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Lie Groups and Subsemigroups with Surjective Exponential Function
Karl H. Hofmann, Technische Hochschule Darmstadt, Germany, and Wolfgang A. F. Ruppert, University of Vienna, Austria
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Memoirs of the American Mathematical Society
1997; 174 pp; softcover
Volume: 130
ISBN-10: 0-8218-0641-6
ISBN-13: 978-0-8218-0641-8
List Price: US$51
Individual Members: US$30.60
Institutional Members: US$40.80
Order Code: MEMO/130/618
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In the structure theory of real Lie groups, there is still information lacking about the exponential function. Most notably, there are no general necessary and sufficient conditions for the exponential function to be surjective. It is surprising that for subsemigroups of Lie groups, the question of the surjectivity of the exponential function can be answered. Under natural reductions setting aside the "group part" of the problem, subsemigroups of Lie groups with surjective exponential function are completely classified and explicitly constructed in this memoir. There are fewer than one would think and the proofs are harder than one would expect, requiring some innovative twists. The main protagonists on the scene are \(SL(2,R)\) and its universal covering group, almost abelian solvable Lie groups (i.e., vector groups extended by homotheties), and compact Lie groups.

Readership

Graduate students and research mathematicians interested in the structure of Lie groups, Lie algebras, and applications like geometric control.

Reviews

"The proof is the heart and bulk of the Memoir and involves extensive use of Lie group and Lie algebra machinery and the development of new Lie theoretic results ... The authors have written a nice summary of their work [4]. There the reader may find motivating examples described and pictured, detailed definitions and statements of problems and theorems, an introduction to the proof methods and strategies, and statements of major intermediate results derived along the way."

-- Semigroup Forum

Table of Contents

  • Introduction
  • The basic theory of exponential semigroups in Lie groups
  • Weyl groups and finiteness properties of Cartan subalgebras
  • Lie semialgebras
  • More examples
  • Test algebras and groups
  • Groups supporting reduced weakly exponential semigroups
  • Roots and root spaces
  • Appendix: The hyperspace of a locally compact space
  • References
  • Index
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