Table Of Contentde Gruyter Expositions in Mathematics 20
Editors
Ο. H. Kegel, Albert-Ludwigs-Universität, Freiburg
V. P. Maslov, Academy of Sciences, Moscow
W. D. Neumann, The University of Melbourne, Parkville
R.O.Wells, Jr., Rice University, Houston
de Gruyter Expositions in Mathematics
1 The Analytical and Topological Theory of Semigroups, Κ. H. Hofmann,
J. D. Lawson, J. S. Pym fEdsJ
2 Combinatorial Homotopy and 4-Dimensional Complexes, H. J. Baues
3 The Stefan Problem, A. M. Meirmanov
4 Finite Soluble Groups, K. Doerk, T. O. Hawkes
5 The Riemann Zeta-Function, A. A. Karatsuba, S. M. Voronin
6 Contact Geometry and Linear Differential Equations, V. R. Nazaikinskii,
V. E. Shatalov, B. Yu. Sternin
7 Infinite Dimensional Lie Superalgebras, Yu. A. Bahturin, A. A. Mikhalev,
V. M. Petrogradsky, Μ. V. Zaicev
8 Nilpotent Groups and their Automorphisms, Ε. I. Khukhro
9 Invariant Distances and Metrics in Complex Analysis, M. Jarnicki, P. Pflug
10 The Link Invariants of the Chern-Simons Field Theory, E. Guadagnini
11 Global Affine Differential Geometry of Hypersurfaces, A. -M. Li, U. Simon,
G. Zhao
12 Moduli Spaces of Abelian Surfaces: Compactification, Degenerations, and
Theta Functions, K. Hulek, C. Kahn, S. H. Weintraub
13 Elliptic Problems in Domains with Piecewise Smooth Boundaries, S.A.
Nazarov, B. A. Plamenevsky
14 Subgroup Lattices of Groups, R. Schmidt
15 Orthogonal Decompositions and Integral Lattices, A. I. Kostrikin, PH.
Tiep
16 The Adjunction Theory of Complex Projective Varieties, M. C. Beltrametti,
A. J. Sommese
17 The Restricted 3-Body Problem: Plane Periodic Orbits, A. D. Bruno
18 Unitary Representation Theory of Exponential Lie Groups, H. Leptin,
J. Ludwig
19 Blow-up in Quasilinear Parabolic Equations, A.A. Samarskii, V.A.
Galaktionov, S. P. Kurdyumov, A. P. Mikhailov
Semigroups in Algebra,
Geometry and Analysis
Editors
Karl H. Hofmann
Jimmie D. Lawson
Ernest B. Vinberg
W
DE
_G
Walter de Gruyter · Berlin · New York 1995
Editors
Karl Η. Hofmann Jimmie D. Lawson Ernest Β. Vinberg
Fachbereich Mathematik Department of Mathematics Chair of Algebra
Technische Hochschule Darmstadt Louisiana State University University of Moscow
Schlossgartenstr. 7 Baton Rouge, LA 70803-0001 119899 Moscow
D-64289 Darmstadt, Germany USA Russia
1991 Mathematics Subject Classification: 14Lxx, 20Gxx, 22Exx
Keywords: Lie groups, symmetric spaces, Lie semigroups, semigroups and big groups,
representation theory
® Printed on acid-free paper which falls within the guidelines of the ANSI to ensure permanence and durability.
Library of Congress Cataloging-in-Publication Data
Semigroups in algebra, geometry, and analysis / editors, Karl H. Hof-
mann, Jimmie D. Lawson, Ernest B. Vinberg.
p. cm. — (De Gruyter expositions in mathematics,
ISSN 0938-6572 ; 20)
Includes bibliographical references.
ISBN 3-11-014319-4 (alk. paper)
1. Semigroups. 2. Lie groups. I. Hofmann, Karl Heinrich.
II. Lawson, Jimmie D. III. Vinberg, Ε. B. (Ernest Borisovich)
IV. Series.
QA182.S45 1995
512'.55-dc20 95-14934
CIP
Die Deutsche Bibliothek - Cataloging-in-Publication Data
Semigroups in algebra, geometry and analysis / ed. Karl H. Hof-
mann ... - Berlin ; New York : de Gruyter, 1995
(De Gruyter expositions in mathematics ; 20)
ISBN 3-11-014319-4
NE: Hofmann, Karl Η. [Hrsg.]; GT
© Copyright 1995 by Walter de Gruyter & Co., D-10785 Berlin.
All rights reserved, including those of translation into foreign languages. No part of this book
may be reproduced or transmitted in any form or by any means, electronic or mechanical, inclu-
ding photocopy, recording, or any information storage or retrieval system, without permission in
writing from the publisher.
Printed in Germany.
Printing: Gerike GmbH, Berlin. Binding: Lüderitz & Bauer GmbH, Berlin.
Cover design: Thomas Bonnie, Hamburg.
Preface
In contemporary mathematics semigroups appear naturally in many fields of inves-
tigation. Indeed semigroup methodology has become a fundamental tool in areas
where no one even suspected a connection fifteen years ago. This methodology has
been significantly enhanced and a whole new range of applications opened up by
the emergence of a comprehensive new theory resulting from the comparatively
recent conjunction of semigroup theory with Lie theory [6, 7, 11, 12, 14], even
though the historical roots for such a theory reach deeper [10, 13].
The general mathematical trend of bringing to bear on a specific theory infor-
mation from quite diverse segments of mathematics is exemplified by developments
in modern geometry. The fundamental ties between the theory of Lie groups [1,
2, 3, 4, 5] and geometry have been recognized for over a century since the days
of SOPHUS LIE and FELIX KLEIN. The natural concept of symmetry is founda-
tional for geometry, as has become accepted since KLEIN'S Erlangen Program.
Geometry and group theory axe, therefore, inextricably linked. It has become in-
creasingly evident that symmetry is no less relevant to physics than to geometry.
Yet in physics and other sciences dealing with real world phenomena, one en-
counters another concept of long standing which until recently was not considered
relevant in geometry, namely, that of irreversible processes and phenomena. A
suitable mathematical concept to accompany the idea of irreversibility is that of a
semigroup. The first genuinely mathematical theory in this direction was the the-
ory of one-parameter semigroups of operators on Banach spaces. Its pioneer was
EINAR HILLE (1894-1980), whose enormously influential book of 1948 [8] planted
a firm tradition which is still very much alive in the theory of partial differential
equations, in probability theory and in other branches of analysis. In as much as
these areas all impinge on physics, semigroups did enter this field. The emphasis
in "semigroup theory" in the sense of HILLE, however, is on spectral theory and
not on the structure theory of the semigroups which occur (with the exception
of ergodic theory, in which the concept of a semigroup compactification is now
a most appropriate tool). Yet semigroups have a rich theory of their own. It is
perhaps astonishing that in physics proper, semigroups modelling irreversibility
have not yet become fully developed. The semigroups which occur in this volume,
at any rate, share the characteristic that in one way or another they are linked
with the theory of Lie groups and Lie algebras and their respresentation theory,
and that their applications lie in various directions involving irreversibility—such
as causality in the theory of relativity, the geometric theory of control systems, or
probability theory.
vi Preface
The first mathematician to draw attention to a connection between semigroup
theory and Lie theory was CHARLES LOEWNER (see [13]), but his work, for rea-
sons not entirely clear, lacked acceptance by the research community and remained
largely without resonance . Postulates for a theory of Lie semigroups proclaimed in
the late sixties and mid seventies, too, remained without noticeable effect [9]. The
theory of Lie semigroups began to take off in the early eighties at different places.
In 1980 VLNBERG wrote his paper [17] on invariant cones in Lie algebras, which
engendered an entire subculture in the Lie theory of semigroups. OL'SHANSKII be-
gan in 1981 [16] a sequence of seminal articles on invariant cones and introduced a
class of Lie semigroups which we now call Ol'shanskii-semigroups. Through these
papers he inaugurated the study of causal symmetric spaces and semigroup ap-
proaches to representation theory. In 1983 there appeared a first overview over
the foundations of a general theory of Lie semigroups by HOFMANN and LAWSON
[11] in a volume reporting on the Conference on Semigroups in Oberwolfach May
24-30, 1981. By the end of the decade, a monograph on Lie semigroups had ap-
peared [6] (1989). From January 29 through February 4, 1989, Mathematisches
Forschungsinstitut Oberwolfach had sponsored a conference on the "Analytical
and Topological Theory of Semigroups," whose principal results were collected in
a book [12]. The individual chapters were written by various authors so as to pro-
vide a self-contained introduction and overview of the latest developments. Further
monographs on Lie semigroups appeared more recently [15] and [7] (1993), so that
a substantial body of the theory was available in book form by 1993.
Yet progress has been rapid and voluminous, and despite some 1500 pages of
monograph literature, the recent state of the theory of semigroups in their rela-
tion to Lie theory remained unavailable in collected form. The collection before
us continues the Oberwolfach concept and presents 14 chapters on "Semigroups
in Algebra, Geometry and Analysis". Their authors take stock of the status of
semigroup and order theory in the various fields they cover; they lead up to latest
results and indicate the directions in which research is likely to develop. Applica-
tions and links to neighboring areas are not neglected.
From October 10 through 16, 1993 the Mathematisches Forschungsinstitut
Oberwolfach hosted a conference under the title "Invariant Ordering in Geom-
etry and Algebra." Invitations to survey particular developments and to write
contributions on these had again been extended by the organizers of the confer-
ence well before the meeting, and the final concept of the book was agreed upon
at the conference.
The book has been grouped into the following divisions:
1. Lie semigroups, ordered symmetric spaces, and causality,
2. Invariant cones, Ol'shanskii semigroups, exponential semigroups,
3. Convexity theorems and representation theory,
4. Semisimple Lie groups and semigroups,
5. Applications: Control,
6. Applications: Probability.
Part 1 contains a chapter by FARAUT and OLAFSSON on symmetric spaces with
causal structure and their harmonic analysis, followed by several chapters con-
Preface vii
cerning causality on manifolds with cone fields, of which the pseudo-Riemannian
geometry of relativity is a special case. HILGERT discusses an effective method
from Lie semigroup theory for considering causality on homogeneous manifolds
with cone fields. GUTS surveys the Russian work on the foundations of axiomatic
relativity and how semigroups play their role in this field, and LEVICHEV gives an
extensive report on I. SEGAL'S chronometry and its consequences in physics and
cosmology.
Part 2 opens with a new and direct approach to the classification of invari-
ant pointed generating cones in Lie algebras by GICHEV; this draws a line from
VINBERG'S seminal work on invariant cones in Lie groups to the highly devloped
state of the theory today. LAWSON presents a comprehensive theory of Ol'shanskii
semigroups in its current state, describing many aspects ranging from their various
decompositions all the way through the natural occurences of these semigroups as
endomorphism semigroups of geometric objects. HOFMANN and RUPPERT describe
the classification of those closed subsemigroups of Lie groups which have a sur-
jective exponential function (and which are reduced in a suitable sense); as is the
case in almost all chapters of this book, a great variety of methods had to be de-
veloped before this classification problem, which used to be called the "divisibility
problem," could be settled.
In Part 3 HILGERT and NEEB provide an extensive discussion of the most up-
to-date results on the convexity theorems orginating from the work of KOSTANT
and survey their link with symplectic geometry; they focus on those aspects of the
theory relating in one way or another to Lie semigroup theory. NEEB discusses the
current state of the theory of holomorphic extensions of unitary representations
of Lie groups on the basis of the theory of Ol'shanskii-semigroups; he shows how
this approach can be used for the study of the highest weight representations
In Part 4 SAN MARTIN reports on his theory of control sets for control systems
on flag manifolds of a semisimple Lie group; these control systems are given in
terms of subsemigroups of the Lie group. This theory has currently reached a highly
developed state. VLNBERG shows how one associates with a connected semisimple
complex Lie group a semigroup which may be considered as a completion of the
given Lie group. It is an irreducible algebraic variety whose dimension is that of
the group, and it may be viewed as describing the structure of the Lie group "at
infinity." This chapter is in the spirit of semigroups on algebraic varieties ä la
PUTCHA and RENNER (who reported on this theory in [12] four years ago).
Part 5 deals with the ever growing interplay between semigroup theory and
geometric control theory. The contribution by MITTENHUBER presents the role of
Pontryagin's maximum principle in Lie semigroup theory; in particular he demon-
strates how this principle is an effective tool in the investigation of the globality of
a Lie wedge. This chapter may also serve as an introduction to the entire complex
of "Lie semigroup theory and geometric control theory." ZELIKIN discusses the
classical optimal control situation from the novel aspect of a Lie group acting on
the phase space leaving the cone field of the control system invariant; the focus is
on the totally extremal submanifolds of the phase space.
viii Preface
Part 6 is the shortest chapter. But it points to a still not fully explored area
of application of semigroup theory to probability theory. While it is widely known
that infinitely divisible laws determine one-parameter convolution semigroups of
measures, it is less known that operator decomposable distributions give rise to
Lie semigroups which are relevant in the central limit problem. JUREK's chapter
introduces the reader to this application of Lie semigroup theory.
The subdivision of this book into various groups, of course, must by necessity
appear somewhat artificial. Its topics are not sequential but are interwoven in a
complicated web of ideas. For instance, invariant cones in Lie algebras do not
only occur in GLCHEV's chapter but are pervasive; they play a substantial role in
the chapter by FARAUT and OLAFSSON and in everything said about OL'shanskii-
semigroups. The chapter on control sets in flag manifolds of semisimple Lie groups
by SAN MARTIN could just as well have been placed into Division 5 on geometric
control theory. This lack of systematic ordering should not be seen as a defect but
as an indication of the complicated network formed by Lie semigroup theory in its
present state of development.
The topics discussed in this monograph show that semigroup theory in geom-
etry and analysis has attained a rich state of maturity, and that its development
rests on fascinating connections with a variety of branches of mainstream mathe-
matics.
Literature
[1] Bourbaki, N., Groupes et algebres de Lie, Chap. 1, Paris, Hermann 1960.
[2] —, Groupes et algebres de Lie, Chap. 2 et 3, Paris, Hermann 1972.
[3] —, Groupes et algebres de Lie, Chap. 7 et 8, Paris, Hermann 1975.
[4] —, Groupes et algebres de Lie, Chap. 9, Paris, Masson 1982.
[5] Helgason, S., Differential Geometry, Lie Groups, and Symmetric Spaces,
Academic Press, Boston etc, 1978.
[6] Hilgert, J., Hofmann, K.H., and J.D. Lawson, Lie groups, convex cones
and semigroups, Oxford Science Publications, Clarendon Press, Oxford
1989, xxxviii+645 pp.
[7] Hilgert, J., and K.-H. Neeb, Lie Semigroups and their Applications, Lec-
ture Notes of Math. 1552, Springer 1993, xii+315 pp.
[8] Hille, Ε., Functional Analysis and Semigroups, Amer. Math. Soc. Coll.
Publ. 31 (1948), xii+ 528 pp.
[9] Hofmann, K.H., Topological semigroups, History, Theory, Applications,
Jahresber. Deutsch. Math.-Verein. 78 (1976), 9-59.
[10] —, Zur Geschichte des Halbgruppenbegriffs, Historia Math. 19 (1992),
40-59.
Preface ix
11] Hofmann, Κ.Η., and J D. Lawson, Foundations of Lie semigroups, in:
Lecture Notes in Math. 998, Springer 1983, 128-201.
12] Hofmann, K.H., J.D. Lawson and J.S. Pym, Eds., The Analytical and
Topological Theory of Semigroups, de Gruyter, Berlin, 1990, xii+398 pp.
13] Lawson, J.D., Historical links to a Lie theory of semigroups, Sem. Sophus
Lie 2 (1992) 263-278.
14] Neeb, K.-H, On the foundations of Lie semigroups, J. Reine Angew. Math.
431 (1992), 165-189.
15] —, Invariant Subsemigroups of Lie Groups, Mem. Amer. Math. Soc. 104
(1993), viii+193 pp.
16] Ol'shanskii, G.I., Invariant cones in in Lie algebras, Lie semigroups, and
the holomorphic discrete series, Functional Anal. Appl. 15 (1981) 275-285.
17] Vinberg, E.B. Invariant cones and orderings in Lie groups, Functional
Anal. Appl. 14 (1980), 1-13.
Acknowledgements
We thank the Mathematisches Forschungsinstitut Oberwolfach for providing firstly
the logistics for the organisation of the conference on which the content of this book
draws, and secondly for the well-known hospitality during the meeting.
To the editors of the series de Gruyter Expositions in Mathematics and to the
mathematics editor of de Gruyter Verlag, Dr. MANFRED KARBE, we express our
gratitude for publishing this collection in the usual superb form.
The editors thank all contributors for having provided IgX-files of their con-
tributions using the macros we supplied. Dr. DIRK MITTENHUBER has invested
a great amount of time, effort and expertise to coordinate these and to make ev-
erybody's macros compatible with those preferred by our publisher. An editor's
life would be so much easier if KNUTH'S ingenious effective and pliable language
TßX had not been diversified into a generation of numerous supposedly improved
and more user-friendly, but certainly more rigid dialects which are all bound to be
used without exception if 14 l^jX-users cooperate.
The editors are extraordinarily grateful to KARL-HERMANN NEEB for the con-
tinuous, active and effective interest he took in the genesis of this book. We thank
him most cordially, but also all other referees who contributed to the final form of
the inputs.
