Lie algebras and applications / Francesco Iachello.

Format:

Book

Author/Creator:

Iachello, F.

Series:

Lecture notes in physics 0075-8450 ; 708.

Lecture notes in physics, 0075-8450 ; 708

Language:

English

Subjects (All):

Lie algebras.

Physical Description:

xiv, 196 pages : illustrations ; 24 cm.

Place of Publication:

Berlin ; New York : Springer, [2006]

Summary:

This book, designed for advanced graduate students and post-graduate researchers, provides an introduction to Lie algebras and some of their applications to the spectroscopy of molecules, atoms, nuclei and hadrons. In the first part, a concise exposition is given of the basic concepts of Lie algebras, their representations and their invariants. The second part contains a description of how Lie algebras are used in practice in the treatment of bosonic and fermionic systems. Physical applications considered include rotations and vibrations of molecules (vibron model), collective modes in nuclei (interacting boson model), the atomic shell model, the nuclear shell model, and the quark model of hadrons. One of the key concepts in the application of Lie algebraic methods in physics, that of spectrum generating algebras and their associated dynamic symmetries, is also discussed. The book contains many examples that help to elucidate the abstract algebraic definitions. It provides a summary of many formulas of practical interest, such as the eigenvalues of Casimir operators and the dimensions of the representations of all classical Lie algebras.

Contents:

1.2 Lie Algebras 1

1.3 Change of Basis 3

1.4 Complex Extensions 4

1.5 Lie Subalgebras 4

1.6 Abelian Algebras 5

1.7 Direct Sum 5

1.8 Ideals (Invariant Subalgebras) 6

1.9 Semisimple Algebras 7

1.10 Semidirect Sum 7

1.11 Killing Form 8

1.12 Compact and Non-Compact Algebras 9

1.13 Derivations 9

1.14 Nilpotent Algebras 10

1.15 Invariant Casimir Operators 10

1.16 Invariant Operators for Non-Semisimple Algebras 12

1.17 Structure of Lie Algebras 12

1.17.1 Algebras with One Element 12

1.17.2 Algebras with Two Elements 12

1.17.3 Algebras with Three Elements 13

2 Semisimple Lie Algebras 15

2.1 Cartan-Weyl Form of a (Complex) Semisimple Lie Algebra 15

2.2 Graphical Representation of Root Vectors 15

2.3 Explicit Construction of the Cartan-Weyl Form 17

2.4 Dynkin Diagrams 19

2.5 Classification of (Complex) Semisimple Lie Algebras 21

2.6 Real Forms of Complex Semisimple Lie Algebras 21

2.7 Isomorphisms of Complex Semisimple Lie Algebras 21

2.8 Isomorphisms of Real Lie Algebras 22

2.9 Enveloping Algebra 23

2.10 Realizations of Lie Algebras 23

2.11 Other Realizations of Lie Algebras 24

3 Lie Groups 27

3.1 Groups of Transformations 27

3.2 Groups of Matrices 27

3.3 Properties of Matrices 28

3.4 Continuous Matrix Groups 29

3.5 Examples of Groups of Transformations 32

3.5.1 The Rotation Group in Two Dimensions, SO(2) 32

3.5.2 The Lorentz Group in One Plus One Dimension, 50(1,1) 33

3.5.3 The Rotation Group in Three Dimensions 34

3.5.4 The Special Unitary Group in Two Dimensions, SU(2) 34

3.5.5 Relation Between SO(3) and SU(2) 35

3.6 Lie Algebras and Lie Groups 37

3.6.1 The Exponential Map 37

3.6.2 Definition of Exp 37

3.6.3 Matrix Exponentials 38

4 Irreducible Bases (Representations) 39

4.2 Abstract Characterization 39

4.3 Irreducible Tensors 40

4.3.1 Irreducible Tensors with Respect to GL(n) 40

4.3.2 Irreducible Tensors with Respect to SU(n) 41

4.3.3 Irreducible Tensors with Respect to O(n). Contractions 41

4.4 Tensor Representations of Classical Compact Algebras 42

4.4.1 Unitary Algebras u(n) 42

4.4.2 Special Unitary Algebras su(n) 42

4.4.3 Orthogonal Algebras so(n), n = Odd 43

4.4.4 Orthogonal Algebras so(n), n = Even 43

4.4.5 Symplectic Algebras sp(n), n = Even 43

4.5 Spinor Representations 44

4.5.1 Orthogonal Algebras so(n), n = Odd 44

4.5.2 Orthogonal Algebras so(n), n = Even 44

4.6 Fundamental Representations 45

4.6.1 Unitary Algebras 45

4.6.2 Special Unitary Algebras 45

4.6.3 Orthogonal Algebras, n = Odd 45

4.6.4 Orthogonal Algebras, n = Even 46

4.6.5 Symplectic Algebras 46

4.7 Chains of Algebras 46

4.8 Canonical Chains 46

4.8.1 Unitary Algebras 47

4.8.2 Orthogonal Algebras 48

4.9 Isomorphisms of Spinor Algebras 49

4.10 Nomenclature for u(n) 50

4.11 Dimensions of the Representations 50

4.11.1 Dimensions of the Representations of u(n) 51

4.11.2 Dimensions of the Representations of su(n) 52

4.11.3 Dimensions of the Representations of A[subscript n] = su(n + 1) 52

4.11.4 Dimensions of the Representations of B[subscript n] = so(2n + 1) 52

4.11.5 Dimensions of the Representations of C[subscript n] = sp(2n) 53

4.11.6 Dimensions of the Representations of D[subscript n] = so(2n) 53

4.12 Action of the Elements of g on the Basis B 53

4.13 Tensor Products 56

4.14 Non-Canonical Chains 58

5 Casimir Operators and Their Eigenvalues 63

5.2 Independent Casimir Operators 63

5.2.1 Casimir Operators of u(n) 63

5.2.2 Casimir Operators of su(n) 64

5.2.3 Casimir Operators of so(n), n = Odd 64

5.2.4 Casimir Operators of so(n), n = Even 64

5.2.5 Casimir Operators of sp(n), n = Even 65

5.2.6 Casimir Operators of the Exceptional Algebras 65

5.3 Complete Set of Commuting Operators 65

5.3.1 The Unitary Algebra u(n) 66

5.3.2 The Orthogonal Algebra so(n), n = Odd 66

5.3.3 The Orthogonal Algebra so(n), n = Even 66

5.4 Eigenvalues of Casimir Operators 66

5.4.1 The Algebras u(n) and su(n) 67

5.4.2 The Orthogonal Algebra so(2n + 1) 69

5.4.3 The Symplectic Algebra sp(2n) 71

5.4.4 The Orthogonal Algebra so(2n) 72

5.5 Eigenvalues of Casimir Operators of Order One and Two 74

6 Tensor Operators 75

6.2 Coupling Coefficients 76

6.3 Wigner-Eckart Theorem 77

6.4 Nested Algebras. Racah's Factorization Lemma 79

6.5 Adjoint Operators 81

6.6 Recoupling Coefficients 83

6.7 Symmetry Properties of Coupling Coefficients 84

6.8 How to Compute Coupling Coefficients 85

6.9 How to Compute Recoupling Coefficients 86

6.10 Properties of Recoupling Coefficients 86

6.11 Double Recoupling Coefficients 87

6.12 Coupled Tensor Operators 88

6.13 Reduction Formula of the First Kind 88

6.14 Reduction Formula of the Second Kind 89

7 Boson Realizations 91

7.1 Boson Operators 91

7.2 The Unitary Algebra u(1) 92

7.3 The Algebras u(2) and su(2) 93

7.3.1 Subalgebra Chains 93

7.4 The Algebras u(n), n [GreaterEqual]3 97

7.4.1 Racah Form 97

7.4.2 Tensor Coupled Form of the Commutators 98

7.4.3 Subalgebra Chains Containing so(3) 99

7.5 The Algebras u(3) and su(3) 99

7.5.1 Subalgebra Chains 100

7.5.2 Lattice of Algebras 103

7.5.3 Boson Calculus of u(3) [Superset] so(3) 103

7.5.4 Matrix Elements of Operators in u(3) [Superset] so(3) 105

7.5.5 Tensor Calculus of u(3) [Superset] so(3) 106

7.5.6 Other Boson Constructions of u(3) 107

7.6 The Unitary Algebra u(4) 108

7.6.1 Subalgebra Chains not Containing so(3) 109

7.6.2 Subalgebra Chains Containing so(3) 109

7.7 The Unitary Algebra u(6) 115

7.7.1 Subalgebra Chains not Containing so(3) 115

7.7.2 Subalgebra Chains Containing so(3) 115

7.8 The Unitary Algebra u(7) 123

7.8.1 Subalgebra Chain Containing g[subscript 2] 124

7.8.2 The Triplet Chains 125

8 Fermion Realizations 131

8.1 Fermion Operators 131

8.2 Lie Algebras Constructed with Fermion Operators 131

8.3 Racah Form 132

8.4 The Algebras u(2j + 1) 133

8.4.1 Subalgebra Chain Containing spin(3) 134

8.4.2 The Algebras u(2) and su(2). Spinors 134

8.4.3 The Algebra u(4) 136

8.4.4 The Algebra u(6) 137

8.5 The Algebra u ([Sum][subscript i] (2j[subscript i] + 1)) 138

8.6 Internal Degrees of Freedom (Different Spaces) 139

8.6.1 The Algebras u(4) and su(4) 139

8.6.2 The Algebras u(6) and su(6) 141

8.7 Internal Degrees of Freedom (Same Space) 142

8.7.1 The Algebra u((2l + 1)(2s + 1)): L-S Coupling 142

8.7.2 The Algebra u ([Sum][subscript j] (2j + 1)): j-j Coupling 145

8.7.3 The Algebra u(([Sum][subscript l](2l + 1)) (2s + 1)): Mixed L-S Configurations 146

9 Differential Realizations 147

9.1 Differential Operators 147

9.2 Unitary Algebras u(n) 147

9.3 Orthogonal Algebras so(n) 148

9.3.1 Casimir Operators.

Laplace-Beltrami Form 150

9.3.2 Basis for the Representations 151

9.4 Orthogonal Algebras so(n, m) 152

9.5 Symplectic Algebras sp(2n) 153

10 Matrix Realizations 155

10.1 Matrices 155

10.2 Unitary Algebras u(n) 155

10.3 Orthogonal Algebras so(n) 158

10.4 Symplectic Algebras sp(2n) 159

10.5 Basis for the Representation 160

10.6 Casimir Operators 161

11 Spectrum Generating Algebras and Dynamic Symmetries 163

11.1 Spectrum Generating Algebras (SGA) 163

11.2 Dynamic Symmetries (DS) 163

11.3 Bosonic Systems 164

11.3.1 Dynamic Symmetries of u(4) 165

11.3.2 Dynamic Symmetries of u(6) 167

11.4 Fermionic Systems 170

11.4.1 Dynamic Symmetry of u(4) 170

11.4.2 Dynamic Symmetry of u(6) 171

12 Degeneracy Algebras and Dynamical Algebras 173

12.1 Degeneracy Algebras 173

12.2 Degeneracy Algebras in v [GreaterEqual] 2 Dimensions 173

12.2.1 The Isotropic Harmonic Oscillator 174

12.2.2 The Coulomb Problem 177

12.3 Degeneracy Algebra in v = 1 Dimension 181

12.4 Dynamical Algebras 182

12.5 Dynamical Algebras in v [GreaterEqual] 2 Dimensions 182

12.5.1 Harmonic Oscillator 182

12.5.2 Coulomb Problem 182

12.6 Dynamical Algebra in v = 1 Dimension 183

12.6.1 Poschl-Teller Potential 183

12.6.2 Morse Potential 185

12.6.3 Lattice of Algebras 187.

Notes:

Includes bibliographical references (pages [189]-192) and index.

ISBN:

3540362363

OCLC:

71336680

Publisher Number:

9783540362364