Showing papers on "Lie group published in 1978"

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 Liouville method of integration of Hamiltonian systems

Abstract: In this paper we shall show that the equations of motion of a solid, and also Liouville's method of integration of Hamiltonian systems, appear in a natural manner when we study the geometry of level surfaces of a finite-dimensional space of integrals that are closed with respect to the Poisson bracket. If V is a finite-dimensional space of integrals on a symplectic manifold X which is closed with respect to the Poisson bracket, and if the integrals at thepoints of common position form linearly independent differentials, then we can refer to a level surface Mp of integrals which forms a manifold at the points of common position. The space V is a Lie algebra, and its associated group @ is a group of Hamiltonian transformations of the symplectic manifold X. The level surface Mp is not invariant under the action of the group ~, but the group ~ contains a subgroup ~ , that leaves invariant the manifold Mp. In the case of a compact Lie algebra V, the subgroup ~ will be commutative, and its algebra H is a Cartan subalgebr~.of V. In the general case the subalgebra H is an annihilator of the covector at the common position. If the action of the group ~ on Mp is free (or if it has a single orbit type), then the factor manifold Y =~fp/~ wili be a symplectic manifold, and the Hamiltonian vector field on the level surface Mp will be projected onto a Hamiltonian vector field on a manifold Y. It is natural tO refer to the latter as Euler's equation for the original Hamiltonian system on the manifold X. A particular case of this scheme is Euler's equation of motion of a solid, or, more generally, of motion of geodesic left-invariant metrics on the Lie group 6The algebra V of integrals consists of integrals of the moment of momentum. This algebra coincides with the Lie algebra of the group @, acting on a cotangent fibering with the aid of left shifts. The level surfaces are invariant under right shifts; therefore the factor manifold Y is homeomorphic to the orbit of a coadjoint representation, and the symplectic form on it coincides with Kirillov's form. Euler's equations of motion of a solid leave these orbits invariant, and they coincide with the projection of a Hamiltonian system onto a cotangent fibering of the group ~.

Euler equations on finite-dimensional Lie groups

Abstract: In this paper, a special class of dynamical systems is studied-the so-called Euler equations (a natural generalization of the classical equations of motion of a rigid body with one fixed point). It turns out that for any finite-dimensional Lie algebra this system has a large collection of integrals which are in involution. For the class of semisimple Lie algebras and for certain series of solvable Lie algebras these integrals turn out to be sufficient for the complete integration (using Liouville's theorem) of the multiparametric family of Euler equations.Bibliography: 8 titles.

Relativistically invariant two-dimensional models of field theory which are integrable by means of the inverse scattering problem method

Abstract: A method is proposed for deriving and classifying relativistically invariant integrable systems that are sufficiently general to encompass all presently known two-dimensional solvable models, and for the construction of a few new ones. The concept of ''gauge equivalence'' introduced in this paper allows one to clarify the relation between several different models of classical field theory, such as the n-field, the sine-Gordon equation, and the Thirring model. We study the model of the principal chiral field for the group SU (N). It is shown that at N=3 this model exhibits nontrivial interactions: decay, fusion and resonant scattering of solitons. New chiral models are proposed with fields taking values in homogeneous spaces of Lie groups and exhibiting a high degree of symmetry. We prove the integrability of these models when the homogeneous space is a Grassmann manifold.

Philosophy of geometry from Riemann to Poincaré

Abstract: 1 / Background.- 1.0.1 Greek Geometry and Philosophy.- 1.0.2 Geometry in Greek Natural Science.- 1.0.3 Modern Science and the Metaphysical Idea of Space.- 1.0.4 Descartes' Method of Coordinates.- 2 / Non-Euclidean Geometries.- 2.1 Parallels.- 2.1.1 Euclid's Fifth Postulate.- 2.1.2 Greek Commentators.- 2.1.3 Wallis and Saccheri.- 2.1.4 Johann Heinrich Lambert.- 2.1.5 The Discovery of Non-Euclidean Geometry.- 2.1.6 Some Results of Bolyai-Lobachevsky Geometry.- 2.1.7 The Philosophical Outlook of the Founders of Non-Euclidean Geometry.- 2.2 Manifolds.- 2.2.1 Introduction.- 2.2.2 Curves and their Curvature.- 2.2.3 Gaussian Curvature of Surfaces.- 2.2.4 Gauss' Theorema Egregium and the Intrinsic Geometry of Surfaces.- 2.2.5 Riemann's Problem of Space and Geometry.- 2.2.6 The Concept of a Manifold.- 2.2.7 The Tangent Space.- 2.2.8 Riemannian Manifolds, Metrics and Curvature.- 2.2.9 Riemann's Speculations about Physical Space.- 2.2.10 Riemann and Herbart. Grassmann.- 2.3 Projective Geometry and Projective Metrics.- 2.3.1 Introduction.- 2.3.2 Projective Geometry: An Intuitive Approach.- 2.3.3 Projective Geometry: A Numerical Interpretation.- 2.3.4 Projective Transformations.- 2.3.5 Cross-ratio.- 2.3.6 Projective Metrics.- 2.3.7 Models.- 2.3.8 Transformation Groups and Klein's Erlangen Programme.- 2.3.9 Projective Coordinates for Intuitive Space.- 2.3.10 Klein's View of Intuition and the Problem of Space-Forms.- 3 / Foundations.- 3.1 Helmholtz's Problem of Space.- 3.1.1 Helmholtz and Riemann.- 3.1.2 The Facts which Lie at the Foundation of Geometry.- 3.1.3 Helmholtz's Philosophy of Geometry.- 3.1.4 Lie Groups.- 3.1.5 Lie's Solution of Helmholtz's Problem.- 3.1.6 Poincare and Killing on the Foundations of Geometry.- 3.1.7 Hilbert's Group-Theoretical Characterization of the Euclidean Plane.- 3.2 Axiomatics.- 3.2.1 The Beginnings of Modern Geometrical Axiomatics.- 3.2.2 Why are Axiomatic Theories Naturally Abstract?.- 3.2.3 Stewart, Grassmann, Plucker.- 3.2.4 Geometrical Axiomatics before Pasch.- 3.2.5 Moritz Pasch.- 3.2.6 Giuseppe Peano.- 3.2.7 The Italian School. Pieri. Padoa.- 3.2.8 Hilbert's Grundlagen.- 3.2.9 Geometrical Axiomatics after Hilbert.- 3.2.10 Axioms and Definitions. Frege's Criticism of Hilbert.- 4 / Empiricism, Apriorism, Conventionalism.- 4.1 Empiricism in Geometry.- 4.1.1 John Stuart Mill.- 4.1.2 Friedrich Ueberweg.- 4.1.3 Benno Erdmann.- 4.1.4 Auguste Calinon.- 4.1.5 Ernst Mach.- 4.2 The Uproar of Boeotians.- 4.2.1 Hermann Lotze.- 4.2.2 Wilhelm Wundt.- 4.2.3 Charles Renouvier.- 4.2.4 Joseph Delboeuf.- 4.3 Russell's Apriorism of 1897.- 4.3.1 The Transcendental Approach.- 4.3.2 The 'Axioms of Projective Geometry'.- 4.3.3 Metrics and Quantity.- 4.3.4 The Axiom of Distance.- 4.3.5 The Axiom of Free Mobility.- 4.3.6 A Geometrical Experiment.- 4.3.7 Multidimensional Series.- 4.4 Henri Poincare.- 4.4.1 Poincare's Conventionalism.- 4.4.2 Max Black's Interpretation of Poincare's Philosophy of Geometry.- 4.4.3 Poincare's Criticism of Apriorism and Empiricism.- 4.4.4 The Conventionality of Metrics.- 4.4.5 The Genesis of Geometry.- 4.5.6 The Definition of Dimension Number.- 1. Mappings.- 2. Algebraic Structures. Groups.- 3. Topologies.- 4. Differentiable Manifolds.- Notes.- To Chapter 1.- To Chapter 2.- 2.1.- 2.2.- 2.3.- To Chapter 3.- 3.1.- 3.2.- To Chapter 4.- 4.1.- 4.2.- 4.3.- 4.4.- References.

Gauge theory of gravity and supergravity on a group manifold

Abstract: The natural arena for the physics of gravity, supergravity and their enlargements appears to be the group manifold of the Poincare group P, the graded Poincare group GP of supersymmetry, and the corresponding enlargements. The dynamics of these theories correspond to geometrical algorithms in P and GP. Differential geometry on Lie groups is reviewed and results applied to P and GP. Curvature, gauge transformations and factorization are introduced. Also reviewed is the general coordinate transformation group and a hybrid gauge transformation, the anholonomized G.C.T. gauge. A study is made of the construction of an action, including the introduction of a set of special 2 forms, the ''pseudo curvatures.'' The possibilities of factorization in supersymmetry are analyzed. The version of supergravity is present which has now become a completely geometrical theory.

On invariant integration over SU(N)

Abstract: We give a graphical algorithm for evaluation of invariant integrals of polynomials in SU(N) group elements. Such integrals occur in strongly coupled lattice gauge theory. The results are expressed in terms of totally antisymmetric tensors and Kronecker delta symbols.

Expansions for spherical functions on noncompact symmetric spaces

Abstract: Section 0 Lot G be a connected noncompact semisimple Lie group with finite center and real rank one; fix a maximal compact subgroup K. Our concern in this paper is Fourier analysis on the Riemannian symmetric space G]K. We shall analyze the local and global behavior of spherical functions, the boundedness of multiplier operators, and the inversion of differential operators. The core of the paper, however, is an analysis of how the size of a function is controlled by the size of its Fourier transform. There is an extensive literature on such topics, centered about the Paley-Wiener and Plancherel theorems. Our work relies heavily on these earlier ideas and techniques, to which detailed reference will be made in the body of the paper. The problems we wish to solve, however, require estimates more precise and of a different nature than are necessary for the Plancherel or Paley-Wienor theorem. Thus the first two sections of this paper are devoted to the construction of various asymptotic expansions for spherical functions; in later sections we show how these expansions may be applied to the Fourier analysis of G/K.

Kirillov's character formula for reductive Lie groups

Abstract: Inventiones math. 48, 2007-220 (1978); on-line version. Kirillov’s famous formula says that the characters X of the irreducible unitary representations of a Lie group G should be given by an equation of the form (Φ) χ(exp x )= p(x) −1 Ω e i(λ,x) dµΩ(λ) where ω =Ω (X )i s aG-orbit in the dual g ∗ ft he Lie algebrag of G, µΩ is Kirillov’s canonical measure on Ω, and p is a certain function on g ,n amely p(x )= det 1/2 {sinh(ad(x/2)) /ad(x/2)} at least for generic orbits Ω [10]. This formula cannot be taken too literally, of course (the integral in (Φ) is usually divergent), but has to be interpreted as an equation of distributions on a certain space of test functions on g. To make this precise, denote by g o an open neighborhoodod of zero in g so that exp : g → G restricts to an invertible analytic map of g o onto an open subset of G. For our purposes, the formula (Φ) should be interpreted as saying that (Φ � )t r g ϕ(x)π(exp(x)) dx = Ω g e i(λ,x) ϕ(x) p(x) −1 } dµΩ(λ) for all C ∞ functions ϕ with compact support in g o . (Here π is the representation of G with character χ.) Of course, Kirillov’s formula does not hold in this generality. It is in fact a major problem in representation theory to determine its exact domain of validity. In this paper we shall show that Kirillov’s formula holds for the characters of a reductive real Lie group which occur in the Plancherel formula. Actually, we shall deal in detail only with the discrete series characters. The formula for the other characters can then be reduced to the formula for the discrete series characters by familiar methods. (Duflo [3]). Kirillov’s formula for the discrete series is a consequence of a formula relating the Fourier transform on g with the Fourier transform on Cartan subalgebras of compact type by means of the invariant integral. This is the form in which Kirillov’s formula will be proved.

The Campbell-Hausdorff Formula and Invariant Hyperfunctions.

Abstract: Let G be a Lie group and g its Lie algebra. We denote by V the underlying vector space of g. There is a canonical isomorphism between the ring Z(g) of the biinvariant differential operators on G and the ring I(g) of the constant coefficient operators on V which are invariant by the adjoint action of G. When g is semi-simple, this is the "Harish-Chandra isomorphism"; for a general Lie algebra, this was established by Duflo I-4]. We shall prove here, that when G is solvable the Duflo isomorphism extends to an isomorphism 9 of the algebra of "local" invariant hyperfunctions under the group convolution and the algebra of invariant hyperfunctions on V under additive convolution (the exact result will be stated below). This gives a partial answer to a conjecture of Rais 1-12]. The existence of such an isomorphism 9 is of importance for the harmonic analysis on G, and for the study of the solvability of biinvariant operators on G (see [7]). It reflects and explains the "orbit method" ([8, 9]), i.e. the correspondence between orbits of G in V*, the dual vector space of V, and unitary irreducible representations of G: let T be an irreducible representation of G, then the infinitesimal character of T is a character of the ring Z(fl). Let t~ be an orbit in V*, the map p~(P)=P(f) (f~ (9) is a character of the ring I(g) (I(g) being identified with the ring of invariant polynomials on V*). The principle of the orbit method is to assign to a (good) orbit tV a representation T~ of G (or g), whose infinitesimal character corresponds to p~ via the isomorphism ~. This is the technique used by M. Duflo to construct the ring isomorphism ~. Furthermore let t~ be (when defined) the distribution on V which is the Fourier transform of the canonical measure on the orbit 0, then t~ is clearly an invariant positive eigendistribution of every operator P in I (g) of eigenvalue p~(P). Kirillov conjectured that the global character of the representation T~ (when

Sequences of Z2⊗Z2 graded Lie algebras and superalgebras

Abstract: Applying methods similar to those used for classical Lie superalgebras (Z2 graded algebras), we construct sequences of Z2⊕Z2 graded Lie superalgebras. In this way one obtains the spl(m,n,r,s), osp(m,n,r,s), P1(m,r), P3(m,n), ospP3(m,n), P1,2(m), and Q (m) series. We also give series of Z2⊕Z2 graded Lie algebras. Closed forms for superdeterminants and determinants of Z2⊕Z2 graded matrices are presented.

Gravity and supergravity as gauge theories on a group manifold

Abstract: We construct generalizations of gravity, including supergravity, by writing the theory on the group manifold (Poincare for gravity, the graded-Poincare group for supergravity). The action involves forms over the group, restricted to a 4-dimensional submanifold. The equations of motion produce a Lorentz gauge in gravity and supergravity, and an additional anholonomic supersymmetric coordinate transformation which reduces to the “local supersymmetry” of supergravity.

Elementary construction of graded Lie groups

Abstract: We show how the definitions of the classical Lie groups have to be modified in the case where Grassmann variables are present. In particular we construct the general linear, the special linear and the orthosymplectic graded Lie groups. Special attention is paid to the question of how to formulate an adequate ’’unitarity condition.’’

The Knapp-Stein dimension theorem for $p$-adic groups

Abstract: The purpose of this paper is to prove for p-adic groups the analogue of a theorem due to Knapp and Stein [3] in the case of real semisimple Lie groups. The Knapp-Stein theorem has precisely-mutatis mutandis-the same statement as we give below. Our proof, which depends upon the Harish-Chandra commuting algebra theorem [4, Theorem 5.5.3.2], carries over, with only slight changes, to the real case too. We wish to thank Nolan Wallach for useful discussions.

Induced and amenable ergodic actions of Lie groups

Abstract: © Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1978, tous droits réservés. L’accès aux archives de la revue « Annales scientifiques de l’É.N.S. » (http://www. elsevier.com/locate/ansens) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Exact spherically symmetric classical solutions for the f-g theory of gravity

Abstract: We find a class of exact spherically symmetric solutions to the coupled classical field equations of $f\ensuremath{-}g$ theory. The $f$ and $g$ metrics each induce a cosmological constant in the field equations of the other, and are both of Schwarzschild-plus-de Sitter type.

Group Symmetries in Nuclear Structure

Abstract: 1: Introduction.- 2: Classification of Symmetries.- 2.1. Space-Time (Geometrical) Symmetries.- 2.2. Exact Dynamical Symmetry (Unknown Origin).- 2.3. Almost Exact Dynamical Symmetry (Unknown Origin).- 2.4. Approximate Dynamical Symmetry.- 2.5. Dynamical Symmetries in Vector Spaces ("Model" Symmetries).- 2.6. Shape Symmetries.- 3: Symmetries and Groups.- 3.1. Groups and Representations of Groups.- 3.2. ?-Particle Model in Light Nuclei.- 3.3. Summary.- 4: Lie Groups and Their Algebras.- 4.1. Definition of a Lie Group.- 4.2. Infinitesimal Operators of a Lie Group.- 4.3. Representations of Lie Groups and Labeling of States.- 4.4. Representations of Lie Groups: Irreducible Tensors.- 4.5. Outer Product and Littlewood Rules.- 4.6. Matrix Groups and Their Representations.- 4.7. Two Theorems Concerning Goodness of Symmetry.- 5: Manifestation of Symmetries.- 5.1. Relationship between Energies.- 5.2. Symmetry Effect in Nuclear Reactions.- 5.3. Selection Rules.- 5.4. The Goodness of Symmetries.- 6: Spectral Distribution Methods.- 6.1. Introduction.- 6.2. The Method.- 6.3. Evaluation of Moments.- 6.4. Normality of the Distribution.- 6.5. Application of Distribution Method to Nuclear Spectroscopy.- 7: The Unitary Group and Its Subgroups.- 7.1. Introduction.- 7.2. Subgroups of U(N).- 7.3. Unitary Decomposition of Operators.- 7.4. Method of Separation.- 7.5. Number Nonconserving Operators.- 7.6. Decomposition by Contraction.- 7.7. Extension to Many Orbits: Configuration Averages.- 7.8. Unitary Group and Hartree-Fock Approximation.- 7.9. Application of Configuration Distributions.- 8: Angular Momentum and Isospin.- 8.1. Introduction.- 8.2. Multipole Sum-Rule Methods.- 8.3. Isospin Distributions.- 8.4. Strength Distributions.- 8.5. Mixing of Isospin Symmetry in Nuclei.- 8.6. Isobaric Mass Formula.- 8.7. Angular Momentum Averaging.- 9: Space-Symmetry Group-Wigner Supermultiplet Scheme.- 9.1. The Group SU(4) and the Supermultiplet Scheme.- 9.2. Casimir Operators of SU(4) and the Space Exchange Operator M.- 9.3. Evidence for Space Symmetry.- 9.4. ?-Particle Spectroscopy.- 9.5. ? Decay and Magnetic Moments of f7/2 Shell Nuclei.- 9.6. Muon Capture in Nuclei.- 9.7. SU(4) Classification of Nuclear Interaction.- 9.8. Study of SU(4) Symmetry Using Spectral Distribution Method.- 9.9. The "Goodness" of SU(4) Symmetry.- 9.10. SU(4)-ST Averaging.- 10: SU(3) Symmetry.- 10.1. Introduction.- 10.2. Brief Summary of Rotational Features in Light Nuclei.- 10.3. Search for the Intermediate Group G.- 10.4. Classification of States within an SU(3) Representation.- 10.5. States in the Projected Representation.- 10.6. Shell Model Calculation in the SU(3) Basis.- 10.7. SU(3) Classification of Interactions in the ds Shell.- 10.8. Mixing of SU(3) Symmetry in the ds Shell.- 10.9. Pseudo-LS and Pseudo-SU(3) Coupling Schemes.- 10.10. Configuration Mixing across Major Shells.- 10.11. "Macroscopic" SU(3) Symmetry.- 11: Seniority and Symplectic Symmetry.- 11.1. Introduction.- 11.2. Seniority in a Single j Shell.- 11.3. Representations of Sp(2j + 1).- 11.4. Casimir Operators and Their Eigenvalues.- 11.5. Goodness of Symmetry.- 11.6. Seniority in the j = 9/2 Shell.- 11.7. Symplectic Symmetry for the 1f7/2 Shell.- 11.8. Quasispin.- 11.9. Quasispin and Its Relation to Seniority.- 11.10. Multishell Seniority.- 11.11. Multishell Seniority Averaging.- 11.12. Multishell Seniority and the Two-Body Interaction.- 11.13. A New Truncation Scheme for Shell-Model Calculations.- 12: Summary and Final Remarks.- References.

Topological excitations in the Abelian Higgs model

Abstract: A lattice version of the Abelian Higgs model is studied in arbitrary Euclidean dimension. Using an exact duality transformation, the theory is rewritten in terms of its topological excitations. The dual form of the theory specifies in a simple way all the allowed topological excitations as well as their interactions. The combination of the scalar Higgs field and the Abelian gauge field produces excitations found neither in the pure gauge theory nor in the pure scalar theory ($\mathrm{XY}$ model). In three dimensions, for example, we find finite vortex strings terminating on monopoles, as well as closed vortex loops. Implications of these singularities for the critical behavior of the theory are briefly discussed.

Processes with independent increments on a Lie group

Abstract: The Levy-Khinchin representation for processes with indepen- dent increments is extended to processes taking values in a Lie group. The basis of the proof is to approximate continuous time processes by Markov chains. The processes involved are handled by the technique, developed by Stroock and Varadhan, of characterizing Markov processes by associated martingales. 1. Introduction. A stochastically continuous real-valued process x(t) with

A note on recoupling coefficients for SU(3)

Abstract: A 6‐ (λμ) coefficient, denoted by Z and different from the usual U coefficient, associated with a specific recoupling of three irreducible representations of SU(3), is defined. A general 9‐ (λμ) coefficient, analogous to the unitary 9‐J coefficient of the angular momentum Racah algebra, is then expressed in terms of the Z coefficient and two U coefficients. In this way problems associated with the existence of outer multiplicities in the products of irreducible representations of SU(3) are circumvented.

Ergodic theorems for noncommutative dynamical systems

Abstract: Recently, E.C. Lance extended the pointwise ergodic theorem to actions of the group of integers on von Neumann algebras. Our purpose is to extend other pointwise ergodic theorems to von Neumann algebra context: the Dunford-Schwartz-Zygmund pointwise ergodic theorem, the pointwise ergodic theorem for connected amenable locally compact groups, the Wiener's local ergodic theorem for ℝ + d and for general Lie groups.

Sharp Inequalities for Weyl Operators and Heisenberg Groups

Abstract: We study inequalities in harmonic analysis in the context of non-commutative non-compact locally compact groups. Our main result is the determination of the best constant in the Hausdorff-Young inequality for Heisenberg groups. We also obtain the somewhat surprising fact that the resulting sharp inequality does not admit any extremal functions. These results are obtained after a detailed study of the operators which occur in the Fourier decomposition of the regular representation of the Heisenberg groups. These are called Weyl operators and are of independent interest. We also obtain bounds for the best constants in the Hausdorff-Young inequality and in Young's inequality on semi-direct product groups, including non-unimodular groups. In particular, for real nilpotent groups of dimension n those best constants are shown to be dominated by the corresponding best constants for IR n. Although some of our preliminary lemmas are valid for all values ofp~(1, 2) the methods we use for our main results require that p belong to the sequence 4/3, 6/5, 8/7 ..... i.e. that p', the conjugate index, be an even integer. The contents of this paper are as follows. In Section 1 we discuss Weyl operators and determine, for p' even, the best constant in a Hausdorff-Young type inequality (Theorem 1). We also show the non-existence of extremal functions for this inequality. In Section 2 we prove some general results for locally compact groups which includes a form of Young's inequality for convolution appropriate for non-unimodular groups. This is applied to arbitrary semi-direct products. Then using a duality argument which relates the inequalities of Young and of Hausdorff-Young we obtain bounds for the Hausdorff-Young inequality (Theorem 2) on unimodular semi-direct product groups (for p' even). An interesting consequence of these results is that for a connected simply connected real nilpotent Lie group of dimension n, the best constants in the inequalities of Young and Hausdorff-Young are dominated by the corresponding best constants for IR n. In Section 3 we show (Theorem 3), using the theory of Weyl operators developed