Introduction / 1: |
Lie groups / 2: |
Matrix groups |
Lie algebras / 4: |
Matrix algebras / 5: |
Operator algebras / 6: |
Exponentiation / 7: |
Structure theory for Lie algebras / 8: |
Structure theory for simple Lie algebras / 9: |
Root spaces and Dykin diagrams / 10: |
Real forms / 11: |
Riemannian symmetric spaces / 12: |
Contraction / 13: |
Hydrogenic atoms / 14: |
Maxwell's equations / 15: |
Lie groups and differential equations / 16: |
References |
Index |
Preface |
The program of Lie / 1.1: |
A result of Galois / 1.2: |
Group theory background / 1.3: |
Approach to solving polynomial equations / 1.4: |
Solution of the quadratic equation / 1.5: |
Solution of the cubic equation / 1.6: |
Solution of the quartic equation / 1.7: |
The quintic cannot be solved / 1.8: |
Example / 1.9: |
Conclusion / 1.10: |
Problems / 1.11: |
Algebraic properties / 2.1: |
Topological properties / 2.2: |
Unification of algebra and topology / 2.3: |
Unexpected simplification / 2.4: |
Preliminaries / 2.5: |
No constraints / 3.2: |
Linear constraints / 3.3: |
Bilinear and quadratic constraints / 3.4: |
Multilinear constraints / 3.5: |
Intersections of groups / 3.6: |
Embedded groups / 3.7: |
Modular groups / 3.8: |
Why bother? / 3.9: |
How to linearize a Lie group / 4.2: |
Inversion of the linearization map: EXP / 4.3: |
Properties of a Lie algebra / 4.4: |
Structure constants / 4.5: |
Regular representation / 4.6: |
Structure of a Lie algebra / 4.7: |
Inner product / 4.8: |
Invariant metric and measure on a Lie group / 4.9: |
Algebras of embedded groups / 4.10: |
Basis vectors / 5.8: |
Boson operator algebras / 5.10: |
Fermion operator algebras / 6.2: |
First order differential operator algebras / 6.3: |
EXPonentiation / 6.4: |
The covering problem / 7.1: |
The isomorphism problem and the covering group / 7.3: |
The parameterization problem and BCH formulas / 7.4: |
EXPonentials and physics / 7.5: |
Some standard forms for the regular representation / 7.6: |
What these forms mean / 8.3: |
How to make this decomposition / 8.4: |
An example / 8.5: |
Objectives of this program / 8.6: |
Eigenoperator decomposition - secular equation / 9.2: |
Rank / 9.3: |
Invariant operators / 9.4: |
Regular elements / 9.5: |
Semisimple Lie algebras / 9.6: |
Canonical commutation relations / 9.7: |
Root spaces and Dynkin diagrams / 9.8: |
Properties of roots / 10.1: |
Root space diagrams / 10.2: |
Dynkin diagrams / 10.3: |
Compact and least compact real forms / 10.4: |
Cartan's procedure for constructing real forms / 11.3: |
Real forms of simple matrix Lie algebras / 11.4: |
Results / 11.5: |
Brief review / 11.6: |
Globally symmetric spaces / 12.2: |
Metric and measure / 12.3: |
Applications and examples / 12.6: |
Pseudo-Riemannian symmetric spaces / 12.7: |
Inonu-Wigner contractions / 12.8: |
Simple examples of Inonu-Wigner contractions / 13.3: |
The contraction U(2) to H[subscript 4] / 13.4: |
Two important principles of physics / 13.5: |
The wave equations / 14.3: |
Quantization conditions / 14.4: |
Geometric symmetry SO(3) / 14.5: |
Dynamical symmetry SO(4) / 14.6: |
Relation with dynamics in four dimensions / 14.7: |
DeSitter symmetry SO(4, 1) / 14.8: |
Conformal symmetry SO(4, 2) / 14.9: |
Spin angular momentum / 14.10: |
Spectrum generating group / 14.11: |
Maxwell's equations / 14.12: |
Review of the inhomogeneous Lorentz group / 15.1: |
Subgroups and their representations / 15.3: |
Representations of the Poincare group / 15.4: |
Transformation properties / 15.5: |
The simplest case / 15.6: |
First order equations / 16.2: |
Additional insights / 16.3: |
Bibliography / 16.5: |