Showing papers on "Lie group published in 1988"

Real reductive groups

Abstract: Intertwining operators completions of admissible (g,K)-modules the theory of the leading term the Harish-Chandra plancherel theorem abstract representation theory the Whittaker plancherel theorem.

The deformation theory of representations of fundamental groups of compact Kähler manifolds

Abstract: Let Γ be the fundamental group of a compact Kahler manifold M and let G be a real algebraic Lie group. Let ℜ(Γ, G) denote the variety of representations Γ → G. Under various conditions on ρ ∈ ℜ(Γ, G) it is shown that there exists a neighborhood of ρ in ℜ(Γ, G) which is analytically equivalent to a cone defined by homogeneous quadratic equations. Furthermore this cone may be identified with the quadratic cone in the space\(Z^1 (\Gamma ,g_{Ad\rho } )\) of Lie algebra-valued l-cocycles on Γ comprising cocyclesu such that the cohomology class of the cup/Lie product square [u, u] is zero in\(H^2 (\Gamma ,g_{Ad\rho } )\). We prove that ℜ(Γ, G) is quadratic at ρ if either (i) G is compact, (ii) ρ is the monodromy of a variation of Hodge structure over M, or (iii) G is the group of automorphisms of a Hermitian symmetric space X and the associated flat X-bundle over M possesses a holomorphic section. Examples are given where singularities of ℜ(Γ, G) are not quadratic, and are quadratic but not reduced. These results can be applied to construct deformations of discrete subgroups of Lie groups.

Harmonic Analysis of Spherical Functions on Real Reductive Groups

Abstract: Contents: The Concept of a Spherical Function Structure of Semisimple Lie Groups and Differential Operators on Them The Elementary Spherical Functions The Harish-Chandra Series for and the c-Function Asymptotic Behaviour of Elementary Spherical Functions The L2-Theory. The Harish-Chandra Transform on the Schwartz Space of G//K LP-Theory of Harish-Chandra Transform. Fourier Analysis on the Spaces CP(G//K) Bibliography Subject Index.

Nonlinear gyrokinetic theory for finite‐beta plasmas

Abstract: A self‐consistent and energy‐conserving set of nonlinear gyrokinetic equations, consisting of the averaged Vlasov and Maxwell’s equations for finite‐beta plasmas, is derived. The method utilized in the present investigation is based on the Hamiltonian formalism and Lie transformation. The resulting formulation is valid for arbitrary values of k⊥ρi and, therefore, is most suitable for studying linear and nonlinear evolution of microinstabilities in tokamak plasmas as well as other areas of plasma physics where the finite Larmor radius effects are important. Because the underlying Hamiltonian structure is preserved in the present formalism, these equations are directly applicable to numerical studies based on the existing gyrokinetic particle simulation techniques.

Free Massless Bosonic Fields of Arbitrary Spin in $d$-dimensional De Sitter Space

Abstract: Free massless bosonic fields of arbitrary spins s>1, corresponding to symmetric representations of SO(d−2) compact subgroup of the d-dimensional massless flat little group, are described in d-dimensional (anti-) de Sitter space in terms of differential forms. The formulation proposed is a generalization to arbitrary d≥4 of that suggested previously in the four dimensional case, which served in Refs. 1 and 2 as a starting point for introducing consistent gravitational interaction for all massless higher spin fields in d=4.

Determinant bundles and Virasoro algebras

Abstract: We consider the interplay of infinite-dimensional Lie algebras of Virasoro type and moduli spaces of curves, suggested by string theory. We will see that the infinitesimal geometry of determinant bundles is governed by Virasoro symmetries. The Mumford forms are just invariants of these symmetries. The representations of Virasoro algebra define (twisted)D-modules on moduli spaces; theseD-modules are equations on correlators in conformal field theory.

Odd symplectic groups.

Abstract: On definit des groupes symplectiques sur des espaces vectoriels de dimension impaire. Ces groupes de Lie ne sont ni simples ni reductifs. On obtient une formule de caracteres pour certaines representations tensorielles de ces groupes

Periodic solutions near equilibria of symmetric Hamiltonian systems

Abstract: We consider the effects of symmetry on the dynamics of a nonlinear hamiltonian system invariant under the action of a compact Lie group T, in the vicinity of an isolated equilibrium: in particular, the local existence and stability of periodic trajectories. The main existence result, an equivariant version of the Weinstein—Moser theorem, asserts the existence of periodic trajectories with certain prescribed symmetries Z c T x S 1 , independently of the precise nonlinearities. We then describe the constraints put on the Floquet operators of these periodic trajectories by the action of T. This description has three ingredients: an analysis of the linear symplectic maps that commute with a symplectic representation, a study of the momentum mapping and its relation to Floquet multipliers, and Krein Theory. We find that for some 2, which we call cylospetral , all eigenvalues of the Floquet operator are forced by the group action to lie on the unit circle; that is, the periodic trajectory is spectrally stable. Similar results for equilibria are described briefly. The results are applied to a number of simple examples such as T = SO(2), 0(2 ), Z n , D n , SU (2) ; and also to the irreducible symplectic actions of O(3) on spaces of complex spherical harmonics, modelling oscillations of a liquid drop.

Lie Algebraic Treatment of Linear and Nonlinear Beam Dynamics

Abstract: The purpose of this paper is to present a summary of new methods employing Lie algebraic tools for characterizing beam dynamics in charged- particle optical systems. These methods are applicable to accelerator design, charged-particle beam transport, electron microscopes, and also light optics. The new methods represent the action of each separate element of a compound optical system, including all departures from paraxial optics by a certain operator. The operators can then be concatenated following well defined rules to obtain a resultant operator that characterizes the entire system. (AIP)

On the infinite‐dimensional symmetry group of the Davey–Stewartson equations

Abstract: The Lie algebra of the group of point transformations, leaving the Davey–Stewartson equations (DSE’s) invariant, is obtained. The general element of this algebra depends on four arbitrary functions of time. The algebra is shown to have a loop structure, a property shared by the symmetry algebras of all known (2+1)‐dimensional integrable nonlinear equations. Subalgebras of the symmetry algebra are classified and used to reduce the DSE’s to various equations involving only two independent variables.

The Homology of Hopf Spaces

Abstract: Hopf Algebras. Classifying Spaces. Localization. The Bockstein Spectral Sequence. The Projective Plane. Reflection Groups and Classifying Spaces. Secondary Operations. The Module of Indecomposables QH * (X F p ) p ODD. The Module of Indecomposables QH * (X F 2 ). K-Theory. The Hopf Algebra H * (X F p ). Power Spaces. Appendices: Lie Groups. The Steenrod Algebra. Brown-Peterson Theory. Bibliography. Index.

Integrability and Nonintegrability in Geometry and Mechanics

Abstract: 1. Some Equations of Classical Mechanics and Their Hamiltonian Properties.- 1. Classical Equations of Motion of a Three-Dimensional Rigid Body.- 1.1. The Euler-Poisson Equations Describing the Motion of a Heavy Rigid Body around a Fixed Point.- 1.2. Integrable Euler, Lagrange, and Kovalevskaya Cases.- 1.3. General Equations of Motion of a Three-Dimensional Rigid Body.- 2. Symplectic Manifolds.- 2.1. Symplectic Structure in a Tangent Space to a Manifold.- 2.2. Symplectic Structure on a Manifold.- 2.3. Hamiltonian and Locally Hamiltonian Vector Fields and the Poisson Bracket.- 2.4. Integrals of Hamiltonian Fields.- 2.5. The Liouville Theorem.- 3. Hamiltonian Properties of the Equations of Motion of a Three-Dimensional Rigid Body.- 4. Some Information on Lie Groups and Lie Algebras Necessary for Hamiltonian Geometry.- 4.1. Adjoint and Coadjoint Representations, Semisimplicity, the System of Roots and Simple Roots, Orbits, and the Canonical Symplectic Structure.- 4.2. Model Example: SL(n, ?) and sl(n, ?).- 4.3. Real, Compact, and Normal Subalgebras.- 2. The Theory of Surgery on Completely Integrable Hamiltonian Systems of Differential Equations.- 1. Classification of Constant-Energy Surfaces of Integrable Systems. Estimation of the Amount of Stable Periodic Solutions on a Constant-Energy Surface. Obstacles in the Way of Smooth Integrability of Hamiltonian Systems.- 1.1. Formulation of the Results in Four Dimensions.- 1.2. A Short List of the Basic Data from the Classical Morse Theory.- 1.3. Topological Surgery on Liouville Tori of an Integrable Hamiltonian System upon Varying Values of a Second Integral.- 1.4. Separatrix Diagrams Cut out Nontrivial Cycles on Nonsingular Liouville Tori.- 1.5. The Topology of Hamiltonian-Level Surfaces of an Integrable System and of the Corresponding One-Dimensional Graphs.- 1.6. Proof of the Principal Classification Theorem 2.1.2.- 1.7. Proof of Claim 2.1.1.- 1.8.Proof of Theorem 2.1.1. Lower Estimates on the Number of Stable Periodic Solutions of a System.- 1.9. Proof of Corollary 2.1.5.- 1.10 Topological Obstacles for Smooth Integrability and Graphlike Manifolds. Not each Three-Dimensional Manifold Can be Realized as a Constant-Energy Manifold of an Integrable System.- 1.11. Proof of Claim 2.1.4.- 2. Multidimensional Integrable Systems. Classification of the Surgery on Liouville Tori in the Neighbourhood of Bifurcation Diagrams.- 2.1. Bifurcation Diagram of the Momentum Mapping for an Integrable System. The Surgery of General Position.- 2.2. The Classification Theorem for Liouville Torus Surgery.- 2.3. Toric Handles. A Separatrix Diagram is Always Glued to a Nonsingular Liouville Torus Tn Along a Nontrivial (n - 1)-Dimensional Cycle Tn-1.- 2.4. Any Composition of Elementary Bifurcations (of Three Types) of Liouville Tori Is Realized for a Certain Integrable System on an Appropriate Symplectic Manifold.- 2.5. Classification of Nonintegrable Critical Submanifolds of Bott Integrals.- 3. The Properties of Decomposition of Constant-Energy Surfaces of Integrable Systems into the Sum of Simplest Manifolds.- 3.1. A Fundamental Decomposition Q = mI +pII +qIII +sIV +rV and the Structure of Singular Fibres.- 3.2. Homological Properties of Constant-Energy Surfaces.- 3. Some General Principles of Integration of Hamiltonian Systems of Differential Equations.- 1. Noncommutative Integration Method.- 1.1. Maximal Linear Commutative Subalgebras in the Algebra of Functions on Symplectic Manifolds.- 1.2. A Hamiltonian System Is Integrable if Its Hamiltonian is Included in a Sufficiently Large Lie Algebra of Functions.- 1.3. Proof of the Theorem.- 2. The General Properties of Invariant Submanifolds of Hamiltonian Systems.- 2.1. Reduction of a System on One Isolated Level Surface.- 2.2. Further Generalizations of the Noncommutative Integration Method.- 3. Systems Completely Integrable in the Noncommutative Sense Are Often Completely Liouville-Integrable in the Conventional Sense.- 3.1. The Formulation of the General Equivalence Hypothesis and its Validity for Compact Manifolds.- 3.2. The Properties of Momentum Mapping of a System Integrable in the Noncommutative Sense.- 3.3. Theorem on the Existence of Maximal Linear Commutative Algebras of Functions on Orbits in Semisimple and Reductive Lie Algebras.- 3.4. Proof of the Hypothesis for the Case of Compact Manifolds.- 3.5. Momentum Mapping of Systems Integrable in the Noncommutative Sense by Means of an Excessive Set of Integrals.- 3.6. Sufficient Conditions for Compactness of the Lie Algebra of Integrals of a Hamiltonian System.- 4. Liouville Integrability on Complex Symplectic Manifolds.- 4.1. Different Notions of Complex Integrability and Their Interrelation.- 4.2. Integrability on Complex Tori.- 4.3. Integrability on K3-Type Surfaces.- 4.4. Integrability on Beauville Manifolds.- 4.5.Symplectic Structures Integrated without Degeneracies.- 4. Integration of Concrete Hamiltonian Systems in Geometry and Mechanics. Methods and Applications.- 1. Lie Algebras and Mechanics.- 1.1. Embeddings of Dynamic Systems into Lie Algebras.- 1.2. List of the Discovered Maximal Linear Commutative Algebras of Polynomials on the Orbits of Coadjoint Representations of Lie Groups.- 2. Integrable Multidimensional Analogues of Mechanical Systems Whose Quadratic Hamiltonians are Contained in the Discovered Maximal Linear Commutative Algebras of Polynomials on Orbits of Lie Algebras.- 2.1. The Description of Integrable Quadratic Hamiltonians.- 2.2. Cases of Complete Integrability of Equations of Various Motions of a Rigid Body.- 2.3. Geometric Properties of Rigid-Body Invariant Metrics on Homogeneous Spaces.- 3. Euler Equations on the Lie Algebra so(4).- 4. Duplication of Integrable Analogues of the Euler Equations by Means of Associative Algebra with Poincare Duality.- 4.1. Algorithm for Constructing Integrable Lie Algebras.- 4.2. Frobenius Algebras and Extensions of Lie Algebras.- 4.3. Maximal Linear Commutative Algebras of Functions on Contractions of Lie Algebras.- 5. The Orbit Method in Hamiltonian Mechanics and Spin Dynamics of Superfluid Helium-3.- 5. Nonintegrability of Certain Classical Hamiltonian Systems.- 1. The Proof of Nonintegrability by the Poincare Method.- 1.1. Perturbation Theory and the Study of Systems Close to Integrable.- 1.2. Nonintegrability of the Equations of Motion of a Dynamically Nonsymmetric Rigid Body with a Fixed Point.- 1.3. Separatrix Splitting.- 1.4. Nonintegrability in the General Case of the Kirchhoff Equations of Motion of a Rigid Body in an Ideal Liquid.- 2. Topological Obstacles for Complete Integrability.- 2.1. Nonintegrability of the Equations of Motion of Natural Mechanical Systems with Two Degrees of Freedom on High-Genus Surfaces.- 2.2. Nonintegrability of Geodesic Flows on High-Genus Riemann Surfaces with Convex Boundary.- 2.3. Nonintegrability of the Problem of n Gravitating Centres for n > 2.- 2.4. Nonintegrability of Several Gyroscopic Systems.- 3. Topological Obstacles for Analytic Integrability of Geodesic Flows on Non-Simply-Connected Manifolds.- 4. Integrability and Nonintegrability of Geodesic Flows on Two-Dimensional Surfaces, Spheres, and Tori.- 4.1. The Holomorphic 1-Form of the Integral of a Geodesic Flow Polynomial in Momenta and the Theorem on Nonintegrability of Geodesic Flows on Compact Surfaces of Genus g > 1 in the Class of Functions Analytic in Momenta.- 4.2. The Case of a Sphere and a Torus.- 4.3. The Properties of Integrable Geodesic Flows on the Sphere.- 6. A New Topological Invariant of Hamiltonian Systems of Liouville-Integrable Differential Equations. An Invariant Portrait of Integrable Equations and Hamiltonians.- 1. Construction of the Topological Invariant.- 2. Calculation of Topological Invariants of Certain Classical Mechanical Systems.- 3. Morse-Type Theory for Hamiltonian Systems Integrated by Means of Non-Bott Integrals.- References.

Regular variation in

Abstract: Researchers investigating certain limit theorems in probability have discovered a multivariable analogue to Karamata's theory of regularly varying functions. The method uses elements of real analysis and Lie groups to analyze the asymptotic behavior of functions and measures on Rk. We present an account here which is independent of probabilistic considerations.

Poisson-Lie groups and complete integrability. I. Drinfeld bigebras, dual extensions and their canonical representations

Abstract: This is the first part of a work on Poisson structures on Lie groups, complete integrability and Drinfeld quantum groups. In sections 1 and 2 we establish the algebraic preliminaries of the theory. Section 1 deals with a new kind of extension of Lie algebras, called twilled extensions (in French, extensions croisees), and with the special case of the dual extensions which are exactly the Drinfeld-Lie algebras. Section 2 deals with the exact Lie bigebras arising from the solutions of the classical and modified Yang-Baxter equations. The case of the non antisymmetric solutions (quasitriangular bigebras) is emphasized. In section 3 we study the equivariant one-forms and equivariant families of vector fields on a Lie group, and we introduce the notion of the Schouten curvature. In section 4 we prove the existence of the canonical representations of a twilled extension in the space of smooth functions on the Lie group factors, when these are connected and simply connected. Part II will study the Poisson and Lie-Poisson structures on groups, while Part III will connect this theory with that of the bihamiltonian structures and complete integrability On considere une notion nouvelle d'extensions d'algebres de Lie que l'on appelle extensions croisees et l'on traite le cas particulier des extensions duales, qui sont exactement les bigebres de Lie de Drinfeld. On etudie les bigebres de Lie exactes definies par des solutions de l'equation de Yang-Baxter classique ou modifiee. Cas des solutions antisymetriques. On etudie les formes equivariantes et les familles de champs de vecteurs equivariantes sur un groupe de Lie et on introduit la notion de courbure de Schouten. On montre l'existence des representations canoniques d'une extension croisee dans les espaces de fonctions lisses sur les groupes de Lie facteurs, lorsque ceux-ci sont connexes et simplement connexes

An infinite class of new conformal field theories with extended algebras

Abstract: A large class of 2D conformal field theories with extended Virasoro algebras related to the GKO construction on the coset SU(2)⊗SU(2)/SU(2) is introduced. Through a Feigin-Fuchs construction the Kac formula is deduced. Characters of the highest-weight irreducible representations are given in terms of the GKO decomposition branching functions. Modular invariant partition functions are constructed and an A-D-E classification based on a triple of simply-laced Lie algebras is analyzed in detail.

Graded Lie algebras and generalized Jordan triple systems

Abstract: One frequently encounters (real) semisimple graded Lie algebras in various branches of differential geometry (e.g. [16], [9], [14], [18]). It is therefore desirable to study semisimple graded Lie algebras, including those which have been studied individually, in a unified way. One of our concerns is to classify (finite-dimensional) semisimple graded Lie algebras in a way that enables us to construct them.

Irreducibility of Discrete Series Representations for Semisimple Symmetric Spaces

Abstract: Publisher Summary This chapter discusses irreducibility of discrete series representations for semi-simple symmetric spaces. It explains the term translation principle that refers to the idea of studying infinite dimensional representations of reductive Lie algebras by investigating their tensor products with the rich, complicated, and well-understood family of finite-dimensional representations of a connected reductive Lie group. The chapter also describes a criterion for a translation functor to take irreducibles to irreducible. A translation functor is a sum of composites of elementary translational functors. To make good use of the translation functors, one needs a way to compute them effectively. This is provided by the conceptually more subtle idea of coherent families.

The analogy between spin glasses and Yang–Mills fluids

Abstract: A dictionary of correspondence is established between the dynamical variables for spin‐glass fluid and Yang‐Mills plasma. The Lie‐algebraic interpretation of these variables is presented for the two theories. The noncanonical Poisson bracket for the Hamiltonian dynamics of an ideal spin glass is shown to be identical to that for the dynamics of a Yang–Mills fluid plasma, although the Hamiltonians differ for the two theories. This Poisson bracket is associated to the dual space of an infinite‐dimensional Lie algebra of semidirect‐product type.

The dynamics of a class of quantum mean‐field theories

Abstract: An infinite quantum system with correctly defined dynamics τQ as an automorphism group of a C*‐algebra C of observables is determined by any continuous unitary representation U(G) of a connected Lie group G, as well as by an arbitrary differentiable real function Q on the dual space g * to the Lie algebra g of G with the canonically defined Poisson flow φQ on g *. For specific choices of Q and G, the system can be obtained as the thermodynamic limit of a net of finite lattice systems with the mean‐field type interaction of Hepp and Lieb [Helv. Phys. Acta 46, 573 (1973)]. A simple nontrivial model of this type is the quasispin BCS model of superconductivity in the strong coupling limit, or a corresponding model of the Josephson junction. A peculiar feature of the considered models is τQ noninvariance of the usually considered C*‐algebra A of quasilocal observables, as well as an important role of classical dynamics of a set of macroscopic (intensive) observables Qφ in the description of τQ. The work is res...