Positivity in Lie theory: open problems

This work 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 ordering 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.

While this 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.;The problem described in this book primarily reflect recent developments.

The intensive work during the 80s 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.