This is the first and only reference to provide a comprehensive treatment of the Lie theory of subsemigroups of Lie groups. The book is uniquely accessible and requires little specialized knowledge. It includes information on the infinitesimal theory of Lie subsemigroups, and a characterization of those cones in a Lie algebra which are invariant under the action of the group of inner automporphisms. It provides full treatment of the local Lie theory for semigroups, and finally, gives the reader a useful account of the global theory for the existence of subsemigroups with a given set of infinitesimal generators.
Author(s): Joachim Hilgert, Karl Heinrich Hofmann, Jimmie D. Lawson
Series: Oxford Mathematical Monographs
Publisher: Oxford University Press, USA
Year: 1989
Language: English
Pages: C+xxxviii+645
Cover
OXFORD MATHEMATICAL MONOGRAPHS
List of Published in this Series
Lie Groups, Convex Cones, and Semigroups
© Joachim Hilgert, Karl Heinrich Hofmann, and Jimmie D. Lawson, 1989
ISBN 0198535694
QA387. H535 1989 512' .55 -dc20
LCCN 89-9289
Preface
Contents
Introduction
The logical interdependence
Chapter I The geometry of cones
1. Cones and their duality
2. Exposed faces
The associated pointed cone
Support hyperplanes
The algebraic interior
Exposed faces of finite dimensional wedges
The semiprojective space of a wedge, bases of cones
Sums of two wedges
The canonical function from C'(W) to II(E1(W*))
3. Mazur's Density Theorem
The Density Theorem
The Theorem of Straszsewicz
Consequences and Refinements
4. Special finite dimensional cones
Polyhedral Wedges
Lorentzian Cones
Round cones
More on quadratic forms and wedges
5. The invariance of cones under flows
Subtangent vectors and tangent vectors
A Lemma in Calculus I
Flows, vector fields
The invariance of wedges and vector fields
Problems for Chapter I
Notes for Chapter I
Chapter II Wedges in Lie algebras
1. Lie wedges and invariant wedges in Lie algebras
2. Lie Semialgebras
The analytic function g(X)
Invariance of vector fields under local translation
Definition and characterization of Lie semialgebras
Faces of Lie semialgebras
Half-space Semialgebras
Almost abelian Lie algebras
The characteristic function of a Lie algebra
Analytic Extension Aspects of Lie Semialgebras
3. Low dimensional and special Lie semialgebras
dim L < 3: The solvable case
dim L = 3: The semisimple case
Examples of Lorentzian cones
More on 4-dimensional solvable examples
The non-solvable 4-dimensional examples
Another special class of solvable Lie algebras
4. Reducing Lie semialgebras, Cartan algebras
5. The base ideal and Lie semialgebras
The base ideal
Special metabelian Lie algebras
Base ideals and Lie semialgebras
Nilpotent ideals
Base ideals and Cartan algebras
Tangent hyperplane subalgebras
6. Lorentzian Lie semialgebras
Lie semialgebras in Lie algebras with invariant quadratic form
Lorentzian Lie algebras
Irreducible Lorentzian Lie algebras
Lorentzian Lie semialgebras
7. Lie algebras with Lie semialgebras
Invariance Theorems
Triviality theorems
Lie semialgebras forcing structure theorems
Problems for Chapter II
Notes for Chapter II
Chapter III Invariant cones
1. The automorphism group of a wedge
The Lie algebra of the automorphism group of a wedge
The special case of a Lie algebra L
2. Compact groups of automorphisms of a wedge
Applications to Lie algebras with invariant cones
Minimal and maximal invariant cones
3. Frobenius-Perron theory for wedges
The case of abelian semigroups
4. The theorems of Kostant and Vinberg
Application to Lie algebras with invariant cones
5. The reconstruction of invariant cones
The orthogonal projection onto a compactly embedded Cartan algebra
Facts on compactly embedded Cartan algebras
The trace of an invariant cone on a Cartan algebra
Reconstructing cones
6. Cartan algebras and invariant cones
Roots and root decompositions
The test subalgebras
Lie algebras with cone potential
Mixed Lie algebras with compactly embedded Cartan algebras
Compact and non-compact roots in quasihermitian Lie algebras
Constructing invariant cones: Reduction to the reductive case
7. Orbits and orbit projections
Orbits generated by root vectors
8. Kostant's Convexity Theorem
9. Invariant cones in reductive Lie algebras
Decomposing the Lie algebra
Invariant cones in hermitian simple Lie algebras
Tracing the maximal invariant wedge
Maximal real positive roots
A suitable Iwasawa decomposition
Exploiting sufficient conditions
The descent procedure
Problems for Chapter III
Notes for Chapter III
Chapter IV The Local Lie theory of semigroups
1. Local semigroups
Germs and local properties
The tangent set at 0
The tangent wedge of a local semigroup
Further invariance properties of Lie wedges
2. Tangent wedges and local wedge semigroups
3. Locally reachable sets
Reachability and attainability
Campbell-Hausdorff multiplication versus addition
Local one-parameter semigroups of sets
4. Lie's Theorem: Pointed cones - split wedges
Lie's Fundamental Theorem for split Wedges
5. Geometric control in a local Lie group
The fundamental differential equation
Invariant vector fields
6. Wedge fields
7. The rerouting technique
Local rerouting
Achieving rerouting
8. The Edge of the Wedge Theorem
Problems for Chapter IV
Notes for Chapter IV
Chapter V Subsemigroups of Lie groups
0. Background on semigroups in groups
Preorders on groups and semigroups of positivity
Green's preorders and relations
Subsemigroups of topological groups
Closed partial orders and order convexity
1. Infinitesimally generated semigroups
Preanalytic semigroups and their tangent objects
Ray semigroups and infinitesimally generated semigroups
2. Groups associated with semigroups
3. Homomorphisms and semidirect products
4. Examples
Semigroups in abelian Lie groups
Semigroups in nilpotent Lie groups
Semigroups in solvable non-nilpotent Lie groups
Semigroups in semisimple Lie groups
Contraction semigroups in Lie groups
5. Maximal Semigroups
Algebraic preliminaries
Topological generalities
Total semigroups
Nilpotent groups
Frobenius-Perron Groups
6. Divisible Semigroups
7. Congruences on open subsemigroups
The Foliation Lemma
Consequences of the Foliation Lemma
The Foliation Theorem
Transporting right congruences
Two-sided congruences
The stratified domain
Problems for Chapter V
Notes for Chapter V
Chapter VI Positivity
1. Cone fields on homogeneous spaces
The homogeneous space G/H
Invariant wedge fields on G and G/H
W -admissible piecewise differentiable curves
2. Positive forms
1-Forms
3. W-admissible chains revisited
4. Ordered groups and homogeneous spaces
Monotone functions and measures
5. Globality and its Applications
The Principal Theorem on Globality
Closed versus exact forms
The tangent bundle of a group
Forms as functions
Tangent bundles and wedge fields
Problems for Chapter VI
Notes for Chapter VI
Chapter VII Embedding semigroups into Lie groups
1. General embedding machinery
Algebraic preliminaries
Local embeddings
Admissible sets and local semigroups
Local homomorphisms
Canonical embeddings
2. Differentiable semigroups
Admissible sets and strong derivatives
Differentiable local semigroups
Differentiable local groups
Differentiable manifolds with generalized boundary
Differentiable semigroups
Applications
3. Cancellative semigroups on manifolds
Left quotients and partial right translations
The double cover and analytic structures
Connected semigroup coverings
The free group on S
Problems for Chapter VII
Notes on Chapter VII
Appendix
1. The Campbell-Hausdorff formalism
2. Compactly embedded subalgebras
Dense analytic subgroups
p-compactness
Compact and p-compact elements
The interior of comp L
Compactly embedded Cartan algebras
The Weyl group
Notes on the Appendix
Reference material
Bibliography
Special symbols
Index
