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
Finite Groups | 1 |
Characters | 12 |
Examples Induced Representations Group Algebras Real | 26 |
Young Diagrams and Frobeniuss | 44 |
Representations of A and GL2F₄ | 63 |
Weyls Construction | 75 |
Lie Groups and Lie Algebras | 89 |
Lie Algebras and Lie Groups | 104 |
Orthogonal Lie Algebras | 267 |
Spin Representations of som C | 299 |
Lie Theory | 317 |
g2 and Other Exceptional Lie Algebras | 339 |
Complex Lie Groups Characters | 366 |
Weyl Character Formula | 399 |
More Character Formulas | 415 |
Real Lie Algebras and Lie Groups | 430 |
Initial Classification of Lie Algebras | 121 |
Lie Algebras in Dimensions One Two and Three | 133 |
Representations of sl2 C | 146 |
Representations of sl3C Part I | 161 |
Mainly Lots of Examples | 175 |
The Classical Lie Algebras and Their Representations | 195 |
sl4C and slC | 211 |
Symplectic Lie Algebras | 238 |
sp C and sp2nC | 253 |
Appendices | 451 |
On Semisimplicity | 478 |
Cartan Subalgebras | 487 |
E Ados and Levis Theorems | 499 |
Hints Answers and References | 516 |
| 536 | |
| 543 | |
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Common terms and phrases
a₁ abelian action ad(X algebra g automorphisms basis Cartan subalgebra coefficients commutator conjugacy classes conjugate corresponding decompose decomposition defined denote determined dimension dimensional direct sum Dynkin diagram e₁ eigenspace eigenvalues eigenvector element endomorphism example Exercise exterior powers fact finite groups follows given GL(V group G H₁ highest weight vector homomorphism identity induced integers invariant irreducible representations isomorphism kernel Killing form lattice Lecture Lemma Lie groups matrix multiplication nilpotent nonzero Note orthogonal partition permutation positive roots proof Proposition prove quotient real Lie representation theory representations of G restriction root spaces scalar Schur semisimple Lie algebra Show simple roots sl₂C sl3C sp2nC spanned standard representation subgroup subrepresentation subspace Sym˛ Sym˛V symmetric powers symplectic tensor product theorem trivial V₁ vector space verify w₁ weight diagram Weyl chamber Weyl character formula Weyl group X₁ Y₁ Young diagram zero Γ₁ λ₁