Symmetries, Lie Algebras and Representations A graduate course for physicists

Title page
Preface
1 Symmetries and conservation laws
1.1 Symmetries
1.2 Continuous parameters and local one-parameter groups
1.3 Classical mechanics: Lagrangian description
1.4 Conservation laws
1.5 Classical mechanics: Hamiltonian description
1.6 Quantum mechanics
1.7 Observation of symmetries
1.8 Gauge 'symmetries'
1.9 Duality symmetries
2 Basic examples
2.1 Angular momentum
2.2 Step operators
2.3 Irreducible representations
2.4 The free scalar field
2.5 The Heisenberg algebra
2.6 Common features
3 The Lie algebra SU(3) and hadron symmetries
3.1 Symmetries of hadrons
3.2 Combining two SL(2)-algebras
3.3 Multiple commutators
3.4 SL(3) and SU(3)
3.5 Orthogonal basis
3.6 The eightfold way
3.7 Quarks
3.8 Abstract Lie algebras
4 Formalization: Algebras and Lie algebras
4.1 Short summary
4.2 Groups, rings, fields
4.3 Vector spaces
4.4 Algebras
4.5 Lie algebras
4.6 Generators
4.7 Homomorphisms, isomorphisms, derivations
4.8 Subalgebras and ideals
4.9* Solvable and nilpotent Lie algebras
4.10 Semisimple and abelian Lie algebras
4.11* Gradations and Lie superalgebras
4.12* Open and non-linear algebras
5 Representations
5.1 Representations and representation matrices
5.2 The adjoint representation
5.3 Constructing more representations
5.4 Schur's Lemma
5.5 Reducible modules
5.6 Matrix Lie algebras
5.7 The representation theory of SL(2) revisited
6 The Cartan-Weyl basis
6.1 Cartan subalgebras
6.2 Roots
6.3 The Killing form
6.4 Some properties of roots and the root system
6.5 Structure constants of the Cartan-Weyl basis
6.6 Positive roots
6.7 Simple roots and the Cartan matrix
6.8 Root and weight lattices. The Dynkin basis
6.9 The metric on weight space
6.10 The Chevalley basis
6.11 Root strings
6.12 Examples
6.13* Non-semisimple Lie algebras
7 Simple and affine Lie algebras
7.1 Chevalley-Serre relations
7.2 The Cartan matrix
7.3 Dynkin diagrams
7.4 Simple Lie algebras
7.5 Quadratic form matrices
7.6 The highest root and the Weyl vector
7.7 The orthonormal basis of weight space
7.8 Kac-Moody algebras
7.9 Affine Lie algebras
7.10 Coxeter labels and dual Coxeter labels
8 Real Lie algebras and real forms
8.1 More about the Killing form
8.2 The Killing form of real Lie algebras
8.3 The compact and the normal real form
8.4 The real forms of simple Lie algebras
9 Lie groups
9.1 Lie group manifolds
9.2 Global vector fields
9.3 Compactness
9.4 The exponential map
9.5* Maurer-Cartan theory
9.6 Spin
9.7 The classical Lie groups
9.8 BCH formulae
10 Symmetries of the root system. The Weyl group
10.1 The Weyl group
10.2 Fundamental reflections. Length and sign
10.3 Weyl chambers
10.4 The Weyl group in the orthogonal basis
10.5 The Weyl vector
10.6 The normalizer of the Cartan subalgebra
10.7 The Weyl group of affine. Lie algebras
10.8* Coxeter groups
11 Automorphisms of Lie algebras
11.1 The group of automorphisms
11.2 Automorphisms of finite order
11.3 Inner and outer automorphisms
11.4 An 5((2) example
11.5 Dynkin diagram symmetries
11.6 Finite order inner automorphisms of simple Lie algebras
11.7 Automorphisms of the root system
11.8 Finite order outer automorphisms of simple Lie algebras
11.9 Involutive automorphisms, conjugations, real forms
11.10* Involutive automorphisms and symmetric spaces
12 Loop algebras and central extensions
12.1 Central extensions
12.2* Ray representations and cohomology
12.3 Loop algebras
12.4 Towards untwisted affine Lie algebras
12.5 The derivation
12.6 The Cartan-Weyl basis
12.7 The root system
12.8 Twisted affine Lie algebras
12.9* The root system of twisted affine Lie algebras
12.10 Heisenberg algebras
12.11* Gradations
12.12 The Virasoro algebra
13 Highest weight representations
13.1 Highest weight representations of 5[(2)
13.2 Highest weight modules of simple Lie algebras
13.3 The weight system
13.4 Unitarity
13.5 Characters and dimensions
13.6 Defining modules and basic modules
13.7* The dual Weyl vector
13.8 Highest weight modules of affine Lie algebras
13.9 Integrable highest weight modules
13.10* Twisted affine Lie algebras
13.11 Triangular decomposition
14 Verma modules, Casimirs, and the character formula
14.1 Universal enveloping algebras
14.2 Quotients of algebras
14.3 The Poincaré-Birkhoff-Witt theorem
14.4 Verma modules and null vectors
14.5 The character of a Verma module
14.6 Null vectors and unitarity
14.7 The quadratic Casimir operator
14.8 Quadratic Casimir eigenvalues
14.9* The second order Dynkin index
14.10 The character formula
14.11* Characters and modular transformations
14.12* Harish-Chandra theorem and higher Casimir operators
15 Tensor products of representations
15.1 Tensor products
15.2* Tensor products for other algebraic structures
15.3 Highest weight modules
15.4 Tensor products for SL(2)
15.5 Conjugacy classes
15.6 Sum rules
15.7 The Racah-Speiser algorithm
15.8 Examples
15.9* Kostant's function and Steinberg's formula
15.10* Affine Lie algebras
16 Clebsch-Gordan coefficients and tensor operators
16.1 Isomorphisms involving tensor products of modules
16.2 Clebsch-Gordan coefficients and 3j-symbols
16.3 Calculation of Clebsch-Gordan coefficients
16.4 Formulre for Clebsch-Gordan coefficients
16.5 Intertwiner spaces
16.6 Racah coefficients and 6j-symbols
16.7 The pentagon and hexagon identities
16.8 Tensor operators
16.9 The Wigner-Eckart theorem
17 Invariant tensors
17.1 Intertwiners and invariant tensors
17.2 Orthogonal and symplectic modules
17.3 Primitive invariants
17.4 Adjoint tensors of SL(n)
17.5 Tensors for spin or modules
17.6 Singlets in tensor products
17.7 Projection operators
17.8 Casimir operators
18 Subalgebras and branching rules
18.1 Subalgebras
18.2 The Dynkin index of an embedding
18.3 Finding regular subalgebras
18.4 S-subalgebras
18.5 Projection maps and defining representations
18.6 Branching rules
18.7* Embeddings of affine Lie algebras
19 Young tableaux and the symmetric group
19.1 Young tableaux
19.2 Dimensions and quadratic Casimir eigenvalues
19.3 Tensor products
19.4* Young tableaux for other Lie algebras
19.5* Brauer-Weyl theory
20 Spinors, Clifford algebras, and supersymmetry
20.1 Spinor representations
20.2 Clifford algebras
20.3 Representation theory of Clifford algebras
20.4 Representations of orthogonal algebras and groups
20.5* SU(2) versus SO(3) revisited
20.6 Zoology of spin ors
20.7 Spinors of the Lorentz group
20.8 The Poincaré algebra. Contractions
20.9 Supersymmetry
20.10* Twistors
21 Representations on function spaces
21.1 Group actions on functions spaces
21.2 Multiplier representations and Lie derivatives
21.3 Multiplier representations of SL(2)
21.4 General function spaces
21.5 Spherical harmonics
21.6 Hypergeometric functions
21.7* Classification of Lie derivatives
21.8 The Haar measure
21.9 The Peter-Weyl theorem
21.10 Group characters and class functions
21.11 Fourier analysis
22 Hopf algebras and representation rings
22.1 Co-products
22.2 Co-algebras and bi-algebras
22.3 Hopf algebras
22.4 Universal enveloping algebras
22.5 Quasi-triangular Hopf algebras
22.6 Deformed enveloping algebras
22.7 Group Hopf algebras
22.8 Functions on compact groups
22.9 Quantum groups
22.10 Quasi-Hopf algebras
22.11 Character rings
22.12 Rational fusion rings
22.13 The extended Racah-Speiser formalism
Epilogue
References
Index