Showing papers on "Lie group published in 1986"

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

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.

Noncommutative Harmonic Analysis

Abstract: Some basic concepts of Lie group representation theory The Heisenberg group The unitary group Compact Lie groups Harmonic analysis on spheres Induced representations, systems of imprimitivity, and semidirect products Nilpotent Lie groups Harmonic analysis on cones $\mathrm {SL}(2,R)$ $\mathrm {SL}(2, \mathbf C)$, and more general Lorentz groups Groups of conformal transformations The symplectic group and the metaplectic group Spinors Semisimple Lie groups.

Theory and Applications of the Poincaré Group

Abstract: I: Elements of Group Theory.- 1. Definition of a Group.- 2. Subgroups, Cosets, and Invariant Subgroups.- 3. Equivalence Classes, Orbits, and Little Groups.- 4. Representations and Representation Spaces.- 5. Properties of Matrices.- 6. Schur's Lemma.- 7. Exercises and Problems.- II: Lie Groups and Lie Algebras.- 1. Basic Concepts of Lie Groups.- 2. Basic Theorems Concerning Lie Groups.- 3. Properties of Lie Algebras.- 4. Properties of Lie Groups.- 5. Further Theorems of Lie Groups.- 6. Exercises and Problems.- III: Theory of the Poincare Group.- 1. Group of Lorentz Transformations.- 2. Orbits and Little Groups of the Proper Lorentz Group.- 3. Representations of the Poincare Group.- 4. Lorentz Transformations of Wave Functions.- 5. Lorentz Transformations of Free Fields.- 6. Discrete Symmetry Operations.- 7. Exercises and Problems.- IV: Theory of Spinors.- 1. SL(2, c) as the Covering Group of the Lorentz Group.- 2. Subgroups of SL(2, c).- 3. SU (2).- 4. 5L(2, c) Spinors and Four-Vectors.- 5. Symmetries of the Dirac Equation.- 6. Exercises and Problems.- V: Covariant Harmonic Oscillator Formalism.- 1. Covariant Harmonic Oscillator Differential Equations.- 2. Normalizable Solutions of the Relativistic Oscillator Equation.- 3. Irreducible Unitary Representations of the Poincare Group.- 4. Transformation Properties of Harmonic Oscillator Wave Functions.- 5. Harmonic Oscillators in the Four-Dimensional Euclidean Space.- 6. Moving O(4) Coordinate System.- 7. Exercises and Problems.- VI: Dirac's Form of Relativistic Quantum Mechanics.- 1. C-Number Time-Energy Uncertainty Relation.- 2. Dirac's Form of Relativistic Theory of "Atom ".- 3. Dirac's Light-Cone Coordinate System.- 4. Harmonic Oscillators in the Light-Cone Coordinate System.- 5. Lorentz-Invariant Uncertainty Relations.- 6. Exercises and Problems.- VII: Massless Particles.- 1. What is the E(2) Group?.- 2. E(2)-like Little Group for Photons.- 3. Transformation Properties of Photon Polarization Vectors.- 4. Unitary Transformation of Photon Polarization Vectors.- 5. Massless Particles with Spin 1/2.- 6. Harmonic Oscillator Wave Functions for Massless Composite Particles.- 7. Exercises and Problems.- VIII: Group Contractions.- 1. SE(2) Group as a Contraction of SO(3).- 2. E(2)-like Little Group as an Infinite-momentum/zero-mass Limit of the O(3)-like Little Group for Massive Particles.- 3. Large-momentum/zero-mass Limit of the Dirac Equation.- 4. Finite-dimensional Non-unitary Representations of the SE(2) Group.- 5. Polarization Vectors for Massless Particles with Integer Spin.- 6. Lorentz and Galilei Transformations.- 7. Group Contractions and Unitary Representations of SE(2).- 8. Exercises and Problems.- IX: SO(2, 1) and SU(1, 1).- 1. Geometry of SL(2, r) and Sp(2).- 2. Finite-dimensional Representations of SO(2, 1).- 3. Complex Angular Momentum.- 4. Unitary Representations of SU(1, 1).- 5. Exercises and Problems.- X: Homogeneous Lorentz Group.- 1. Statement of the Problem.- 2. Finite-dimensional Representations of the Homogeneous Lorentz Group.- 3. Transformation Properties of Electric and Magnetic Fields.- 4. Pseudo-unitary Representations for Dirac Spinors.- 5. Harmonic Oscillator Wave Functions in the Lorentz Coordinate System.- 6. Further Properties of the Homogeneous Lorentz Group.- 7. Concluding Remarks.- XI: Hadronic Mass Spectra.- 1. Quark Model.- 2. Three-particle Symmetry Classifications According to the Method of Dirac.- 3. Construction of Symmetrized Wave Functions.- 4. Symmetrized Products of Symmetrized Wave Functions.- 5. Spin Wave Functions for the Three-Quark System.- 6. Three-quark Unitary Spin and SU(6) Wave Functions.- 7. Three-body Spatial Wave Functions.- 8. Totally Symmetric Baryonic Wave Functions.- 9. Baryonic Mass Spectra.- 10. Mesons.- 11. Exercises and Problems.- XII: Lorentz-Dirac Deformation in High-Energy Physics.- 1. Lorentz-Dirac Deformation of Hadronic Wave Functions.- 2. Form Factors of Nucleons.- 3. Calculation of the Form Factors.- 4. Scaling Phenomenon and the Parton Picture.- 5. Covariant Harmonic Oscillators and the Parton Picture.- 6. Calculation of the Parton Distribution Function for the Proton.- 7. Jet Phenomenon.- 8. Exercises and Problems.- References.

Deformation Spaces Associated to Compact Hyperbolic Manifolds

Abstract: In this paper we take a first step toward understanding representations of cocompact lattices in SO(n,1) into arbitrary Lie groups by studying the deformations of rational representations — see Proposition 5.1 for a rather general existence result. This proposition has a number of algebraic applications. For example, we remark that such deformations show that the Margulis Super-Rigidity Theorem, see [30], cannot be extended to the rank 1 case. We remark also that if Γ ⊂ SO(n,1) is one of the standard arithmetic examples described in Section 7 then Γ has a faithful representation ρ′ in SO(n+1), the Galois conjugate of the uniformization representation, and Proposition 5.1 may be used to deform the direct sum of ρ′ and the trivial representation in SO(n+2) thereby constructing non-trivial families of irreducible orthogonal representations of Γ. However, most of this paper is devoted to studying certain spaces of representations which are of interest in differential geometry in a sense which we now explain.

Group Structure of Gauge Theories

Abstract: This monograph provides an account of the structure of gauge theories from a group theoretical point of view. The first part of the text is devoted to a review of those aspects of compact Lie groups (the Lie algebras, the representation theory, and the global structure) which are necessary for the application of group theory to the physics of particles and fields. The second part describes the way in which compact Lie groups are used to construct gauge theories. Models that describe the known fundamental interactions and the proposed unification of these interactions (grand unified theories) are considered in some detail. The book concludes with an up to date description of the group structure of spontaneous symmetry breakdown, which plays a vital role in these interactions. This book will be of interest to graduate students and to researchers in theoretical physics and applied mathematics, especially those interested in the applications of differential geometry and group theory in physics.

Integrable graded magnets

Abstract: Solutions of the graded Yang-Baxter equations are constructed which are invariant relative to the general linear and orthosymplectic supergroups. The Hamiltonians and other higher integrals (the transfer matrix) of spin systems on a finite lattice connected with the solutions found are diagonalized. A generalization of the Yang-Baxter equation to the case of the δ-commutative, G-graded Zamolodchikov algebra is presented.

Calculus on Manifolds

Abstract: The equations of mathematical physics are typically ordinary or partial differential equations for vector or tensor fields over Riemannian manifolds whose group of isometries is a Lie group. It is taken as axiomatic that the equations be independent of the observer, in a sense we shall make precise below; and the consequence of this axiom is that the equations are invariant with respect to the group action. The action of a Lie group on tensor fields over a manifold is thus of primary importance. The action of a Lie group on a manifold M induces in a natural way automorphisms of the algebra of C∞ functions over M and on the algebra of tensor fields over M. The one parameter subgroups of the group induce one parameter subgroups of automorphisms of the tensor fields. The infinitesimal generators of these groups of automorphisms are the Lie derivatives of the action.

Classification of ten-dimensional kinematical groups with space isotropy

Abstract: All the abstract ten‐dimensional real Lie algebras that contain as a subalgebra the algebra of the three‐dimensional rotation group (generators J) and decompose under the rotation group into three three‐vector representation spaces (J itself, K, and P) and a scalar (generator H) are classified. In all cases, the existence of a homogeneous space of dimension 4 is shown.

Symmetry classes of pp-waves.

Abstract: The isometry groups admitted by plane-fronted gravitational waves with parallel rays are determined without use of any field equations. New groups with 5, 6, and 7 parameters arise which cannot occur for (nontrivial) exact solutions of Einstein's vacuum field equations. For all 17 possible cases the functional form of the free function in the metric is given. We apply the classification to Einstein-Maxwell fields and also determine, in Riemann-Cartan geometry, the form of the torsion tensor by assuming the vanishing of its Lie derivative with respect to the generators of the isometry groups.

A canonical expansion for nonlinear systems

Abstract: The importance of differential geometry, in particular, Lie brackets of vector fields, in the study of nonlinear systems is well established. Under very mild assumptions, we show that a real-analytic nonlinear system has an expansion in which the coefficients are computed in terms of Lie brackets. This expansion occurs in a special coordinate system. We also explain the concept of a pure feedback system. For control design involving a nonlinear system, one approach is to put the system in its canonical expansion and approximate by that part having only feedback paths.

Lie group invariance properties of radiation hydrodynamics equations and their associated similarity solutions

Abstract: The Lie group invariance properties of the one‐dimensional radiation hydrodynamic equations with the equilibrium diffusion approximation, a local thermodynamical equilibrium assumption, and an arbitrary material equation of state are derived. These properties are used systematically to generate similarity solutions of these equations for a given form of the equation of state. A comprehensive list of allowed similarity solutions for a perfect gas is presented. Several special cases that have been found previously by other authors appear in the list. Many other cases not reported previously are also presented. An example numerical solution is given for a piston‐driven shock with a thermal precursor.

Invariant star products and representations of compact semisimple Lie groups

Abstract: Starting from the work by F. A. Berezin, and earlier paper by the author defined an invariant star product on every nonexceptional Kahler symmetric space. In this Letter a recursion formula is obtained to calculate the corresponding invariant Hochschild 2-cochains for spaces of types II and III. An invariant star product is defined on every integral symplectic (Kahler) homogeneous space of simply-connected compact Lie groups (on every integral orbit of the coadjoint representation). The invariant 2-cochains are obtained from the Bochner-Calabi function of the space. The leading term of the lth-2-cochain is determined by the l-power of the Laplace operator.

Instantons on $\mathbf{CP}_2$

Abstract: On decrit et on classe les instantons sur CP 2 pour les groupes de Lie simples compacts unitaires et classiques

Analytic extension of the holomorphic discrete series

Abstract: Introduction. The embedding R ' C naturally divides C into two half-planes, each with R as boundary. These domains together with their function theory play an important role in the harmonic analysis of R. As but one example there is the theorem of Paley and Wiener that describes any square integrable function on R as a sum of boundary values of holomorphic functions (in Hardy spaces) on these domains. Much of the theory has natural extensions to R", to tube domains in C" based on cones, and even to more general Siegel domains. However, the dominant influence is abelian harmonic analysis. On the other hand, if G is a semisimple Lie group over R it may often be embedded in its natural complexification Gc. Does any of the abelian

Oscillatory singular integrals and harmonic analysis on nilpotent groups.

Abstract: Several related classes of operators on nilpotent Lie groups are considered. These operators involve the following features: (i) oscillatory factors that are exponentials of imaginary polynomials, (ii) convolutions with singular kernels supported on lower-dimensional submanifolds, (iii) validity in the general context not requiring the existence of dilations that are automorphisms.

Fixed points of Lie group actions on surfaces

Abstract: Let G be a connected finite-dimensional Lie group and M a compact surface. We investigate whether, for a given G and M , every continuous action of G on M must have a fixed (stationary) point. It is shown that when G is nilpotent and M has non-zero Euler characteristic that every action of G on M must have a fixed point. On the other hand, it is shown that the non-abelian 2-dimensional Lie group (affine group of the line) acts without fixed points on every compact surface. These results make it possible to complete this investigation for Lie groups of dimension at most 3.

Spinor structures on spheres and projective spaces

Abstract: An explicit construction of spinor structures on real, complex, and quaternionic projective spaces is given for all cases when they exist The construction is based on a theorem describing the bundle of orthonormal frames of a homogeneous Riemannian manifold This research is motivated by a remarkable coincidence of spinor connections on low‐dimensional spheres with simple, topologically nontrivial gauge configurations

Analytic continuation of local representations of lie groups

Abstract: We consider a symmetric space (G, K, σ) where G is a Lie group, K a closed subgroup, and σ the involutive automorphism defining the space. A local representation π is defined for g in a neighborhood of e in G\ and the operator π(g) is unbounded and defined on a dense subspace in a Hubert space where the identity holds. We study analytic continuations of π to unitary representations of a group G* which is dual to G.

Symmetric submanifolds of compact symmetric spaces

Abstract: This paper is the finalreport for the author's anouncement of the same title,appeared in Lect. Notes in Math., 1090, Springer-Verlag ([15]). It contains the results of the anouncement and their detailed proofs, and some further results. Now symmetric submanifold is defined analogously to riemannian symmetric space. Namely, for riemannian symmetric space itis assumed, the existence of the (intrinsic)symmetry at each point. And for symmetric submanifold itis assumed, the existence of the extrinsic symmetry at each point in the submanifold. If the ambient spaces are riemannian symmetric spaces, symmetric submanifolds are locally characterized as submanifolds with parallelsecond fundamental form which satisfy some condition on the normal spaces. (See Theorem 1.3.) This characterization corresponds to the characterization that riemannian symmetric spaces are riemannian manifolds with parallel curvature tensor locally.If the ambient spaces are rank-one symmetric spaces, submanifolds with parallel second fundamental form have already been classified by several mathematicians. (See [1], [4], [5], [9], [10], [13], [14], [17], [18], [21], [22].) Hence we can take up symmetric submanifolds of their spaces. But if the ambient spaces are other riemannian symmetric spaces, the symmetric submanifolds are almost unknown except Tsukada [23]. In this paper we consider the classification for the case when the ambient spaces are compact simply connected riemannian symmetric spaces. Firstly we will show that symmetric submanifolds of compact riemannian manifolds are equivariant for certain Lie groups acting transitively on the submanifolds, that is, the inclusions are induced from Lie group homomorphisms of the Lie groups into the isometry groups of the ambient spaces. (See Theorem 2.5.) This result implies that our classification may be reduced into that of certain algebraic objects associated with Lie group or Lie algebra. Next for symmetric submanifolds we will define the totally geodesic symmetric submanifolds tangent to the original symmetric submanifolds, and divide our classification problem into the following two steps. The first step is to classify the associated totally geodesic symmetric submanifolds. This is reduced to the local classification of non-

Wigner quantum systems. Two particles interacting via a harmonic potential. I. Two‐dimensional space

Abstract: A noncanonical quantum system, consisting of two nonrelativistic particles, interacting via a harmonic potential, is considered. The center‐of‐mass position and momentum operators obey the canonical commutation relations, whereas the internal variables are assumed to be the odd generators of the Lie superalgebra sl(1,2). This assumption implies a set of constraints in the phase space, which are explicitly written in the paper. All finite‐dimensional irreducible representations of sl(1,2) are considered. Particular attention is paid to the physical representations, i.e., the representations corresponding to Hermitian position and momentum operators. The properties of the physical observables are investigated. In particular, the operators of the internal Hamiltonian, the relative distance, the internal momentum, and the orbital momentum commute with each other. The spectrum of these operators is finite. The distance between the constituents is preserved in time. It can take no more than three different values. For any non‐negative integer or half‐integer l there exists a representation, where the orbital momentum is l (in units of 2ℏ). The position of any one of the particles cannot be localized, since the operators of the coordinates do not commute with each other. The constituents are smeared with a certain probability within a finite surface, which moves with a constant velocity together with the center of mass.

Central‐limit theorems on groups

Abstract: The probability density pN of the product of N statistically independent (and identically distributed, each with probability density p1) elements of a group is studied in the limit N→∞. It is shown, for the compact groups R(2) and R(3), that pN→1 as N→∞, independently of p1. It is made plausible that a similar behavior is to be expected for other compact groups. For noncompact groups, the case of SU(1,1)which is of interest to the physics of disordered conductors, is studied. The case in which p1 is isotropic, i.e., independent of the phases, is analyzed in detail. When p1 is fixed and N≫1, a Gaussian distribution in the appropriate variable is found. When the original variables are rescaled by 1/N and the limit N→∞ is taken, keeping the ratio of the length of the conductor to the localization length fixed, an explicit integral representation for the resulting probability density is found. It is also exhibited that the latter satisfies a ‘‘diffusion’’ equation on the group manifold.