# Lie Groups and Subsemigroups with Surjective Exponential Function

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*Karl H. Hofmann; Wolfgang A. F. Ruppert*

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.

#### Reviews & Endorsements

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

# Table of Contents

## Lie Groups and Subsemigroups with Surjective Exponential Function

- Contents v6 free
- Chapter 1. Introduction 110 free
- Chapter 2. The Basic Theory of Exponential Semigroups in Lie Groups 1221
- Chapter 3. Weyl Groups and Finiteness Properties of Cartan Subalgebras 3847
- Chapter 4. Lie Semialgebras 7584
- 1. Semialgebras Revisited 7584
- 2. Dispersion of Weakly Exponential Semigroups 7988
- 3. More about Subtangent Vectors 8089
- 4. Reductions by Factoring Normal Subgroups 8392
- 5. Porcupine Varieties and Lie Semialgebras 8695
- 6. Groups with a Unique Maximal Compact Subgroup 9099
- 7. More about Lean Sets 92101

- Chapter 5. More Examples 95104
- Chapter 6. Test Algebras and Groups 109118
- Chapter 7. Groups Supporting Reduced Weakly Exponential Semigroups 126135
- Chapter 8. Roots and Root Spaces 141150
- Chapter 9. Appendix: The Hyperspace of a Locally Compact Space 153162
- References 169178
- Index 173182