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Subsemigroups of finite-dimensional Lie groups that are
generated by one-parameter semigroups are the subject of
this book. It covers basic Lie theory for such semigroups
and some closely related topics. These include ordered
homogeneous manifolds, where the order is defined by a field
of cones, invariant cones in Lie algebras and associated
Ol'shanskii semigroups. Applications to representation
theory, symplectic geometry and Hardy spaces are also given.
The book is written as an efficient guide for those
interested in subsemigroups of Lie groups and their
applications in various fields of mathematics (see the
User's guide at the end of the Introduction). Since it is
essentially self-contained and leads directly to the core of
the theory, the first part of the book can also serve as an
introduction to the subject.
The reader is merely expected to be familiar with the basic
theory of Lie groups and Lie algebras.
Lie semigroups and their tangent wedges.- Examples.- Geometry and topology of Lie semigroups.- Ordered homogeneous spaces.- Applications of ordered spaces to Lie semigroups.- Maximal semigroups in groups with cocompact radical.- Invariant Cones and Ol'shanskii semigroups.- Compression semigroups.- Representation theory.- The theory for Sl(2).