Potential Theory and Geometry on Lie Groups: (Series Number 38 New Mathematical Monographs)

This book provides a complete and reasonably self-contained account of a new classification of connected Lie groups into two classes. The first part describes the use of tools from potential theory to establish the classification and to show that the analytic and algebraic approaches to the classification are equivalent. Part II covers geometric theory of the same classification and a proof that it is equivalent to the algebraic approach. Part III is a new approach to the geometric classification that requires more advanced geometric technology, namely homotopy, homology and the theory of currents. Using these methods, a more direct, but also more sophisticated, approach to the equivalence of the geometric and algebraic classification is made. Background material is introduced gradually to familiarise readers with ideas from areas such as Lie groups, differential topology and probability, in particular, random walks on groups. Numerous open problems inspire students to explore further.

Table of Contents:
Preface; 1. Introduction; Part I. The Analytic and Algebraic Classification: 2. The classification and the first main theorem; 3. NC-groups; 4. The B–NB classification; 5. NB-groups; 6. Other classes of locally compact groups; Appendix A. Semisimple groups and the Iwasawa decomposition; Appendix B. The characterisation of NB-algebras; Appendix C. The structure of NB-groups; Appendix D. Invariant differential operators and their diffusion kernels; Appendix E. Additional results. Alternative proofs and prospects; Part II. The Geometric Theory: 7. The geometric theory. An introduction; 8. The geometric NC-theorem; 9. Algebra and geometries on C-groups; 10. The end game in the C-theorem; 11. The metric classification; Appendix F. Retracts on general NB-groups (not necessarily simply connected); Part III. Homology Theory: 12. The homotopy and homology classification of connected Lie groups; 13. The polynomial homology for simply connected soluble groups; 14. Cohomology on Lie groups; Appendix G. Discrete groups; Epilogue; References; Index.

About the Author :
N. Th. Varopoulos was for many years a professor at Université de Paris VI. He is a member of the Institut Universitaire de France.

Review :
'The motivated reader will find this book fascinating. It presents, in a somewhat idiosyncratic but readable way, a personal, substantial, and interesting mathematical journey.' Laurent Saloff-Coste, Bulletin of the American Mathematical Society 'The results presented in the book are original, deep and interesting. They straddle a large number of distinct areas of mathematics, such as Lie theory, probability theory, analysis, potential theory, geometry, and topology. The author makes a valiant attempt to present the material in a self-contained and understandable way. The text mentions a number of significant open questions that emerge from the work.' Laurent Saloff-Coste, MathSciNet