Lie Algebras and Locally Compact Groups
9780226424538
Lie Algebras and Locally Compact Groups
This volume presents lecture notes based on the author’s courses on Lie algebras and the solution of Hilbert’s fifth problem. In chapter 1, "Lie Algebras," the structure theory of semi-simple Lie algebras in characteristic zero is presented, following the ideas of Killing and Cartan. Chapter 2, "The Structure of Locally Compact Groups," deals with the solution of Hilbert’s fifth problem given by Gleason, Montgomery, and Zipplin in 1952.
Table of Contents
PREFACE
Chapter I. LIE ALGEBRAS
1. Definitions and examples
2. Solvable and nilpotent algebras
3. Semi-simple algebras
4. Cartan subalgebras
5. Transition to a geometric problem
(characteristic 0)
6. The geometric classification
7. Transition to a geometric problem
(characteristic p)
8. Transition to a geometric problem
(characteristic p), continued
Chapter II. THE STRUCTURE OF LOCALLY COMPACT GROUPS
1. NSS groups
2. Existence of one-parameter subgroups
3. Differentiable functions
4. Functions constructed from a single Q
5. Functions constructed from a sequence of Q’s
6. Proof that i/n. is bounded
7. Existence of proper differentiable functions
8. The vector space of one-parameter subgroups
9. Proof that K is a neighborhood of 1
10. Approximation by NSS groups
11. Further developments
BIBLIOGRAPHY
INDEX
Chapter I. LIE ALGEBRAS
1. Definitions and examples
2. Solvable and nilpotent algebras
3. Semi-simple algebras
4. Cartan subalgebras
5. Transition to a geometric problem
(characteristic 0)
6. The geometric classification
7. Transition to a geometric problem
(characteristic p)
8. Transition to a geometric problem
(characteristic p), continued
Chapter II. THE STRUCTURE OF LOCALLY COMPACT GROUPS
1. NSS groups
2. Existence of one-parameter subgroups
3. Differentiable functions
4. Functions constructed from a single Q
5. Functions constructed from a sequence of Q’s
6. Proof that i/n. is bounded
7. Existence of proper differentiable functions
8. The vector space of one-parameter subgroups
9. Proof that K is a neighborhood of 1
10. Approximation by NSS groups
11. Further developments
BIBLIOGRAPHY
INDEX