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Lie semigroups and their applications. (English) Zbl 0807.22001
Lecture Notes in Mathematics. 1552. Berlin: Springer-Verlag. xii, 315 p. (1993).
Closed subsemigroups of Lie groups which are generated by their one- parameter subsemigroups are called Lie semigroups and have arisen in such varied contexts as geometric function theory, representation theory, the study of causality in mathematical relativity, and geometric control theory. In the last fifteen years there has occurred a rapid development of a full-blown theory of such semigroups. Major achievements of the theory up to 1989 appear in the monograph of J. Hilgert, K. H. Hofmann, and J. D. Lawson [Lie groups, convex cones, and semigroups (Oxford 1989; Zbl 0701.22001)]. It has become clear that certain classes of Lie semigroups play a vital role in representation theory and the harmonic analysis of symmetric spaces, and the main purpose of the monograph under review is to lead the reader through the structure theory of Lie semigroups up to these applications of the theory. There is some overlap with the earlier work, since the authors make some effort for the book to be relatively self-contained, but for the most part the book contains new developments, improved treatments and sharpened versions of previous theory, and topics not treated previously.
The first three chapters treat the general Lie theory of semigroups, appropriately illustrated with a variety of examples. Chapters 4 and 5 develop the close ties between ordered homogeneous spaces and Lie semigroup theory, a theory which bears certain resemblance to causality theory in mathematical physics, but which is also currently the framework for new developments in harmonic analysis. Chapter 6 treats the theory of maximal semigroups, which are a useful tool for considering reachability and globality questions (questions that involve determining the semigroup associated with a wedge of tangent vectors in the Lie algebra). Chapter 7 introduces the centrally important class of semigroups called Ol’shanskij semigroups, and these are studied in some detail in Chapter 8 in the context of compression semigroups. Finally applications of Lie semigroup theory to representation theory are explored in Chapter 9.
Overall the authors have done an excellent job in selecting, organizing and presenting the major and central developments of Lie semigroup theory in a concise and accessible manner. The monograph should serve as a valuable central reference for finding out or referencing key results in the area and as most appropriate background source for those interested in applications of the theory.

MSC:
22-02 Research exposition (monographs, survey articles) pertaining to topological groups
22E15 General properties and structure of real Lie groups
22A15 Structure of topological semigroups
22A22 Topological groupoids (including differentiable and Lie groupoids)
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