李群 内容简介

《李群(英文版)》讲述了：Part I covers standard general properties of representations of compactgroups （including Lie groups and other compact groups, such as finite or p-adie ones）. These include Schur orthogonality, properties of matrix coefficientsand the Peter-Weyl Theorem.Part II covers the fundamentals of Lie gronps, by which I mean those sub-jects that I think are most urgent for the student to learn. These include thefollowing topics for compact groups: the fundamental group, the conjngacyof maximal tori （two proofs）, and the Weyl character formula. For noncom-pact groups, we start with complex analytic groups that are obtained bycomplexification of compact Lie groups, obtaining the lwasawa and Bruhatdecompositions. These arc the reductive complex groups. They are of course aspecial case, bnt a good place to start in the noncompact world. More generalnoncompact Lie groups with a Cartan decomposition are studied in the lastfew chapters of Part II. Chapter 31, on symmetric spaces, alternates exampleswith theory, discussing the embedding of a noncompact symmetric space inits compact dnal, the boundary components and Bergman-Shilov boundaryof a symmetric tube domain, anti Cartan's classification. Chapter 32 con-structs the relative root system, explains Satake diagrams and gives examplesillustrating the various phenomena that can occur, and reproves the Iwasawadecomposition, formerly obtained for complex analytic groups, in this moregeneral context. Finally, Chapter 33 surveys the different ways Lie groups canbe embedded in oue another.

李群 本书特色

This book aims to be a course in Lie groups that can be covered in one year with a group of good graduate students. I have attempted to address a problem that anyone teaching this subject must have, which is that the amount of essential material is too much to cover.

李群 目录

preface
part i: compact groups
1 haar measure
2 schur orthogonality
3 compact operators
4 the peter-weyl theorem
part ii: lie group fundamentals
5 lie subgroups of gl(n, c)
6 vector fields
7 left-invariant vector fields
8 the exponential map
9 tensors and universal properties
10 the universal enveloping algebra
11 extension of scalars
12 representations of s1(2, c)
13 the universal cover
14 the local frobenius theorem
15 tori
16 geodesics and maximal tori
17 topological proof of cartan's theorem
18 the weyl integration formula
19 the root system
20 examples of root systems
21 abstract weyl groups
22 the fundamental group
23 semisimple compact groups
24 highest-weight vectors
25 the weyl character formula
26 spin
27 complexification
28 coxeter groups
29 the iwasawa decomposition
30 the bruhat decomposition
31 symmetric spaces
32 relative root systems
33 embeddings of lie groups
part iii: topics
34 mackey theory
35 characters of gl(n,c)
36 duality between sk and gl(n,c)
……
references
index