Lie group

About: Lie group is a research topic. Over the lifetime, 18359 publications have been published within this topic receiving 381012 citations.

Applications of Lie Groups to Differential Equations

Abstract: 1 Introduction to Lie Groups- 11 Manifolds- Change of Coordinates- Maps Between Manifolds- The Maximal Rank Condition- Submanifolds- Regular Submanifolds- Implicit Submanifolds- Curves and Connectedness- 12 Lie Groups- Lie Subgroups- Local Lie Groups- Local Transformation Groups- Orbits- 13 Vector Fields- Flows- Action on Functions- Differentials- Lie Brackets- Tangent Spaces and Vectors Fields on Submanifolds- Frobenius' Theorem- 14 Lie Algebras- One-Parameter Subgroups- Subalgebras- The Exponential Map- Lie Algebras of Local Lie Groups- Structure Constants- Commutator Tables- Infinitesimal Group Actions- 15 Differential Forms- Pull-Back and Change of Coordinates- Interior Products- The Differential- The de Rham Complex- Lie Derivatives- Homotopy Operators- Integration and Stokes' Theorem- Notes- Exercises- 2 Symmetry Groups of Differential Equations- 21 Symmetries of Algebraic Equations- Invariant Subsets- Invariant Functions- Infinitesimal Invariance- Local Invariance- Invariants and Functional Dependence- Methods for Constructing Invariants- 22 Groups and Differential Equations- 23 Prolongation- Systems of Differential Equations- Prolongation of Group Actions- Invariance of Differential Equations- Prolongation of Vector Fields- Infinitesimal Invariance- The Prolongation Formula- Total Derivatives- The General Prolongation Formula- Properties of Prolonged Vector Fields- Characteristics of Symmetries- 24 Calculation of Symmetry Groups- 25 Integration of Ordinary Differential Equations- First Order Equations- Higher Order Equations- Differential Invariants- Multi-parameter Symmetry Groups- Solvable Groups- Systems of Ordinary Differential Equations- 26 Nondegeneracy Conditions for Differential Equations- Local Solvability- In variance Criteria- The Cauchy-Kovalevskaya Theorem- Characteristics- Normal Systems- Prolongation of Differential Equations- Notes- Exercises- 3 Group-Invariant Solutions- 31 Construction of Group-Invariant Solutions- 32 Examples of Group-Invariant Solutions- 33 Classification of Group-Invariant Solutions- The Adjoint Representation- Classification of Subgroups and Subalgebras- Classification of Group-Invariant Solutions- 34 Quotient Manifolds- Dimensional Analysis- 35 Group-Invariant Prolongations and Reduction- Extended Jet Bundles- Differential Equations- Group Actions- The Invariant Jet Space- Connection with the Quotient Manifold- The Reduced Equation- Local Coordinates- Notes- Exercises- 4 Symmetry Groups and Conservation Laws- 41 The Calculus of Variations- The Variational Derivative- Null Lagrangians and Divergences- Invariance of the Euler Operator- 42 Variational Symmetries- Infinitesimal Criterion of Invariance- Symmetries of the Euler-Lagrange Equations- Reduction of Order- 43 Conservation Laws- Trivial Conservation Laws- Characteristics of Conservation Laws- 44 Noether's Theorem- Divergence Symmetries- Notes- Exercises- 5 Generalized Symmetries- 51 Generalized Symmetries of Differential Equations- Differential Functions- Generalized Vector Fields- Evolutionary Vector Fields- Equivalence and Trivial Symmetries- Computation of Generalized Symmetries- Group Transformations- Symmetries and Prolongations- The Lie Bracket- Evolution Equations- 52 Recursion Operators, Master Symmetries and Formal Symmetries- Frechet Derivatives- Lie Derivatives of Differential Operators- Criteria for Recursion Operators- The Korteweg-de Vries Equation- Master Symmetries- Pseudo-differential Operators- Formal Symmetries- 53 Generalized Symmetries and Conservation Laws- Adjoints of Differential Operators- Characteristics of Conservation Laws- Variational Symmetries- Group Transformations- Noether's Theorem- Self-adjoint Linear Systems- Action of Symmetries on Conservation Laws- Abnormal Systems and Noether's Second Theorem- Formal Symmetries and Conservation Laws- 54 The Variational Complex- The D-Complex- Vertical Forms- Total Derivatives of Vertical Forms- Functionals and Functional Forms- The Variational Differential- Higher Euler Operators- The Total Homotopy Operator- Notes- Exercises- 6 Finite-Dimensional Hamiltonian Systems- 61 Poisson Brackets- Hamiltonian Vector Fields- The Structure Functions- The Lie-Poisson Structure- 62 Symplectic Structures and Foliations- The Correspondence Between One-Forms and Vector Fields- Rank of a Poisson Structure- Symplectic Manifolds- Maps Between Poisson Manifolds- Poisson Submanifolds- Darboux' Theorem- The Co-adjoint Representation- 63 Symmetries, First Integrals and Reduction of Order- First Integrals- Hamiltonian Symmetry Groups- Reduction of Order in Hamiltonian Systems- Reduction Using Multi-parameter Groups- Hamiltonian Transformation Groups- The Momentum Map- Notes- Exercises- 7 Hamiltonian Methods for Evolution Equations- 71 Poisson Brackets- The Jacobi Identity- Functional Multi-vectors- 72 Symmetries and Conservation Laws- Distinguished Functionals- Lie Brackets- Conservation Laws- 73 Bi-Hamiltonian Systems- Recursion Operators- Notes- Exercises- References- Symbol Index- Author Index

Differential Geometry, Lie Groups, and Symmetric Spaces

Abstract: Elementary differential geometry Lie groups and Lie algebras Structure of semisimple Lie algebras Symmetric spaces Decomposition of symmetric spaces Symmetric spaces of the noncompact type Symmetric spaces of the compact type Hermitian symmetric spaces Structure of semisimple Lie groups The classification of simple Lie algebras and of symmetric spaces Solutions to exercises Some details Bibliography List of notational conventions Symbols frequently used Index Reviews for the first edition.

Generalized Coherent States and Their Applications

Abstract: I Generalized Coherent States for the Simplest Lie Groups.- 1. Standard System of Coherent States Related to the Heisenberg-Weyl Group: One Degree of Freedom.- 1.1 The Heisenberg-Weyl Group and Its Representations.- 1.1.1 The Heisenberg-Weyl Group.- 1.1.2 Representations of the Heisenberg-Weyl Group.- 1.1.3 Concrete Realization of the Representation T?(g).- 1.2 Coherent States.- 1.3 The Fock-Bargmann Representation.- 1.4 Completeness of Coherent-State Subsystems.- 1.5 Coherent States and Theta Functions.- 1.6 Operators and Their Symbols.- 1.7 Characteristic Functions.- 2. Coherent States for Arbitrary Lie Groups.- 2.1 Definition of the Generalized Coherent State.- 2.2 General Properties of Coherent-State Systems.- 2.3 Completeness and Expansion in States of the CS System.- 2.4 Selection of Generalized CS Systems with States Closest to Classical.- 3. The Standard System of Coherent States Several Degrees of Freedom.- 3.1 General Properties.- 3.2 Coherent States and Theta Functions for Several Degrees of Freedom.- 4. Coherent States for the Rotation Group of Three-Dimensional Space.- 4.1 Structure of the Groups SO(3) and SU(2).- 4.2 Representations of SU(2).- 4.3 Coherent States.- 5. The Most Elementary Noneompact, Non-Abelian Simple Lie Group: SU(1,1).- 5.1 Group SU(1,1) and Its Representations.- 5.1.1 Fundamental Properties ofU(1,1) 67.- 5.1.2 Discrete Series.- 5.1.3 Principal (Continuous) Series.- 5.2 Coherent States.- 5.2.1 Discrete Series.- 5.2.2 Principal (Continuous) Series.- 6. The Lorentz Group: SO(3,1).- 6.1 Representations of the Lorentz Group.- 6.2 Coherent States.- 7. Coherent States for the SO(n, 1) Group: Class-1 Representations of the Principal Series.- 7.1 Class-I Representations of SO(n,1).- 7.2 Coherent States.- 8. Coherent States for a Bosonic System with Finite Number of Degrees of Freedom.- 8.1 Canonical Transformations.- 8.2 Coherent States.- 8.3 Operators in the Space ?B(+).- 9. Coherent States for a Fermionic System with Finite Number of Degrees of Freedom.- 9.1 Canonical Transformations.- 9.2 Coherent States.- 9.3 Operators in the Space ?F(+).- II General Case.- 10. Coherent States for Nilpotent Lie Groups.- 10.1 Structure of Nilpotent Lie Groups.- 10.2 Orbits of Coadjoint Representation.- 10.3 Orbits of Nilpotent Lie Groups.- 10.4 Representations of Nilpotent Lie Groups.- 10.5 Coherent States.- 11. Coherent States for Compact Semisimple Lie Groups.- 11.1 Elements of the Theory of Compact Semisimple Lie Groups..- 11.2 Representations of Compact Simple Lie Groups.- 11.3 Coherent States.- 12. Discrete Series of Representations: The General Case.- 12.1 Discrete Series.- 12.2 Bounded Domains.- 12.3 Coherent States.- 13. Coherent States for Real Semisimple Lie Groups: Class-I Representations of Principal Series.- 13.1 Class-I Representations.- 13.2 Coherent States.- 13.3 Horocycles in Symmetric Space.- 13.4 Rank-1 Symmetric Spaces.- 13.5 Properties of Rank-1 CS Systems.- 13.6 Complex Homogeneous Bounded Domains.- 13.6.1 Type-I Tube Domains.- 13.6.2 Type-II Tube Domains.- 13.6.3 Type-III Tube Domains.- 13.6.4 Type-IV Domains.- 13.6.5 The Exceptional Domain Dv.- 13.7 Properties of the Coherent States.- 14. Coherent States and Discrete Subgroups: The Case of SU(1,1).- 14.1 Preliminaries.- 14.2 Incompleteness Criterion for CS Subsystems Related to Discrete Subgroups.- 14.3 Growth of a Function Analytical in a Disk Related to the Distribution of Its Zeros.- 14.4 Completeness Criterion for CS Subsystems.- 14.5 Discrete Subgroups of SU(1,1) and Automorphic Forms.- 15. Coherent States for Discrete Series and Discrete Subgroups: General Case.- 15.1 Automorphic Forms.- 15.2 Completeness of Some CS Subsystems.- 16. Coherent States and Berezin's Quantization.- 16.1 Classical Mechanics.- 16.2 Quantization.- 16.3 Quantization on the Lobachevsky Plane.- 16.3.1 Description of Operators.- 16.3.2 The Correspondence Principle.- 16.3.3 Operator Th in Terms of a Laplacian.- 16.3.4 Representation of Group of Motions of the Lobachevsky Plane in Space ?h.- 16.3.5 Quantization by Inversions Analog to Weyl Quantization.- 16.4 Quantization on a Sphere.- 16.5 Quantization on Homogeneous Kahler Manifolds.- III Physical Applications.- 17. Preliminaries.- 18. Quantum Oscillators.- 18.1 Quantum Oscillator Acted on by a Variable External Force..- 18.2 Parametric Excitation of a Quantum Oscillator.- 18.3 Quantum Singular Oscillator.- 18.3.1 The Stationary Case.- 18.3.2 The Nonstationary Case.- 18.3.3 The Case of N Interacting Particles.- 18.4 Oscillator with Variable Frequency Acted on by an External Force.- 19. Particles in External Electromagnetic Fields.- 19.1 Spin Motion in a Variable Magnetic Field.- 19.2 Boson Pair Production in a Variable Homogeneous External Field.- 19.2.1 Dynamical Symmetry for Scalar Particles.- 19.2.2 The Multidimensional Case: Coherent States.- 19.2.3 The Multidimensional Case: Nonstationary Problem..- 19.3 Fermion Pair Production in a Variable Homogeneous External Field.- 19.3.1 Dynamical Symmetry for Spin-1/2 particles.- 19.3.2 Heisenberg Representation.- 19.3.3 The Multidimensional Case: Coherent States.- 20. Generating Function for Clebsch-Gordan Coefficients of the SU(2) group.- 21. Coherent States and the Quasiclassical Limit.- 22. 1/N Expansion for Gross-Neveu Models.- 22.1 Description of the Model.- 22.2 Dimensionality of Space ?N= ?O in the Fermion Case.- 22.3 Quasiclassical Limit.- 23. Relaxation to Thermodynamic Equilibrium.- 23.1 Relaxation of Quantum Oscillator to Thermodynamic Equilibrium.- 23.1.1 Kinetic Equation.- 23.1.2 Characteristic Functions and Quasiprobability Distributions.- 23.1.3 Use of Operator Symbols.- 23.2 Relaxation of a Spinning Particle to Thermodynamic Equilibrium in the Presence of a Magnetic Field.- 24. Landau Diamagnetism.- 25. The Heisenberg-Euler Lagrangian.- 26. Synchrotron Radiation.- 27. Classical and Quantal Entropy.- Appendix A. Proof of Completeness for Certain CS Subsystems.- Appendix B. Matrix Elements of the Operator D(y).- Appendix C. Jacobians of Group Transformations for Classical Domains.- Further Applications of the CS Method.- References.- Subject-Index.- Addendum. Further Applications of the CS Method.- References.- References to Addendum.- Subject-Index.

Differentiable dynamical systems

Abstract: This is a survey article on the area of global analysis defined by differentiable dynamical systems or equivalently the action (differentiable) of a Lie group G on a manifold M. An action is a homomorphism G→Diff(M) such that the induced map G×M→M is differentiable. Here Diff(M) is the group of all diffeomorphisms of M and a diffeo- morphism is a differentiable map with a differentiable inverse. Everything will be discussed here from the C ∞ or C r point of view. All manifolds maps, etc. will be differentiable (C r , 1 ≦ r ≦ ∞) unless stated otherwise.

Representation Theory: A First Course

Abstract: This volume represents a series of lectures which aims to introduce the beginner to the finite dimensional representations of Lie groups and Lie algebras. Following an introduction to representation theory of finite groups, the text explains how to work out the representations of classical groups.