Potential Theory and Geometry on Lie Groups

Complete account of a new classification of connected Lie groups in two classes, including open problems to motivate further study.

N. Th. Varopoulos (Author)

9781107036499, Cambridge University Press

Hardback, published 22 October 2020

611 pages, 20 b/w illus. 130 exercises
16 x 23.5 x 4.5 cm, 1.08 kg

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.

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.

Subject Areas: Probability & statistics [PBT], Topology [PBP], Geometry [PBM], Calculus & mathematical analysis [PBK]