Positivity in Lie Theory: Open Problems

Walter de Gruyter GmBH & Co.

Key words: Causality, Compression Semigroups, Conformal Geometry, Control Systems, Discrete Series Representations, Exponential Function, Harmonic Analysis, Holomorphic Representations, Invariant Cones, Jordan Algebras, Lie Algebras, Lie Semigroups, Linear Monoids, One-parameter Groups of Probability Measures, Reductive Groups, Reproducing Kernels, Singular Representations, Spherical Functions, Stein Manifolds, Symmetric Spaces, Total Positivity, Tube Domains

• Table of Contents
• Preface
• List of Contributors
• Solutions (last change: September 26, 1999)

This book consists of 15 articles, each of which is an introduction to a set of open research problems in Lie theory. The unifying theme is ``positivity'', which means orderings on the level of manifolds, semigroups on the level of groups, and cones on the level of linear spaces and Lie algebras. The topics range from geometric and algebraic structure theory through harmonic analysis, representation theory as far as control theory and probability

A characteristic feature of the field called ``Positivity in Lie Theory'' is that notions of positivity in Lie theory occur in quite diverse settings, are motivated by a wide variety of problems and applications, and are approached from quite varying mathematical viewpoints. The link between the topics treated under the heading of ``positivity" in Lie theory is the presence of orderings at the level of manifolds, semigroups at the level of groups, and cones at the level of vector spaces and Lie algebras.

While the diversity is attractive for the specialists, it is often difficult for newcomers to the field to see the relation between the various aspects and to pick the right problems. The editors of this book have tried to put together a collection of problems with commentary that would serve as an invitation and a guide to the field.

In each chapter the reader is introduced to a specific open problem or circle of problems that the author considers important for further development. The level of presentation is chosen in such a way that a graduate student with a sound knowledge of basic Lie theory should be able to grasp the gist of the problem. The main definitions are explained, preliminary results are quoted, and the relevant references are listed. Moreover, the authors often tried to formulate smaller problems that might serve as approaches to the main problems.

The problems described in this book primarily reflect recent developments. The intensive work during the eighties on the structure of subsemigroups of Lie groups as well as semigroup closures of linear algebraic groups laid the foundation for more specialized and application-oriented research conducted in recent years. New results on structure theory of Lie semigroups and causal spaces are now usually motivated by and obtained in the context of either control theory, harmonic analysis or representation theory.

Joachim Hilgert (E-Mail: hilgert@math.tu-clausthal.de)


Homepage Joachim Hilgert

September 16, 1999