Showing papers on "Lie group published in 1990"

Integrable systems of classical mechanics and Lie algebras

Abstract: 1. Preliminaries.- 1.1 A Simple Example: Motion in a Potential Field.- 1.2 Poisson Structure and Hamiltonian Systems.- 1.3 Symplectic Manifolds.- 1.4 Homogeneous Symplectic Spaces.- 1.5 The Moment Map.- 1.6 Hamiltonian Systems with Symmetry.- 1.7 Reduction of Hamiltonian Systems with Symmetry.- 1.8 Integrable Hamiltonian Systems.- 1.9 The Projection Method.- 1.10 The Isospectral Deformation Method.- 1.11 Hamiltonian Systems on Coadjoint Orbits of Lie Groups.- 1.12 Constructions of Hamiltonian Systems with Large Families of Integrals of Motion.- 1.13 Completeness of Involutive Systems.- 1.14 Hamiltonian Systems and Algebraic Curves.- 2. Simplest Systems.- 2.1 Systems with One Degree of Freedom.- 2.2 Systems with Two Degrees of Freedom.- 2.3 Separation of Variables.- 2.4 Systems with Quadratic Integrals of Motion.- 2.5 Motion in a Central Field.- 2.6 Systems with Closed Trajectories.- 2.7 The Harmonic Oscillator.- 2.8 The Kepler Problem.- 2.9 Motion in Coupled Newtonian and Homogeneous Fields.- 2.10 Motion in the Field of Two Newtonian Centers.- 3. Many-Body Systems.- 3.1 Lax Representation for Many-Body Systems.- 3.2 Completely Integrable Many-Body Systems.- 3.3 Explicit Integration of the Equations of Motion for Systems of Type I and V via the Projection Method.- 3.4 Relationship Between the Solutions of the Equations of Motion for Systems of Type I and V.- 3.5 Explicit Integration of the Equations of Motion for Systems of Type II and III.- 3.6 Integration of the Equations of Motion for Systems with Two Types of Particles.- 3.7 Many-Body Systems as Reduced Systems.- 3.8 Generalizations of Many-Body Systems of Type I-III to the Case of the Root Systems of Simple Lie Algebras.- 3.9 Complete Integrability of the Systems of Section 3.8.- 3.10 Anisotropic Harmonic Oscillator in the Field of a Quartic Central Potential (the Garnier System).- 3.11 A Family of Integrable Quartic Potentials Related to Symmetric Spaces.- 4. The Toda Lattice.- 4.1 The Ordinary Toda Lattice. Lax Representation. Complete Integrability.- 4.2 The Toda Lattice as a Dynamical System on a Coadjoint Orbit of the Group of Triangular Matrices.- 4.3 Explicit Integration of the Equations of Motion for the Ordinary Nonperiodic Toda Lattice.- 4.4 The Toda Lattice as a Reduced System.- 4.5 Generalized Nonperiodic Toda Lattices Related to Simple Lie Algebras.- 4.6 Toda-like Systems on Coadjoint Orbits of Borel Subgroups.- 4.7 Canonical Coordinates for Systems of Toda Type.- 4.8 Integrability of Toda-like Systems on Generic Orbits.- 5. Miscellanea.- 5.1 Equilibrium Configurations and Small Oscillations of Some Integrable Hamiltonian Systems.- 5.2 Motion of the Poles of Solutions of Nonlinear Evolution Equations and Related Many-Body Problems.- 5.3 Motion of the Zeros of Solutions of Linear Evolution Equations and Related Many-Body Problems.- 5.4 Concluding Remarks.- Appendix A.- Examples of Symplectic Non-Kahlerian Manifolds.- Appendix B.- Solution of the Functional Equation (3.1.9).- Appendix C.- Semisimple Lie Algebras and Root Systems.- Appendix D.- Symmetric Spaces.- References.

Representations of nilpotent Lie groups and their applications. Part I: Basic theory and examples

Abstract: Preface 1. Elementary theory of nilpotent Lie groups and Lie algebras 2. Kirillov theory 3. Parametrization of coadjoint orbits 4. Plancherel formula and related topics 5. Discrete cocompact subgroups Appendix Bibliography Symbol index Subject index.

Central extensions of quantum current groups

Abstract: We describe Hopf algebras which are central extensions of quantum current groups. For a special value of the central charge, we describe Casimir elements in these algebras. New types of generators for quantum current algebra and its central extension for quantum simple Lie groups, are obtained. The application of our construction to the elliptic case is also discussed.

Group-Theoretical Methods in Image Understanding

Abstract: This book presents the mathematics relevant to image understanding by computer vision and gives examples of actual applications. Group representation theory, Lie groups and Lie algebras, the theory of invariance, tensor calculus, differential geometry and projective geometry are used for three-dimensional shape and motion analysis from images, making use of techniques such as shape from motion, shape from texture, shape from angle and shape from surface. Although the mathematics itself may be well known to mathematicians, people working in areas related to computer science, image understanding, computer vision and image processing have usually never studied such mathematics, and so may be surprised to learn that abstract mathematical concepts can be of enormous help in building intelligent computer vision systems.

Quantum Mechanics: Symmetries

Abstract: 1. Symmetries in Quantum Mechanics.- 1.1 Symmetries in Classical Physics.- 1.2 Spatial Translations in Quantum Mechanics.- 1.3 The Unitary Translation Operator.- 1.4 The Equation of Motion for States Shifted in Space.- 1.5 Symmetry and Degeneracy of States.- 1.6 Time Displacements in Quantum Mechanics.- 1.7 Mathematical Supplement: Definition of a Group.- 1.8 Mathematical Supplement: Rotations and their Group Theoretical Properties.- 1.9 An Isomorphism of the Rotation Group.- 1.9.1 Infinitesimal and Finite Rotations.- 1.9.2 Isotropy of Space.- 1.10 The Rotation Operator for Many-Particle States.- 1.11 Biographical Notes.- 2. Angular Momentum Algebra Representation of Angular Momentum Operators - Generators of SO(3).- 2.1 Irreducible Representations of the Rotation Group.- 2.2 Matrix Representations of Angular Momentum Operators.- 2.3 Addition of Two Angular Momenta.- 2.4 Evaluation of Clebsch-Gordan Coefficients.- 2.5 Recursion Relations for Clebsch-Gordan Coefficients.- 2.6 Explicit Calculation of Clebsch-Gordan Coefficients.- 2.7 Biographical Notes.- 3. Mathematical Supplement: Fundamental Properties of Lie Groups.- 3.1 General Structure of Lie Groups.- 3.2 Interpretation of Commutators as Generalized Vector Products, Lie's Theorem, Rank of Lie Group.- 3.3 Invariant Subgroups, Simple and Semisimple Lie Groups, Ideals.- 3.4 Compact Lie Groups and Lie Algebras.- 3.5 Invariant Operators (Casimir Operators).- 3.6 Theorem of Racah.- 3.7 Comments on Multiplets.- 3.8 Invariance Under a Symmetry Group.- 3.9 Construction of the Invariant Operators.- 3.10 Remark on Casimir Operators of Abelian Lie Groups.- 3.11 Completeness Relation for Casimir Operators.- 3.12 Review of Some Groups and Their Properties.- 3.13 The Connection Between Coordianate Transformations and Transformations of Functions.- 3.14 Biographical Notes.- 4. Symmetry Groups and Their Physical Meaning -General Considerations.- 4.1 Biographical Notes.- 5. The Isospin Group (Isobaric Spin).- 5.1 Isospin Operators for a Multi-Nucleon System.- 5.2 General Properties of Representations of a Lie Algebra.- 5.3 Regular (or Adjoint) Representation of a Lie Algebra.- 5.4 Transformation Law for Isospin Vectors.- 5.5 Experimental Test of Isospin Invariance.- 5.6 Biographical Notes.- 6. The Hypercharge.- 6.1 Biographical Notes.- 7. The SU(3) Symmetry.- 7.1 The Groups U(n) and SU(n).- 7.1.1. The Generators of U(n) and SU(n).- 7.2 The Generators of SU(3).- 7.3 The Lie Algebra of SU(3).- 7.4 The Subalgebras of the SU(3)-Lie Algebra and the Shift Operators.- 7.5 Coupling of T-, U- and V-Multiplets.- 7.6 Quantitative Analysis of Our Reasoning.- 7.7 Further Remarks About the Geometric Form of an SU(3) Multiplet.- 7.8 The Number of States on Mesh Points on Inner Shells.- 8. Quarks and SU(3).- 8.1 Searching for Quarks.- 8.2 The Transformation Properties of Quark States.- 8.3 Construction of all SU(3) Multiplets from the Elementary Representations [3] and 3.- 8.4 Construction of the Representation D(p, q) from Quarks and Antiquarks.- 8.4.1. The Smallest SU(3) Representations.- 8.5 Meson Multiplets.- 8.6 Rules for the Reduction of Direct Product of SU(3) Multiplets.- 8.7 U-spin Invariance.- 8.8 Test of U-spin Invariance.- 8.9 The Gell-Mann-Okubo Mass Formula.- 8.10 The Clebsch-Gordan Coefficients of the SU(3).- 8.11 Quark Models with Inner Degrees of Freedom.- 8.12 The Mass Formula in SU(6).- 8.13 Magnetic Moments in the Quark Model.- 8.14 Excited Meson and Baryon States.- 8.14.1 Combinations of More Than Three Quarks.- 8.15 Excited States with Orbital Angular Momentum.- 9. Representations of the Permutation Group and Young Tableaux.- 9.1 The Permutation Group and Identical Particles.- 9.2 The Standard Form of Young Diagrams.- 9.3 Standard Form and Dimension of Irreducible Representations of the Permutation Group SN.- 9.4 The Connection Between SU(2) and S2.- 9.5 The Irreducible Representations of SU(n).- 9.6 Determination of the Dimension.- 9.7 The SU(n - 1) Subgroups of SU(n).- 9.8 Decomposition of the Tensor Product of Two Multiplets.- 10. Mathematical Excursion. Group Characters.- 10.1 Definition of Group Characters.- 10.2 Schur's Lemmas.- 10.2.1 Schur's First Lemma.- 10.2.2 Schur's Second Lemma.- 10.3 Orthogonality Relations of Representations and Discrete Groups.- 10.4 Equivalence Classes.- 10.5 Orthogonality Relations of the Group Characters for Discrete Groups and Other Relations.- 10.6 Orthogonality Relations of the Group Characters for the Example of the Group D3.- 10.7 Reduction of a Representation.- 10.8 Criterion for Irreducibility.- 10.9 Direct Product of Representations.- 10.10 Extension to Continuous, Compact Groups.- 10.11 Mathematical Excursion: Group Integration.- 10.12 Unitary Groups.- 10.13 The Transition from U(N) to SU(N) for the Example SU(3).- 10.14 Integration over Unitary Groups.- 10.15 Group Characters of Unitary Groups.- 11. Charm and SU(4).- 11.1 Particles with Charm and the SU(4).- 11.2 The Group Properties of SU(4).- 11.3 Tables of the Structure Constants fijk and the Coefficients dijk for SU(4).- 11.4 Multiplet Structure of SU(4).- 11.5 Advanced Considerations.- 11.5.1 Decay of Mesons with Hidden Charm.- 11.5.2 Decay of Mesons with Open Charm.- 11.5.3 Baryon Multiplets.- 11.6 The Potential Model of Charmonium.- 11.7 The SU(4) [SU(8)] Mass Formula.- 11.8 The ? Resonances.- 12. Mathematical Supplement.- 12.1 Introduction.- 12.2 Root Vectors and Classical Lie Algebras.- 12.3 Scalar Products of Eigenvalues.- 12.4 Cartan-Weyl Normalization.- 12.5 Graphic Representation of the Root Vectors.- 12.6 Lie Algebra of Rank 1.- 12.7 Lie Algebras of Rank 2.- 12.8 Lie Algebras of Rank l > 2.- 12.9 The Exceptional Lie Algebras.- 12.10 Simple Roots and Dynkin Diagrams.- 12.11 Dynkin's Prescription.- 12.12 The Cartan Matrix.- 12.13 Determination of all Roots from the Simple Roots.- 12.14 Two Simple Lie Algebras.- 12.15 Representations of the Classical Lie Algebras.- 13. Special Discrete Symmetries.- 13.1 Space Reflection (Parity Transformation).- 13.2 Reflected States and Operators.- 13.3 Time Reversal.- 13.4 Antiunitary Operators.- 13.5 Many-Particle Systems.- 13.6 Real Eigenfunctions.- 14. Dynamical Symmetries.- 14.1 The Hydrogen Atom.- 14.2 The Group SO(4).- 14.3 The Energy Levels of the Hydrogen Atom.- 14.4 The Classical Isotropic Oscillator.- 14.4.1 The Quantum Mechanical Isotropic Oscillator.- 15. Mathematical Excursion: Non-compact Lie Groups.- 15.1 Definition and Examples of Non-compact Lie Groups.- 15.2 The Lie Group SO(2,l).- 15.3 Application to Scattering Problems.

Matched pairs of lie groups associated to solutions of the yang-baxter equations

Abstract: Two groups G, H are said to be a matched pair if they act on each other and these actions, (a, /?), obey a certain compatibility condition In such a situation one may form a bicrossproduct group, denoted Gβ cχiQ H Also in this situation one may form a bicrossproduct Hopf, Hopf-von Neumann or Kac algebra obtained by simultaneous cross product and cross coproduct We show that every compact semi-simple simply-connected Lie group G is a member of a matched pair, denoted (G, G*)9 in a natural way As an example we construct the matched pair in detail in the case (SU(2), SU(2)*) where

Infinite Dimensional Algebras and a Trigonometric Basis for the Classical Lie Algebras

Abstract: This paper explores features of the infinite‐dimensional algebras that have been previously introduced. In particular, it is shown that the classical simple Lie algebras (AN, BN, CN, DN) may be expressed in an ‘‘egalitarian’’ basis with trigonometric structure constants. The transformation to the standard Cartan–Weyl basis, and the particularly transparent N→∞ limit that this formulation allows is provided.

Three-dimensional quantum groups from contractions of SU(2) q

Abstract: Contractions of Lie algebras and of their representations are generalized to define new quantum groups. An explicit and complete exposition is made for the one‐dimensional Heisenberg H(1)q and the two‐dimensional Euclidean quantum group E(2)q obtained by contracting SU(2)q.

Consistent multiparameter quantisation of GL(n)

Abstract: The author describes a manifold of quantum group structures on the vector space of the universal enveloping algebra of gl(n) and on its dual, the space of polynomials in n2 variables. The dimension of the manifold is (n2-n+2)/2.

Solvable Lie algebras of dimension six

Abstract: Six‐dimensional solvable Lie algebras over the field of real numbers that possess nilradicals of dimension four are classified into equivalence classes. This completes Mubarakzyanov’s classification of the real six‐dimensional solvable Lie algebras.

From geometric quantization to conformal field theory

Abstract: Investigation of 2d conformal field theory in terms of geometric quantization is given. We quantize the so-called model space of the compact Lie group, Virasoro group and Kac-Moody group. In particular, we give a geometrical interpretation of the Virasoro discrete series and explain that this type of geometric quantization reproduces the chiral part of CFT (minimal models, 2d-gravity, WZNW theory). In the appendix we discuss the relation between classical (constant)r-matrices and this geometrical approach.

Unitary representations of the quantum group SU q (1,1): Structure of the dual space of U q ( sl (2))

Abstract: Real forms of the quantum universal enveloping algebraU q (sl(2)) and a topological quantum group associated with this algebra are discussed.

Spectral Theory of Automorphic Functions: and Its Applications

Abstract: 1. Introduction.- 2. What Does One Need Automorphic Functions for? Some Remarks or a Pragmatic Reader.- 3. Harmonic Analysis of Periodic Functions. The Hardy-Vorono? Formula.- 4. Expansion in Eigenfunctions of the Automorphic Laplacian on the Lobachevsky Plane.- 5. Harmonic Analysis of Automorphic Functions. Estimates for Fourier Coefficients of Parabolic Forms of Weight Zero.- 6. The Selberg Trace Formula for Fuchsian Groups of the First Kind.- 7. The Theory of the Selberg Zeta-Function.- 8. Problems in the Theory of the Discrete Spectrum of Automorphic Laplacians.- 9. The Spectral Moduli Problem.- 10. Automorphic Functions and the Kummer Problem.- 11. The Selberg Trace Formula on the Reductive Lie Groups.- 12. Automorphic Functions, Representations and L-functions.- 13. Remarks and Comments. Annotations to the Cited Literature.- References.- Appendix 1. Monodromy Groups and Automorphic Functions.- Appendix 2. Automorphic Functions for Effective Solutions of Certain Issues of the Riemann-Hilbert Problem.- Author Index.

Hamiltonian Dynamics of a Rigid Body in a Central Gravitational Field

Abstract: This paper concerns the dynamics of a rigid body of finite extent moving under the influence of a central gravitational field. A principal motivation behind this paper is to reveal the hamiltonian structure of the n-body problem for masses of finite extent and to understand the approximation inherent to modeling the system as the motion of point masses. To this end, explicit account is taken of effects arising because of the finite extent of the moving body. In the spirit of Arnold and Smale, exact models of spin-orbit coupling are formulated, with particular attention given to the underlying Lie group framework. Hamiltonian structures associated with such models are carefully constructed and shown to benon-canonical. Special motions, namely relative equilibria, are investigated in detail and the notion of anon-great circle relative equilibrium is introduced. Non-great circle motions cannot arise in the point mass model. In our analysis, a variational characterization of relative equilibria is found to be very useful.

Generalized Lax pairs, the modified classical Yang-Baxter equation, and affine geometry of Lie groups

Abstract: We generalize the usual Lax equationd/dt L=[M, L] byd/dt L=−ϱ(M)L, where ϱ is an arbitrary representation of a Lie algebra g (the values ofM) in a representation spaceV (the values ofL). The usual classicalr-matrix programme for Hamiltonian integrable systems is generalized tor-matrices taking values in g⊗V. Ther-matrices are then considered as left invariant torsion-free covariant derivatives on a Lie groupK (with Lie algebraV*). The Classical Yang-Baxter Equation (CYBE) is equivalent to the flatness ofK whereas the Modified CYBE implies thatK is an affine locally symmetric space. An example is discussed.

Lie Group Convolution Algebras as Deformation Quantizations of Linear Poisson Structures

Abstract: Introduction. Let L be a finite dimensional Lie algebra over the real numbers, R, and let L* be its dual vector space. It is well-known [24] that the Lie algebra structure on L defines a natural Poisson structure on L*-in fact this was already known to Lie [24]-and these Poisson structures are exactly what are now called the linear Poisson structures. Given a manifold M equipped with a Poisson structure, { , }, one can seek deformation quantizations "in the direction of { , }", as first studied in [3]. These are, loosely speaking, one-parameter families, {*h}lER, of deformations of the pointwise multiplication on C<(M) (or an appropriate subalgebra), such that

On Gelfand Pairs Associated with Solvable Lie Groups

Abstract: Let G be a locally compact group, and let K be a compact subgroup of Aut(G) , the group of automorphisms of G. There is a natural action of K on the convolution algebra L (G), and we denote by LK(G) the subalgebra of those elements in L (G) that are invariant under this action. The pair (K, G) is called a Gelfand pair if LI(G) is commutative. In this paper we consider the case where G is a connected, simply connected solvable Lie group and K C Aut(G) is a compact, connected group. We characterize such Gelfand pairs (K, G), and determine a moduli space for the associated K-spherical functions.

Elliptic genera, involutions, and homogeneous spin manifolds

Abstract: We study the normalized elliptic genera Φ(X)=ϕ(X)/e k/2 for 4k-dimensional homogeneous spin manifolds X and show that they are constant as modular functions. The basic tool is a reduction formula relating Φ(X) to that of the self-intersection of the fixed point set of an involution γ on X. When Φ(X) is a constant it equals the signature of X. We derive a general formula for sign(G/H), G⊃H compact Lie groups, and determine its value in some cases by making use of the theory of involutions in compact Lie groups.

Similarity solutions of the two-dimensional unsteady boundary-layer equations

Abstract: The method of Lie group transformations is used to derive all group-invariant similarity solutions of the unsteady two-dimensional laminar boundary-layer equations. A new method of nonlinear superposition is then used to generate further similarity solutions from a group-invariant solution. Our results are shown to include all the existing solutions as special cases. A detailed analysis is given to several classes of solutions which are also solutions to the full Navier–Stokes equations and which exhibit flow separation.

K symmetries and tau symmetries of evolution equations and their Lie algebras

Abstract: On the basis of Tu's work (1951), the author establishes a more general skeleton on K symmetries and tau symmetries of evolution equations and their Lie algebraic structures and discuss carefully the corresponding symmetries and Lie algebras for KdV and Jaulent-Miodek hierarchies.

Tableaux in the Representation Theory of the Classical Lie Groups

Abstract: A remote control circuit and apparatus for exploding explosives. Electric power generated at a resonance circuit when it is tuned to a control electromagnetic field is used to charge an ignition condenser. Provision is made for a control circuit including a controlled rectifier called a triac. The control circuit is adapted to generate a starting pulse at an instant when the control electromagnetic field is extinguished. The starting pulse serves to make the controlled rectifier conductive and hence discharge the ignition condenser through a detonator, thereby initiating it and exploding blasting explosives used to break rock and other solid material.

Twistor Theory for Riemannian Symmetric Spaces: With Applications to Harmonic Maps of Riemann Surfaces

Abstract: Homogeneous geometry- Harmonic maps and twistor spaces- Symmetric spaces- Flag manifolds- The twistor space of a Riemannian symmetric space- Twistor lifts over Riemannian symmetric spaces- Stable Harmonic 2-spheres- Factorisation of harmonic spheres in Lie groups

The free energy of Maxwell--Vlasov equilibria

Abstract: A previously derived expression [Phys Rev A 40, 3898 (1989)] for the energy of arbitrary perturbations about arbitrary Vlasov–Maxwell equilibria is transformed into a very compact form The new form is also obtained by a canonical transformation method for solving Vlasov’s equation, which is based on Lie group theory This method is simpler than the one used before and provides better physical insight Finally, a procedure is presented for determining the existence of negative‐energy modes In this context the question of why there is an accessibility constraint for the particles, but not for the fields, is discussed

Quantization of Lie Groups and Lie Algebras

Abstract: Publisher Summary This chapter focuses on the quantization of lie groups and lie algebras. The Algebraic Bethe Ansatz—the quantum inverse scattering method—emerges as a natural development of the various directions in mathematical physics: the inverse scattering method for solving nonlinear equations of evolution, the quantum theory of magnets, the method of commuting transfer-matrices in classical statistical mechanics, and factorizable scattering theory. The chapter discusses quantum formal groups, a finite-dimensional example, an infinite-dimensional example, and reviews the deformation theory and quantum groups.