Lie Semigroups and their Applications
Subsemigroups of finite-dimensional Lie groups that are generated by one-parameter semigroups are the subject of this book. It covers basic Lie theory for such semigroups and some closely related topics. These include ordered homogeneous manifolds, where
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Joachim Hilgert Karl-Hermann Neeb
Lie Semigroups and their Applications
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest
Authors Joachim Hilgert Mathematisches Institut der Universitat Erlangen Bismarckstr, 1 1/2 0-91054 Erlangen, Germany Karl-Hermann Neeb Fachbereich Mathematik Technische Hochschule Darmstadt SchloBgartenstr. 7 0-64289 Darmstadt, Germany
Mathematics Subject Classification (1991): 22A 15, 22A25, 22E46, 22E30, 53C30, 53C50, 53C75
ISBN 3-540-56954-5 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-56954-5 Springer-Verlag New York Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1993 Printed in Germany 46/3140-543210 Printed on acid-free paper
Table of Contents 1. Lie semigroups and their tangent wedges 1.1 Geometry of wedges 1.2 Wedges in K -modules 1.3 The characteristic function of a cone Endomorphisms of a cone ", 1.4 Lie wedges and Lie semigroups
1 9
, . , , .. , ,
,
The ordered space of Lorentzian cones Affine compressions of a ball 1.5 1.6 1.7 1.8
, . 11
,
" .. ,., , "."
,., .. , , .. ,.,
"" ,.", .. ', " .. ",." , ,
,
, 18 ,. 19
, 21 ,., .. 23
Functorial relations between Lie semigroups and Lie wedges , , , 25 Globality of Lie wedges , , " 28 Monotone functions and semigroups .. , , , .. , , , , .. , , , 29 Smooth and analytic monotone functions on a Lie group .. , . , . , . , . , . , .. , , 32
1.9 W -positive functions and globality . , , 1.10 Globality criteria ,
, .. ,
, .. 38 42
, ,
2. Examples 2,1 Semigroups in the Heisenberg group
,
,
, 48
2.2 The groups Sl(2) and PS1(2, IR) 2,3 The hyperboloid and its order structure 2.4 The Olshanskif semigroup in Sl(2,([;) .. , 2.5 Affine compression semi groups
49
, .. ",.,., .. 56 , 59 ,,.,.,
2.6 The euclidean compression and contraction semigroups ,
61
, ".,.,
2.7 Codel's cosmological model and the universal covering of Sl(2, IR) 2.8 The causal action of SU(n, n) on U(n) , The action of SU(n, n) on the euclidean contraction semigroup , 2.9 Almost abelian groups ,., , , 2.10 The whirlpot and the parking ramp .. , .. , , '.,., 2.11 The oscillator group
,. 63 ,.65
68 69
73 73 76
vi
3. Geometry and topology of Lie semigroups 3.1 3.2 3.3 3.4 3.5
Faces of Lie semi groups The interior of Lie semigroups Non generating Lie semigroups with interior points The universal covering semi group 5 The free group on S
3.6 Groups with directed orders
81 86 88 90 101 107
4. Ordered
