Synopses & Reviews
Loop groups, the simplest class of infinite dimensional Lie groups, have recently been the subject of intense study. This book gives a complete and self-contained account of what is known about them from a geometrical and analytical point of view, drawing together the many branches of mathematics from which current theory developed--algebra, geometry, analysis, combinatorics, and the mathematics of quantum field theory. The authors discuss loop groups' applications to simple particle physics and explain how the mathematics used in connection with loop groups is itself interesting and valuable, thereby making this work accessible to mathematicians in many fields.
"The authors have done an important service in giving the first coherent account of the geometric representation theory and structure of loop groups and their central extensions....Besides mathematicians working on infinite-dimensional groups and manifolds, the book is to be recommended for theoretical physicists working in quantum field theory, completely integrable systems and string theory." --Mathematical Reviews
"This book is the first account, from a mathematical point of view, of what is known about the global analysis of loop groups--a particular kind of infinite dimensional group consisting of maps from a circle to a fixed (finite dimensional) group. These objects arise naturally in one dimensional field theories and 'string' models. Loop Groups is remarkably comprehensive and coherent. . . . Although many of the results are at the cutting edge of research, the exposition and proofs are elegant and intelligible. The future influence of the book may be hard to overstate." --American Scientist
This book gives a complete and self-contained account of what is known about this subject and is written from a geometrical and analytical point of view, with quantum field theory very much in mind.
Includes bibliographical references (p. -310).
Table of Contents
2. Finite Dimensional Lie Groups
3. Groups of Smooth Maps
4. Central Extensions
5. The Root System: Kac-Moody Algebras
6. Loop Groups as Groups of Operators in Hilbert Space
7. The Grasmannian of Hilbert Space and the Determinant Time Bundle
8. The Fundamental Homogeneous Space
9. Representation Theory
10. The Fundamental Representation
11. The Borel-Weil Theory
12. The Spin Representation
13. 'Blips' or 'Vertex-Operators'
14. The Kac Character Formula and the Bernstein-Gelfand-Gelfand Resolution