Lie groups, convex cones and semigroups

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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