Representations of Compact Lie GroupsSpringer Science & Business Media, 14 Mar 2013 - 316 halaman This book is based on several courses given by the authors since 1966. It introduces the reader to the representation theory of compact Lie groups. We have chosen a geometrical and analytical approach since we feel that this is the easiest way to motivate and establish the theory and to indicate relations to other branches of mathematics. Lie algebras, though mentioned occasionally, are not used in an essential way. The material as well as its presentation are classical; one might say that the foundations were known to Hermann Weyl at least 50 years ago. Prerequisites to the book are standard linear algebra and analysis, including Stokes' theorem for manifolds. The book can be read by German students in their third year, or by first-year graduate students in the United States. Generally speaking the book should be useful for mathematicians with geometric interests and, we hope, for physicists. At the end of each section the reader will find a set of exercises. These vary in character: Some ask the reader to verify statements used in the text, some contain additional information, and some present examples and counter examples. We advise the reader at least to read through the exercises. |
Isi
| 1 | |
The Exponential | 24 |
Homogeneous Spaces and Quotient Groups | 30 |
Invariant Integration | 48 |
Clifford Algebras and Spinor Groups | 54 |
40 | 60 |
Linear Algebra and Representations | 74 |
Representations of SU2 SO3 U2 and O3 | 84 |
CHAPTER V | 183 |
Roots and Weyl Chambers | 189 |
Root Systems | 197 |
Bases and Weyl Chambers | 202 |
Dynkin Diagrams | 209 |
The Roots of the Classical Groups | 216 |
The Fundamental Group the Center and the Stiefel Diagram | 223 |
The Structure of the Compact Groups | 232 |
Real and Quaternionic Representations | 93 |
The Character Ring and the Representation Ring | 102 |
Representations of Lie Algebras | 111 |
CHAPTER III | 123 |
Some Analysis on Compact Groups | 129 |
The Theorem of Peter and Weyl | 136 |
Induced Representations | 143 |
TannakaKrein Duality | 146 |
The Complexification of Compact Lie Groups | 151 |
CHAPTER IV | 157 |
Consequences of the Conjugation Theorem | 164 |
The Maximal Tori and Weyl Groups of the Classical Groups | 169 |
Cartan Subgroups of Nonconnected Compact Groups | 176 |
CHAPTER VI | 239 |
The Dominant Weight and the Structure of the Representation Ring | 249 |
The Multiplicities of the Weights of an Irreducible Representation | 257 |
Representations of Real or Quaternionic Type | 261 |
Representations of the Classical Groups | 265 |
Representations of the Spinor Groups | 278 |
Representations of the Orthogonal Groups | 292 |
Bibliography | 299 |
| 304 | |
| 307 | |
| 308 | |
Edisi yang lain - Lihat semua
Istilah dan frasa umum
a₁ abelian adjoint representation automorphism basis bijection canonical Cartan subgroup closed subgroup coefficients compact connected Lie compact Lie group complex compute conjugation connected Lie group contained corresponding defined Definition denote det(w diffeomorphism differentiable direct sum dominant weight e₁ element exponential map finite finite-dimensional follows G-module given GL(n hence homomorphism induces injective inner product integral forms integral lattice invariant inverse roots Irr(G irreducible characters irreducible representations isomorphism left-invariant Lemma Let G Lie algebra Lie group G linear LT(n manifold matrix maximal torus module morphism multiplication operates orthogonal polynomial positive roots PROOF Proposition quaternionic representation of G representation ring root system self-conjugate semisimple Show SO(n Sp(n Spin(2n Spin(m structure SU(n submodules subspace summands surjective symmetric theorem topology trivial U₁ V₁ vector space W-invariant Weyl chamber Weyl group