Representation Theory: A First CourseThe primary goal of these lectures is to introduce a beginner to the finite dimensional representations of Lie groups and Lie algebras. Since this goal is shared by quite a few other books, we should explain in this Preface how our approach differs, although the potential reader can probably see this better by a quick browse through the book. Representation theory is simple to define: it is the study of the ways in which a given group may act on vector spaces. It is almost certainly unique, however, among such clearly delineated subjects, in the breadth of its interest to mathematicians. This is not surprising: group actions are ubiquitous in 20th century mathematics, and where the object on which a group acts is not a vector space, we have learned to replace it by one that is {e. g. , a cohomology group, tangent space, etc. }. As a consequence, many mathematicians other than specialists in the field {or even those who think they might want to be} come in contact with the subject in various ways. It is for such people that this text is designed. To put it another way, we intend this as a book for beginners to learn from and not as a reference. This idea essentially determines the choice of material covered here. As simple as is the definition of representation theory given above, it fragments considerably when we try to get more specific. |
Contents
Preface | 1 |
Young Diagrams and Frobeniuss | 4 |
Lie Groups and Lie Algebras | 89 |
Lie Groups | 95 |
Lie Algebras and Lie Groups | 104 |
Initial Classification of Lie Algebras | 121 |
Lie Algebras in Dimensions One Two and Three | 133 |
Representations of sl₂C | 148 |
Spin Representations of som C | 299 |
Lie Theory | 317 |
Complex Lie Groups Characters | 366 |
Weyl Character Formula | 399 |
More Character Formulas | 415 |
Real Lie Algebras and Lie Groups | 430 |
Appendices | 451 |
On Semisimplicity | 478 |
Representations of sl3 C Part I | 161 |
Mainly Lots of Examples | 175 |
The Classical Lie Algebras and Their Representations | 195 |
sl₁C and slC | 211 |
Symplectic Lie Algebras | 238 |
sp6C and sp₂C | 253 |
Orthogonal Lie Algebras | 267 |
so C soC and soC | 282 |
Cartan Subalgebras | 487 |
E Ados and Levis Theorems | 499 |
Hints Answers and References | 516 |
104 | 529 |
| 536 | |
| 543 | |
| 551 | |
Other editions - View all
Common terms and phrases
a₁ a₂ abelian action ad(X adjoint representation algebra g automorphisms basis Cartan subalgebra coefficients commutator complex Lie conjugacy classes conjugate construction corresponding decompose decomposition defined denote determinant dimension direct sum Dynkin diagram e₁ eigenspace eigenvalues eigenvector element endomorphism example Exercise exterior powers fact finite follows g₂ gl(V group G H₁ highest weight vector homomorphism ideal identity integers invariant irreducible representation isomorphism kernel Killing form L₂ Lecture Lemma Lie groups linear matrix multiplication nilpotent nonzero Note partition permutations positive roots proof Proposition quadric real Lie representations of G root spaces root system scalar Schur semisimple Lie algebra Show simple roots sl,C sl₂C sl3C spanned standard representation subgroup subspace Sym² symmetric polynomials symmetric powers symplectic tensor product theorem V₁ vector space verify w₁ weight diagram weight space Weyl chamber Weyl character formula Weyl group X₁ Y₁ Y₂ Young diagram zero