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Lie Groups, Lie Algebras, and Representations: An Elementary Introduction 2nd ed. 2015 [Kõva köide]

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  • Formaat: Hardback, 449 pages, kõrgus x laius: 235x155 mm, kaal: 9132 g, 7 Illustrations, color; 72 Illustrations, black and white; XIII, 449 p. 79 illus., 7 illus. in color., 1 Hardback
  • Sari: Graduate Texts in Mathematics 222
  • Ilmumisaeg: 22-May-2015
  • Kirjastus: Springer International Publishing AG
  • ISBN-10: 3319134663
  • ISBN-13: 9783319134666
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Lie Groups, Lie Algebras, and Representations: An Elementary Introduction 2nd ed. 2015Suurem pilt
  • Formaat: Hardback, 449 pages, kõrgus x laius: 235x155 mm, kaal: 9132 g, 7 Illustrations, color; 72 Illustrations, black and white; XIII, 449 p. 79 illus., 7 illus. in color., 1 Hardback
  • Sari: Graduate Texts in Mathematics 222
  • Ilmumisaeg: 22-May-2015
  • Kirjastus: Springer International Publishing AG
  • ISBN-10: 3319134663
  • ISBN-13: 9783319134666
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9783319134666.html
Märksõnad:
This textbook treats Lie groups, Lie algebras and their representations in an elementary but fully rigorous fashion requiring minimal prerequisites. In particular, the theory of matrix Lie groups and their Lie algebras is developed using only linear algebra, and more motivation and intuition for proofs is provided than in most classic texts on the subject.

In addition to its accessible treatment of the basic theory of Lie groups and Lie algebras, the book is also noteworthy for including:





a treatment of the BakerCampbellHausdorff formula and its use in place of the Frobenius theorem to establish deeper results about the relationship between Lie groups and Lie algebras motivation for the machinery of roots, weights and the Weyl group via a concrete and detailed exposition of the representation theory of sl(3;C) an unconventional definition of semisimplicity that allows for a rapid development of the structure theory of semisimple Lie algebras a self-contained construction of the representations of compact groups, independent of Lie-algebraic arguments

The second edition of Lie Groups, Lie Algebras, and Representations contains many substantial improvements and additions, among them: an entirely new part devoted to the structure and representation theory of compact Lie groups; a complete derivation of the main properties of root systems; the construction of finite-dimensional representations of semisimple Lie algebras has been elaborated; a treatment of universal enveloping algebras, including a proof of the PoincaréBirkhoffWitt theorem and the existence of Verma modules; complete proofs of the Weyl character formula, the Weyl dimension formula and the Kostant multiplicity formula.

Review of the first edition:

This is an excellent book. It deserves to, and undoubtedly will, become the standard text for early graduate courses in Lie group theory ... an important addition tothe textbook literature ... it is highly recommended.

The Mathematical Gazette

Arvustused

The first edition of this book was very good; the second is even better, and more versatile. This text remains one of the most attractive sources available from which to learn elementary Lie group theory, and is highly recommended. (Mark Hunacek, The Mathematical Gazette, Vol. 101 (551), July, 2017)

Part I General Theory
1 Matrix Lie Groups
3(28)
1.1 Definitions
3(2)
1.2 Examples
5(11)
1.3 Topological Properties
16(5)
1.4 Homomorphisms
21(4)
1.5 Lie Groups
25(1)
1.6 Exercises
26(5)
2 The Matrix Exponential
31(18)
2.1 The Exponential of a Matrix
31(3)
2.2 Computing the Exponential
34(2)
2.3 The Matrix Logarithm
36(4)
2.4 Further Properties of the Exponential
40(2)
2.5 The Polar Decomposition
42(4)
2.6 Exercises
46(3)
3 Lie Algebras
49(28)
3.1 Definitions and First Examples
49(4)
3.2 Simple, Solvable, and Nilpotent Lie Algebras
53(2)
3.3 The Lie Algebra of a Matrix Lie Group
55(2)
3.4 Examples
57(3)
3.5 Lie Group and Lie Algebra Homomorphisms
60(5)
3.6 The Complexification of a Real Lie Algebra
65(2)
3.7 The Exponential Map
67(3)
3.8 Consequences of Theorem 3.42
70(3)
3.9 Exercises
73(4)
4 Basic Representation Theory
77(32)
4.1 Representations
77(4)
4.2 Examples of Representations
81(3)
4.3 New Representations from Old
84(6)
4.4 Complete Reducibility
90(4)
4.5 Schur's Lemma
94(2)
4.6 Representations of sl(2; C)
96(5)
4.7 Group Versus Lie Algebra Representations
101(2)
4.8 A Nonmatrix Lie Group
103(2)
4.9 Exercises
105(4)
5 The Baker--Campbell--Hausdorff Formula and Its Consequences
109(32)
5.1 The "Hard" Questions
109(1)
5.2 An Illustrative Example
110(3)
5.3 The Baker--Campbell--Hausdorff Formula
113(1)
5.4 The Derivative of the Exponential Map
114(3)
5.5 Proof of the BCH Formula
117(1)
5.6 The Series Form of the BCH Formula
118(1)
5.7 Group Versus Lie Algebra Homomorphisms
119(7)
5.8 Universal Covers
126(2)
5.9 Subgroups and Subalgebras
128(7)
5.10 Lie's Third Theorem
135(1)
5.11 Exercises
135(6)
Part II Semisimple Lie Algebras
6 The Representations of sl(3; C)
141(28)
6.1 Preliminaries
141(1)
6.2 Weights and Roots
142(4)
6.3 The Theorem of the Highest Weight
146(2)
6.4 Proof of the Theorem
148(5)
6.5 An Example: Highest Weight (1,1)
153(1)
6.6 The Weyl Group
154(4)
6.7 Weight Diagrams
158(1)
6.8 Further Properties of the Representations
159(6)
6.9 Exercises
165(4)
7 Semisimple Lie Algebras
169(28)
7.1 Semisimple and Reductive Lie Algebras
169(5)
7.2 Cartan Subalgebras
174(2)
7.3 Roots and Root Spaces
176(6)
7.4 The Weyl Group
182(1)
7.5 Root Systems
183(2)
7.6 Simple Lie Algebras
185(3)
7.7 The Root Systems of the Classical Lie Algebras
188(5)
7.8 Exercises
193(4)
8 Root Systems
197(44)
8.1 Abstract Root Systems
197(4)
8.2 Examples in Rank Two
201(3)
8.3 Duality
204(2)
8.4 Bases and Weyl Chambers
206(6)
8.5 Weyl Chambers and the Weyl Group
212(4)
8.6 Dynkin Diagrams
216(2)
8.7 Integral and Dominant Integral Elements
218(3)
8.8 The Partial Ordering
221(7)
8.9 Examples in Rank Three
228(4)
8.10 The Classical Root Systems
232(4)
8.11 The Classification
236(2)
8.12 Exercises
238(3)
9 Representations of Semisimple Lie Algebras
241(24)
9.1 Weights of Representations
242(2)
9.2 Introduction to Verma Modules
244(2)
9.3 Universal Enveloping Algebras
246(4)
9.4 Proof of the PBW Theorem
250(4)
9.5 Construction of Verma Modules
254(3)
9.6 Irreducible Quotient Modules
257(3)
9.7 Finite-Dimensional Quotient Modules
260(3)
9.8 Exercises
263(2)
10 Further Properties of the Representations
265(42)
10.1 The Structure of the Weights
265(4)
10.2 The Casimir Element
269(4)
10.3 Complete Reducibility
273(2)
10.4 The Weyl Character Formula
275(6)
10.5 The Weyl Dimension Formula
281(6)
10.6 The Kostant Multiplicity Formula
287(7)
10.7 The Character Formula for Verma Modules
294(1)
10.8 Proof of the Character Formula
295(8)
10.9 Exercises
303(4)
Part III Compact Lie Groups
11 Compact Lie Groups and Maximal Tori
307(36)
11.1 Tori
308(4)
11.2 Maximal Tori and the Weyl Group
312(3)
11.3 Mapping Degrees
315(6)
11.4 Quotient Manifolds
321(5)
11.5 Proof of the Torus Theorem
326(4)
11.6 The Weyl Integral Formula
330(3)
11.7 Roots and the Structure of the Weyl Group
333(6)
11.8 Exercises
339(4)
12 The Compact Group Approach to Representation Theory
343(28)
12.1 Representations
343(3)
12.2 Analytically Integral Elements
346(5)
12.3 Orthonormality and Completeness for Characters
351(6)
12.4 The Analytic Proof of the Weyl Character Formula
357(4)
12.5 Constructing the Representations
361(5)
12.6 The Case in Which S is Not Analytically Integral
366(3)
12.7 Exercises
369(2)
13 Fundamental Groups of Compact Lie Groups
371
13.1 The Fundamental Group
371(2)
13.2 Fundamental Groups of Compact Classical Groups
373(4)
13.3 Fundamental Groups of Noncompact Classical Groups
377(1)
13.4 The Fundamental Groups of K and T
377(6)
13.5 Regular Elements
383(6)
13.6 The Stiefel Diagram
389(5)
13.7 Proofs of the Main Theorems
394(5)
13.8 The Center of A
399(4)
13.9 Exercises
403
Erratum
1(442)
A Linear Algebra Review
407(12)
A.1 Eigenvectors and Eigenvalues
407(1)
A.2 Diagonalization
408(1)
A.3 Generalized Eigenvectors and the SN Decomposition
409(2)
A.4 The Jordan Canonical Form
411(1)
A.5 The Trace
411(1)
A.6 Inner Products
412(2)
A.7 Dual Spaces
414(1)
A.8 Simultaneous Diagonalization
415(4)
B Differential Forms
419(6)
C Clebsch--Gordan Theory and the Wigner--Eckart Theorem
425(10)
C.1 Tensor Products of sl(2; C) Representations
425(3)
C.2 The Wigner-Eckart Theorem
428(4)
C.3 More on Vector Operators
432(3)
D Completeness of Characters
435(8)
References 443(2)
Index 445
Brian Hall is Professor of Mathematics at the University of Notre Dame, IN.