Memoirs of the American Mathematical Society 1997; 174 pp; softcover Volume: 130 ISBN10: 0821806416 ISBN13: 9780821806418 List Price: US$54 Individual Members: US$32.40 Institutional Members: US$43.20 Order Code: MEMO/130/618
 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
