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
Author(s): Yvette Kosmann-Schwarzbach
Series: Universitext
Edition: 2
Publisher: Springer
Year: 2022
Language: English
Pages: 251
City: Cham
Tags: Groups, Representations, Lie Groups, Lie Algebras, Spherical Harmonics, Quarks, Spinors
Contents
Introduction
Acknowledgments
1 General Facts About Groups
1 Review of Definitions
2 Examples of Finite Groups
2.1 Cyclic Groups
2.2 Symmetric Groups
2.3 Dihedral Groups
2.4 Crystallographic Groups
3 Examples of Infinite Groups
4 Group Actions and Conjugacy Classes
References
Topics in History
Exercises
2 Representations of Finite Groups
1 Representations
1.1 General Facts
1.2 Irreducible Representations
1.3 Direct Sum of Representations
1.4 Intertwining Operators and Schur's Lemma
2 Characters and Orthogonality Relations
2.1 Matrix Coefficients
2.2 Characters of Representations and Orthogonality Relations
2.3 Character Table
2.4 Application to the Decomposition of Representations
3 The Regular Representation
3.1 Definition
3.2 Character of the Regular Representation
3.3 Isotypic Decomposition
3.4 Basis of the Vector Space of Class Functions
4 Projection Operators
5 Induced Representations
5.1 Definition
5.2 Geometric Interpretation
References
Topics in History
Exercises
3 Representations of Compact Groups
1 Compact Groups
2 Haar Measure
3 Representations of Topological Groups andSchur's Lemma
3.1 General Facts
3.2 Coefficients of a Representation
3.3 Intertwining Operators
3.4 Operations on Representations
3.5 Schur's Lemma
4 Representations of Compact Groups
4.1 Complete Reducibility
4.2 Orthogonality Relations
5 Summary of Chapter 3
References
Topics in history
Exercises
4 Lie Groups and Lie Algebras
1 Lie Algebras
1.1 Definition and Examples
1.2 Morphisms
1.3 Commutation Relations and Structure Constants
1.4 Real Forms
1.5 Representations of Lie Algebras
2 Review of the Exponential Map
3 One-Parameter Subgroups of GL(n,mathbbK)
4 Lie Groups
5 The Lie Algebra of a Lie Group
6 The Connected Component of the Identity
7 Morphisms of Lie Groups and of Lie Algebras
7.1 Differential of a Lie Group Morphism
7.2 Differential of a Lie Group Representation
7.3 The Adjoint Representation
References
Topics in history
Exercises
5 Lie Groups SU(2) and SO(3)
1 The Lie Algebras mathfraksu(2) and mathfrakso(3)
1.1 Bases of mathfraksu(2)
1.2 Bases of mathfrakso(3)
1.3 Bases of mathfraksl(2,mathbbC)
2 The Covering Morphism of SU(2) onto SO(3)
2.1 The Lie Group SO(3)
2.2 The Lie Group SU(2)
2.3 Projection of SU(2) onto SO(3)
References
Topics in History
Exercises
6 Representations of SU (2) and SO(3)
1 Irreducible Representations of mathfraksl(2, mathbbC)
1.1 The Representations Dj
1.2 The Casimir Operator
1.3 Hermitian Nature of the Operators J3 and J2
2 Representations of SU(2)
2.1 The Representations mathcalDj
2.2 Characters of the Representations mathcalDj
3 Representations of SO(3)
References
Topics in History
Exercises
7 Spherical Harmonics
1 Review of L2(S2)
2 Harmonic Polynomials
2.1 Representations of Groups on Function Spaces
2.2 Spaces of Harmonic Polynomials
2.3 Representations of SO(3) on Spaces of Harmonic Polynomials
3 Definition of Spherical Harmonics
3.1 Representations of SO(3) on Spaces of Spherical Harmonics
3.2 The Casimir Operator
3.3 Eigenfunctions of the Casimir Operator
3.4 Bases of the Spaces of Spherical Harmonics
3.5 Explicit Formulas
References
Topics in History
Exercises
8 Representations of SU(3) and Quarks
1 Representations of mathfraksl(3,mathbbC) and SU(3)
1.1 Review of mathfraksl(n,mathbbC)
1.2 The Case of mathfraksl(3,mathbbC)
1.3 The Bases (I3,Y) and (I3,T8) of mathfrakh
1.4 Representations of mathfraksl(3,mathbbC) and SU(3)
2 The Adjoint Representation and Roots
3 The Fundamental Representation and Its Dual
3.1 The Fundamental Representation
3.2 The Dual of the Fundamental Representation
4 Highest Weight of a Finite-Dimensional Representation
4.1 Highest Weight
4.2 Weights as Linear Combinations of the λi
4.3 Finite-Dimensional Representations and Weights
4.4 Another Example: the Representation 6
4.5 One More Example: the Representation 10
5 Tensor Products of Representations
6 The Eightfold Way
6.1 Baryons (B=1)
6.2 Mesons (B=0)
6.3 Baryon Resonances
7 Quarks and Antiquarks
References
Topics in History
Exercises
9 Spin Groups and Spinors
1 Clifford Algebras
1.1 Definition
1.2 Universal Property
1.3 Complex and Real Clifford Algebras
2 The Groups Pin(n) and Spin(n)
2.1 The Group Pin(n)
2.2 Adjunction and Conjugation
2.3 Orthogonal Transformations are Products of Reflections
2.4 The Group Morphism from Pin(n) to O(n)
2.5 Definition and Properties of the Group Spin(n)
2.6 The Groups Spin(1), Spin(2), and Spin(3)
3 Spinor Representations of the Clifford Algebras
3.1 Representations of Algebras
3.2 Spinor Representations of the Complex Clifford Algebras
3.3 The Real Case
4 Representations of the Spin Groups
4.1 The Complex Spin Groups
4.2 The Groups Spin(p,q)
4.3 Representations of the Spin Groups and Spinors
4.4 Spinors in 3 Dimensions
4.5 Spinors in 4 Dimensions and the Dirac Equation
4.6 Important Remark
References
Topics in History
Exercises
Problems and Solutions
1 Restriction of a Representation to a Finite Group
2 The Group O(2)
3 Representations of the Dihedral and Quaternion Groups
4 Representations of SU(2) and of mathfrakS3
5 Pseudo-Unitary and Pseudo-Orthogonal Groups
6 Irreducible Representations of SU(2) timesSU(2)
7 Projection Operators
8 Symmetries of Fullerene Molecules
9 Matrix Coefficients and Spherical Harmonics
Endnote
BibliographyThe literature on finite groups, Lie algebras, Lie groups and representation theory is immense. We present a list containing classical and recent works which seem most useful for a first approach to the topics dealt with in this text, and we mark them with the symbol (*), books for further study, and texts of historical significance.*18pt
