Lie Algebras and Applications Book

Basic Concepts     1
Definitions     1
Lie Algebras     1
Change of Basis     3
Complex Extensions     4
Lie Subalgebras     4
Abelian Algebras     5
Direct Sum     5
Ideals (Invariant Subalgebras)     6
Semisimple Algebras     7
Semidirect Sum     7
Killing Form     8
Compact and Non-Compact Algebras     9
Derivations     9
Nilpotent Algebras     10
Invariant Casimir Operators     10
Invariant Operators for Non-Semisimple Algebras     12
Structure of Lie Algebras     12
Algebras with One Element     12
Algebras with Two Elements     12
Algebras with Three Elements     13
Semisimple Lie Algebras     15
Cartan-Weyl Form of a (Complex) Semisimple Lie Algebra     15
Graphical Representation of Root Vectors     15
Explicit Construction of the Cartan-Weyl Form     17
Dynkin Diagrams     19
Classification of (Complex) Semisimple Lie Algebras     21
Real Forms of Complex Semisimple Lie Algebras     21
Isomorphisms of ComplexSemisimple Lie Algebras     21
Isomorphisms of Real Lie Algebras     22
Enveloping Algebra     23
Realizations of Lie Algebras     23
Other Realizations of Lie Algebras     24
Lie Groups     27
Groups of Transformations     27
Groups of Matrices     27
Properties of Matrices     28
Continuous Matrix Groups     29
Examples of Groups of Transformations     32
The Rotation Group in Two Dimensions, SO(2)     32
The Lorentz Group in One Plus One Dimension, 50(1,1)     33
The Rotation Group in Three Dimensions     34
The Special Unitary Group in Two Dimensions, SU(2)     34
Relation Between SO(3) and SU(2)     35
Lie Algebras and Lie Groups     37
The Exponential Map     37
Definition of Exp     37
Matrix Exponentials     38
Irreducible Bases (Representations)     39
Definitions     39
Abstract Characterization     39
Irreducible Tensors     40
Irreducible Tensors with Respect to GL(n)     40
Irreducible Tensors with Respect to SU(n)     41
Irreducible Tensors with Respect to O(n). Contractions     41
Tensor Representations of Classical Compact Algebras     42
Unitary Algebras u(n)     42
Special Unitary Algebras su(n)     42
Orthogonal Algebras so(n), n = Odd     43
Orthogonal Algebras so(n), n = Even     43
Symplectic Algebras sp(n), n = Even     43
Spinor Representations     44
Orthogonal Algebras so(n), n = Odd     44
Orthogonal Algebras so(n), n = Even     44
Fundamental Representations     45
Unitary Algebras     45
Special Unitary Algebras     45
Orthogonal Algebras, n = Odd     45
Orthogonal Algebras, n = Even     46
Symplectic Algebras     46
Chains of Algebras     46
Canonical Chains     46
Unitary Algebras     47
Orthogonal Algebras     48
Isomorphisms of Spinor Algebras     49
Nomenclature for u(n)     50
Dimensions of the Representations     50
Dimensions of the Representations of u(n)     51
Dimensions of the Representations of su(n)     52
Dimensions of the Representations of A[subscript n] = su(n + 1)     52
Dimensions of the Representations of B[subscript n] = so(2n + 1)     52
Dimensions of the Representations of C[subscript n] = sp(2n)     53
Dimensions of the Representations of D[subscript n] = so(2n)     53
Action of the Elements of g on the Basis B     53
Tensor Products     56
Non-Canonical Chains     58
Casimir Operators and Their Eigenvalues     63
Definitions     63
Independent Casimir Operators     63
Casimir Operators of u(n)     63
Casimir Operators of su(n)     64
Casimir Operators of so(n), n = Odd     64
Casimir Operators of so(n), n = Even     64
Casimir Operators of sp(n), n = Even     65
Casimir Operators of the Exceptional Algebras     65
Complete Set of Commuting Operators     65
The Unitary Algebra u(n)     66
The Orthogonal Algebra so(n), n = Odd     66
The Orthogonal Algebra so(n), n = Even     66
Eigenvalues of Casimir Operators     66
The Algebras u(n) and su(n)     67
The Orthogonal Algebra so(2n + 1)     69
The Symplectic Algebra sp(2n)     71
The Orthogonal Algebra so(2n)     72
Eigenvalues of Casimir Operators of Order One and Two      74
Tensor Operators     75
Definitions     75
Coupling Coefficients     76
Wigner-Eckart Theorem     77
Nested Algebras. Racah's Factorization Lemma     79
Adjoint Operators     81
Recoupling Coefficients     83
Symmetry Properties of Coupling Coefficients     84
How to Compute Coupling Coefficients     85
How to Compute Recoupling Coefficients     86
Properties of Recoupling Coefficients     86
Double Recoupling Coefficients     87
Coupled Tensor Operators     88
Reduction Formula of the First Kind     88
Reduction Formula of the Second Kind     89
Boson Realizations     91
Boson Operators     91
The Unitary Algebra u(1)     92
The Algebras u(2) and su(2)     93
Subalgebra Chains     93
The Algebras u(n), n [GreaterEqual]3     97
Racah Form     97
Tensor Coupled Form of the Commutators     98
Subalgebra Chains Containing so(3)     99
The Algebras u(3) and su(3)     99
Subalgebra Chains     100
Lattice of Algebras     103
Boson Calculus of u(3) [Superset] so(3)     103
Matrix Elements of Operators in u(3) [Superset] so(3)     105
Tensor Calculus of u(3) [Superset] so(3)     106
Other Boson Constructions of u(3)     107
The Unitary Algebra u(4)     108
Subalgebra Chains not Containing so(3)     109
Subalgebra Chains Containing so(3)     109
The Unitary Algebra u(6)     115
Subalgebra Chains not Containing so(3)     115
Subalgebra Chains Containing so(3)     115
The Unitary Algebra u(7)     123
Subalgebra Chain Containing g[subscript 2]     124
The Triplet Chains     125
Fermion Realizations     131
Fermion Operators     131
Lie Algebras Constructed with Fermion Operators     131
Racah Form     132
The Algebras u(2j + 1)     133
Subalgebra Chain Containing spin(3)     134
The Algebras u(2) and su(2). Spinors     134
The Algebra u(4)     136
The Algebra u(6)     137
The Algebra u ([Sum][subscript i] (2j[subscript i] + 1))     138
Internal Degrees of Freedom (Different Spaces)     139
The Algebras u(4) and su(4)     139
The Algebras u(6) and su(6)     141
Internal Degrees of Freedom (Same Space)     142
The Algebra u((2l + 1)(2s + 1)): L-S Coupling     142
The Algebra u ([Sum][subscript j] (2j + 1)): j-j Coupling     145
The Algebra u(([Sum][subscript l](2l + 1)) (2s + 1)): Mixed L-S Configurations     146
Differential Realizations     147
Differential Operators     147
Unitary Algebras u(n)     147
Orthogonal Algebras so(n)     148
Casimir Operators. Laplace-Beltrami Form     150
Basis for the Representations     151
Orthogonal Algebras so(n, m)     152
Symplectic Algebras sp(2n)     153
Matrix Realizations     155
Matrices     155
Unitary Algebras u(n)     155
Orthogonal Algebras so(n)     158
Symplectic Algebras sp(2n)     159
Basis for the Representation     160
Casimir Operators     161
Spectrum Generating Algebras and Dynamic Symmetries     163
Spectrum Generating Algebras (SGA)     163
Dynamic Symmetries (DS)     163
Bosonic Systems     164
Dynamic Symmetries of u(4)     165
Dynamic Symmetries of u(6)      167
Fermionic Systems     170
Dynamic Symmetry of u(4)     170
Dynamic Symmetry of u(6)     171
Degeneracy Algebras and Dynamical Algebras     173
Degeneracy Algebras     173
Degeneracy Algebras in v [GreaterEqual] 2 Dimensions     173
The Isotropic Harmonic Oscillator     174
The Coulomb Problem     177
Degeneracy Algebra in v = 1 Dimension     181
Dynamical Algebras     182
Dynamical Algebras in v [GreaterEqual] 2 Dimensions     182
Harmonic Oscillator     182
Coulomb Problem     182
Dynamical Algebra in v = 1 Dimension     183
Poschl-Teller Potential     183
Morse Potential     185
Lattice of Algebras     187
References     189
Index     193