathematics Subject Classification ID: 32U05
Mathematics Subject Classification ID: 43A35
Mathematics Subject Classification ID: 43A65
Mathematics Subject Classification ID: 81R05
Mathematics Subject Classification ID: 81R30
zbMATH DE Number: 1377484
zbMATH Keywords: representation theory
zbMATH Keywords: Lie group
zbMATH Keywords: Lie algebra
zbMATH Keywords: semigroup
zbMATH Keywords: complex analysis
zbMATH Keywords: operator algebra
zbMATH Keywords: holomorphic and unitary representations
zbMATH Keywords: involutive semigroups
zbMATH Keywords: positive definite functions
zbMATH Keywords: Laplace transforms
zbMATH Keywords: theory of convex functions
zbMATH Keywords: coadjoint orbits
zbMATH Keywords: admissible Lie algebras
zbMATH Keywords: highest weight representations
zbMATH Keywords: Hardy spaces
zbMATH Keywords: Ol'shanskii semigroups
zbMATH Keywords: coherent state orbits
zbMATH Keywords: oscillator representation
zbMATH Keywords: oscillator group
zbMATH Keywords: Heisenberg's uncertainty principle
author: Karl-Hermann Neeb
reviewed by: A. K. Guts
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1377484