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El. knyga: Groups and Symmetries: From Finite Groups to Lie Groups

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  • Formatas: EPUB+DRM
  • Serija: Universitext
  • Išleidimo metai: 16-Jul-2022
  • Leidėjas: Springer Nature Switzerland AG
  • Kalba: eng
  • ISBN-13: 9783030943608
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Groups and Symmetries: From Finite Groups to Lie GroupsDidesnis paveikslėlis
  • Formatas: EPUB+DRM
  • Serija: Universitext
  • Išleidimo metai: 16-Jul-2022
  • Leidėjas: Springer Nature Switzerland AG
  • Kalba: eng
  • ISBN-13: 9783030943608
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Groups and Symmetries: From Finite Groups to Lie Groups presents an introduction to the theory of group representations and its applications in quantum mechanics. Accessible to advanced undergraduates in mathematics and physics as well as beginning graduate students, the text deals with the theory of representations of finite groups, compact groups, linear Lie groups and their Lie algebras, concisely and in one volume. Prerequisites include calculus and linear algebra.





This new edition contains an additional chapter that deals with Clifford algebras, spin groups, and the theory of spinors, as well as new sections entitled Topics in history comprising notes on the history of the material treated within each chapter. (Taken together, they constitute an account of the development of the theory of groups from its inception in the 18th century to the mid-20th.)





References for additional resources and further study are provided in each chapter. All chapters end with exercises of varying degree of difficulty, some of which introduce new definitions and results. The text concludes with a collection of problems with complete solutions making it ideal for both course work and independent study.





Key Topics include:









Brisk review of the basic definitions of group theory, with examples Representation theory of finite groups: character theory Representations of compact groups using the Haar measure Lie algebras and linear Lie groups Detailed study of SO(3) and SU(2), and their representations Spherical harmonics Representations of SU(3), roots and weights, with quark theory as a consequence of the mathematical properties of this symmetry group Spin groups and spinors

Recenzijos

The book is very clear and easy to read linearly . The reader can also cement their understanding with exercises at the end of each chapter, and the book concludes with with an epilogue of worked problems which allow a deeper dive in various directions. the book is also a great starting point for those with some training in mathematics or physics who wish to broaden their horizons. (Alastair Litterick, zbMATH 1514.20003, 2023)

Introduction xiii
Acknowledgments xix
1 General Facts About Groups
1(12)
1 Review of Definitions
1(2)
2 Examples of Finite Groups
3(1)
2.1 Cyclic Groups
3(1)
2.2 Symmetric Groups
3(1)
2.3 Dihedral Groups
3(1)
2.4 Crystallographic Groups
4(1)
3 Examples of Infinite Groups
4(2)
4 Group Actions and Conjugacy Classes
6(7)
References
7(1)
Topics in History
7(3)
Exercises
10(3)
2 Representations of Finite Groups
13(28)
1 Representations
13(5)
1.1 General Facts
13(2)
1.2 Irreducible Representations
15(3)
1.3 Direct Sum of Representations
18(1)
1.4 Intertwining Operators and Schur's Lemma
18(2)
2 Characters and Orthogonality Relations
18(1)
2.1 Matrix Coefficients
18(1)
2.2 Characters of Representations and Orthogonality Relations
19(4)
2.3 Character Table
23(1)
2.4 Application to the Decomposition of Representations
23(1)
3 The Regular Representation
24(4)
3.1 Definition
24(1)
3.2 Character of the Regular Representation
25(1)
3.3 Isotypic Decomposition
26(1)
3.4 Basis of the Vector Space of Class Functions
26(2)
4 Projection Operators
28(1)
5 Induced Representations
29(12)
5.1 Definition
30(1)
5.2 Geometric Interpretation
30(1)
References
31(1)
Topics in History
31(4)
Exercises
35(6)
3 Representations of Compact Groups
41(18)
1 Compact Groups
41(1)
2 Haar Measure
42(2)
3 Representations of Topological Groups and Schur's Lemma
44(3)
3.1 General Facts
44(1)
3.2 Coefficients of a Representation
45(1)
3.3 Intertwining Operators
45(1)
3.4 Operations on Representations
46(1)
3.5 Schur's Lemma
47(1)
4 Representations of Compact Groups
47(4)
4.1 Complete Reducibility
47(1)
4.2 Orthogonality Relations
48(3)
5 Summary of
Chapter 3
51(8)
References
52(1)
Topics in History
52(3)
Exercises
55(4)
4 Lie Groups and Lie Algebras
59(30)
1 Lie Algebras
59(5)
1.1 Definition and Examples
59(2)
1.2 Morphisms
61(1)
1.3 Commutation Relations and Structure Constants
61(1)
1.4 Real Forms
62(1)
1.5 Representations of Lie Algebras
62(2)
2 Review of the Exponential Map
64(2)
3 One-Parameter Subgroups of GL(n, K)
66(2)
4 Lie Groups
68(1)
5 The Lie Algebra of a Lie Group
69(3)
6 The Connected Component of the Identity
72(1)
7 Morphisms of Lie Groups and of Lie Algebras
73(16)
7.1 Differential of a Lie Group Morphism
73(2)
7.2 Differential of a Lie Group Representation
75(2)
7.3 The Adjoint Representation
77(2)
References
79(1)
Topics in History
79(4)
Exercises
83(6)
5 Lie Groups SU(2) and SO(3)
89(14)
1 The Lie Algebras su(2) and so(3)
89(3)
1.1 Bases of su(2)
89(2)
1.2 Bases of so(3)
91(1)
1.3 Bases of sI(2, C)
92(1)
2 The Covering Morphism of SU(2) onto SO(3)
92(11)
2.1 The Lie Group SO(3)
93(2)
2.2 The Lie Group SU(2)
95(1)
2.3 Projection of SU(2) onto SO(3)
96(1)
References
97(1)
Topics in History
97(4)
Exercises
101(2)
6 Representations of SU(2) and SO(3)
103(16)
1 Irreducible Representations of sI(2, C)
103(5)
1.1 The Representations Dj
103(3)
1.2 The Casimir Operator
106(1)
1.3 Hermitian Nature of the Operators J3 and J2
106(2)
2 Representations of SU(2)
108(4)
2.1 The Representations Dj
108(3)
2.2 Characters of the Representations Dj
111(1)
3 Representations of SO(3)
112(7)
References
113(1)
Topics in History
113(3)
Exercises
116(3)
7 Spherical Harmonics
119(16)
1 Review of L2(SP)
119(1)
2 Harmonic Polynomials
120(3)
2.1 Representations of Groups on Function Spaces
120(1)
2.2 Spaces of Harmonic Polynomials
120(1)
2.3 Representations of SO(3) on Spaces of Harmonic Polynomials
121(2)
3 Definition of Spherical Harmonics
123(12)
3.1 Representations of SO(3) on Spaces of Spherical Harmonics
123(2)
3.2 The Casimir Operator
125(1)
3.3 Eigenfunctions of the Casimir Operator
125(1)
3.4 Bases of the Spaces of Spherical Harmonics
126(3)
3.5 Explicit Formulas
129(1)
References
130(1)
Topics in History
130(3)
Exercises
133(2)
8 Representations of SU(3) and Quarks
135(26)
1 Representations of sI(3, C) and SU(3)
135(3)
1.1 Review of sI(n, C)
135(1)
1.2 The Case of sI(3, C)
136(2)
1.3 The Bases (I3, Y) and (I3, T3) of h
138(1)
1.4 Representations of sI(3, C) and SU(3)
138(1)
2 The Adjoint Representation and Roots
138(2)
3 The Fundamental Representation and Its Dual
140(2)
3.1 The Fundamental Representation
140(1)
3.2 The Dual of the Fundamental Representation
141(1)
4 Highest Weight of a Finite-Dimensional Representation
142(5)
4.1 Highest Weight
142(1)
4.2 Weights as Linear Combinations of the λi
143(1)
4.3 Finite-Dimensional Representations and Weights
144(1)
4.4 Another Example: the Representation 6
145(1)
4.5 One More Example: the Representation 10
146(1)
5 Tensor Products of Representations
147(3)
6 The Eightfold Way
150(3)
6.1 Baryons (B = 1)
151(1)
6.2 Mesons (B = 0)
151(1)
6.3 Baryon Resonances
152(1)
7 Quarks and Antiquarks
153(8)
References
154(1)
Topics in History
154(3)
Exercises
157(4)
9 Spin Groups and Spinors
161(18)
1 Clifford Algebras
161(2)
1.1 Definition
161(1)
1.2 Universal Property
162(1)
1.3 Complex and Real Clifford Algebras
162(1)
2 The Groups Pin(n) and Spin(n)
163(5)
2.1 The Group Pin(n)
163(1)
2.2 Adjunction and Conjugation
164(1)
2.3 Orthogonal Transformations are Products of Reflections
164(2)
2.4 The Group Morphism from Pin(n) to O(n)
166(1)
2.5 Definition and Properties of the Group Spin(n)
166(2)
2.6 The Groups Spin(1), Spin(2), and Spin(3)
168(1)
3 Spinor Representations of the Clifford Algebras
168(2)
3.1 Representations of Algebras
168(1)
3.2 Spinor Representations of the Complex Clifford Algebras
169(1)
3.3 The Real Case
170(1)
4 Representations of the Spin Groups
170(1)
4.1 The Complex Spin Groups
170(1)
4.2 The Groups Spin(p, q)
171(1)
4.3 Representations of the Spin Groups and Spinors
172(1)
4.4 Spinors in 3 Dimensions
172(1)
4.5 Spinors in 4 Dimensions and the Dirac Equation
172(1)
4.6 Important Remark
173(1)
References
173(1)
Topics in History
173(3)
Exercises
176(3)
Problems and Solutions
179(58)
1 Restriction of a Representation to a Finite Group
179(3)
2 The Group 0(2)
182(3)
3 Representations of the Dihedral and Quaternion Groups
185(9)
4 Representations of SU(2) and of C3
194(4)
5 Pseudo-Unitary and Pseudo-Orthogonal Groups
198(6)
6 Irreducible Representations of SU(2) × SU(2)
204(8)
7 Projection Operators
212(7)
8 Symmetries of Fullerene Molecules
219(11)
9 Matrix Coefficients and Spherical Harmonics
230(7)
Endnote 237(4)
Bibliography 241(6)
Index 247
A former student of the École Normale Supérieure in Paris, Yvette Kosmann-Schwarzbach holds a Doctorat dÉtat in mathematics as well as a degree in physics from the University of Paris. She has been a professor of mathematics at the University of Lille, at Brooklyn College of the City University of New York, and at the École Polytechnique (France). She has organized numerous conferences, and has held visiting positions and lectured on four continents.

The author of the historical study, The Noether Theorems, Invariance and Conservation Laws in the Twentieth Century (Sources and Studies in the History of Mathematics and Physical Sciences), she has published over 90 research articles in differential geometry, algebra and mathematical physics, and has co-edited The Verdier Memorial Conference on Integrable Systems (Progress in Mathematics), Integrability of Nonlinear Systems (Lecture Notes in Physics) and Discrete Integrable Systems (LectureNotes in Physics).
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