Introduction to Lie Algebras and Representation TheorySpringer Science & Business Media, 2012. dec. 6. - 173 oldal This book is designed to introduce the reader to the theory of semisimple Lie algebras over an algebraically closed field of characteristic 0, with emphasis on representations. A good knowledge of linear algebra (including eigenvalues, bilinear forms, euclidean spaces, and tensor products of vector spaces) is presupposed, as well as some acquaintance with the methods of abstract algebra. The first four chapters might well be read by a bright undergraduate; however, the remaining three chapters are admittedly a little more demanding. Besides being useful in many parts of mathematics and physics, the theory of semisimple Lie algebras is inherently attractive, combining as it does a certain amount of depth and a satisfying degree of completeness in its basic results. Since Jacobson's book appeared a decade ago, improvements have been made even in the classical parts of the theory. I have tried to incor porate some of them here and to provide easier access to the subject for non-specialists. For the specialist, the following features should be noted: (I) The Jordan-Chevalley decomposition of linear transformations is emphasized, with "toral" subalgebras replacing the more traditional Cartan subalgebras in the semisimple case. (2) The conjugacy theorem for Cartan subalgebras is proved (following D. J. Winter and G. D. Mostow) by elementary Lie algebra methods, avoiding the use of algebraic geometry. |
Tartalomjegyzék
II | 15 |
The simple algebras | 19 |
ROOT SYSTEMS | 42 |
ISOMORPHISM AND CONJUGACY THEOREMS | 73 |
EXISTENCE THEOREM | 89 |
6 | 91 |
7 | 98 |
REPRESENTATION THEORY | 107 |
17 | 108 |
19 | 119 |
Formulas of Weyl Kostant and Steinberg | 128 |
CHEVALLEY ALGEBRAS AND GROUPS | 145 |
Kostants Theorem | 151 |
| 165 | |
| 172 | |
Más kiadások - Összes megtekintése
Gyakori szavak és kifejezések
a+ß A₁ A₂ abelian adjoint admissible lattice arbitrary automorphism base Borel called Cartan matrix char F Chevalley commutes conjugate construction Corollary Coxeter graph CSA's define denote direct sum dominant weights dual Dynkin diagram eigenvalues elements endomorphisms euclidean space Exercise exists finite dimensional fixed follows formula G₂ gl(V H₁ H₂ hence highest weight homomorphism ideal implies induction integers irreducible L-module isomorphism Jordan decomposition Killing form L-submodule Lemma linear maximal toral subalgebra maximal vector module morphism multiplication nilpotent nondegenerate nonnegative nonzero orthogonal particular polynomial positive roots Proposition Prove representation resp root system satisfying scalar semisimple Lie algebra sends shows simple roots solvable spans ẞe standard cyclic subgroup submodule subspace tensor unique V₁ vector space verify weight spaces Weyl chambers Weyl group α α αεφ λε