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El. knyga: Introduction to Lie Groups and Lie Algebras

3.40/5 (10 ratings by Goodreads)
(State University of New York, Stony Brook)
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With roots in the nineteenth century, Lie theory has since found many and varied applications in mathematics and mathematical physics, to the point where it is now regarded as a classical branch of mathematics in its own right. This graduate text focuses on the study of semisimple Lie algebras, developing the necessary theory along the way. The material covered ranges from basic definitions of Lie groups to the classification of finite-dimensional representations of semisimple Lie algebras. Written in an informal style, this is a contemporary introduction to the subject which emphasizes the main concepts of the proofs and outlines the necessary technical details, allowing the material to be conveyed concisely. Based on a lecture course given by the author at the State University of New York, Stony Brook, the book includes numerous exercises and worked examples, and is ideal for graduate courses on Lie groups and Lie algebras.

Recenzijos

' the exposition is very clear and logical. It has the advantage of giving the basic facts about Lie algebra theory with enough arguments but skipping the tedious technical details of proofs. Another excellent feature of the book is that many of the basic notions, properties and results are illustrated by a great number of exercises and examples. In my opinion this book is a nice addition to the landmarks in the field I strongly recommend it to anyone wishing to enter into the beautiful and exciting field of Lie algebras and their applications.' Journal of Geometry and Symmetry in Physics ' very readable I strongly recommend this book as a possible selection for graduate courses, as well as for independent study, or individual reading.' MAA Reviews 'There are many exercises The last appendix contains a useful detailed sample syllabus for a one-semester graduate course (two lectures a week).' EMS Newsletter 'The book is a very concise and nice introduction to Lie groups and Lie algebras. It seems to be well suited for a course on the subject.' Mathematical Reviews

Daugiau informacijos

This book is an introduction to semisimple Lie algebras. It is concise and informal, with numerous exercises and examples.
Preface xi
1 Introduction 1
2 Lie groups: basic definitions 4
2.1. Reminders from differential geometry
4
2.2. Lie groups, subgroups, and cosets
5
2.3. Lie subgroups and homomorphism theorem
10
2.4. Action of Lie groups on manifolds and representations
10
2.5. Orbits and homogeneous spaces
12
2.6. Left, right, and adjoint action
14
2.7. Classical groups
16
2.8. Exercises
21
3 Lie groups and Lie algebras 25
3.1. Exponential map
25
3.2. The commutator
28
3.3. Jacobi identity and the definition of a Lie algebra
30
3.4. Subalgebras, ideals, and center
32
3.5. Lie algebra of vector fields
33
3.6. Stabilizers and the center
36
3.7. Campbell—Hausdorff formula
38
3.8. Fundamental theorems of Lie theory
40
3.9. Complex and real forms
44
3.10. Example: so (3, R), su(2), and sl(2, C)
46
3.11. Exercises
48
4 Representations of Lie groups and Lie algebras 52
4.1. Basic definitions
52
4.2. Operations on representations
54
4.3. Irreducible representations
57
4.4. Intertwining operators and Schur's lemma
59
4.5. Complete reducibility of unitary representations: representations of finite groups
61
4.6. Haar measure on compact Lie groups
62
4.7. Orthogonality of characters and Peter—Weyl theorem
65
4.8. Representations of sl(2, C)
70
4.9. Spherical Laplace operator and the hydrogen atom
75
4.10. Exercises
80
5 Structure theory of Lie algebras 84
5.1. Universal enveloping algebra
84
5.2. Poincare—Birkhoff—Witt theorem
87
5.3. Ideals and commutant
90
5.4. Solvable and nilpotent Lie algebras
91
5.5. Lie's and Engel's theorems
94
5.6. The radical. Semisimple and reductive algebras
96
5.7. Invariant bilinear forms and semisimplicity of classical Lie algebras
99
5.8. Killing form and Cartan's criterion
101
5.9. Jordan decomposition
104
5.10. Exercises
106
6 Complex semisimple Lie algebras 108
6.1. Properties of semisimple Lie algebras
108
6.2. Relation with compact groups
110
6.3. Complete reducibility of representations
112
6.4. Semisimple elements and toral subalgebras
116
6.5. Cartan subalgebra
119
6.6. Root decomposition and root systems
120
6.7. Regular elements and conjugacy of Cartan subalgebras
126
6.8. Exercises
130
7 Root systems 132
7.1. Abstract root systems
132
7.2. Automorphisms and the Weyl group
134
7.3. Pairs of roots and rank two root systems
135
7.4. Positive roots and simple roots
137
7.5. Weight and root lattices
140
7.6. Weyl chambers
142
7.7. Simple reflections
146
7.8. Dynkin diagrams and classification of root systems
149
7.9. Serre relations and classification of semisimple Lie algebras
154
7.10. Proof of the classification theorem in simply-laced case
157
7.11. Exercises
160
8 Representations of semisimple Lie algebras 163
8.1. Weight decomposition and characters
163
8.2. Highest weight representations and Verma modules
167
8.3. Classification of irreducible finite-dimensional representations
171
8.4. Bernstein–Gelfand–Gelfand resolution
174
8.5. Weyl character formula
177
8.6. Multiplicities
182
8.7. Representations of sl(n, C)
183
8.8. Harish–Chandra isomorphism
187
8.9. Proof of Theorem 8.25
192
8.10. Exercises
194
Overview of the literature 197
Basic textbooks
197
Monographs
198
Further reading
198
Appendix A Root systems and simple Lie algebras 202
A.1. An = sl(n + 1,C), n > or = to 1
202
A.2. Bn = so(2n + 1,C), n > or = to 1
204
A.3. Cn = sp(n, C), n > or = to 1
206
A.4. Dn = so(2n,C), n > or = to 2
207
Appendix B Sample syllabus 210
List of notation 213
Bibliography 216
Index 220
Alexander Kirillov, Jr, is an Associate Professor in the Mathematics Department, State University of New York, Stony Brook. His research interests are representation theory, Lie algebras, quantum groups, affine Lie algebras and conformal field theory.
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