Showing papers on "Lie group published in 1981"

The modified Lax and two-dimensional Toda lattice equations associated with simple Lie algebras

Abstract: We associate to each complex simple Lie algebra g a hierarchy of evolution equations; in the simplest case g = sl(2) they are the modified KdV equations. These new equations are related to the two-dimensional Toda lattice equations associated with g in the same way that the modified KdV equations are related to the sinh-Gordon equation.

Dimension and Character Formulas for Lie Supergroups

Abstract: A character formula is derived for Lie supergroups. The basic technique is that of symmetrization and antisymmetrization associated with Young tableaux generalized to supergroups. We rewrite the characters of the ordinary Lie groups U(N), O(N), and Sp(2N) in terms of traces in the fundamental representation. It is then shown that by simply replacing traces with supertraces the characters of certain representations for U(N/M) and OSP(N/2M) are obtained. Dimension formulas are derived by calculating the characters of a special diagonal supergroup element with (+1) and (−1) eigenvalues. Formulas for the eigenvalues of the quadratic Casimir operators are given. As applications, the decomposition of a representation into representations of subgroups is discussed. Examples are given for the Lie supergroup SU(6/4) which has physical applications as a dynamical supersymmetry in nuclei.

Horseshoes and Arnold Diffusion for Hamiltonian Systems on Lie Groups

Abstract: : The Melnikov theory of perturbations of Hamiltonian systems containing homoclinic orbits is extended to systems containing canonical variables belonging to the coadjoint orbits of a Lie group. This is applied to the free rigid body with attachments and to the nearly symmetric top. These systems are thereby shown to have transverse homoclinic manifolds in an appropriate return map and therefore have complex dynamics. In particular, the heavy top and rigid body with one attachment are shown to contain horseshoes and therefore have no additional analytic integrals, while the rigid body with several attachments exhibits Arnold diffusion.

Massless Particles, Conformal Group and De Sitter Universe

Abstract: We first review a recent result on the uniqueness of the extension to the conformal group of massless representations of the Poincar\'e group. By restricting these representations to SO(3,2) we obtain a unique definition of massless particles in de Sitter space. This definition is compared with the concept of masslessness that arises from considerations of gauge invariance. Next, we recall the startling fact that the direct product of two Dirac singleton representations of SO(3,2) decomposes into a direct sum of the massless representations of SO(3,2). A theory of interacting singleton fields is developed and a simple expression is given for the intertwining operator between massless fields and two-singleton fields. Finally, we discuss the behavior of these massless representations with respect to the contraction of the deSitter group to the Poincar\'e group.

Subadditivity of homogeneous norms on certain nilpotent Lie groups

Abstract: Let N be a Lie group with its Lie algebra generated by the leftinvariant vector fields Xi,.. . ,Xk on N. An explicit fundamental solution for the (hypoelliptic) operator L = Xx + ■ ■ ■ + Xk on N has been obtained for the Heisenberg group by Folland [1] and for the nilpotent (Iwasawa) groups of isometries of rank-one symmetric spaces by Kaplan and Putz [2]. Recently Kaplan [3] introduced a (still larger) class of step-2 nilpotent groups N arising from Clifford modules for which similar explicit solutions exist. As in the case of L being the ordinary Laplacian on N = R*, these solutions are of the form g t—» const|J ^|j2—m, g e N, where the "norm" function || || satisfies a certain homogeneity condition. We prove that the above norm is also subadditive. Let u, b be real finite-dimensional vector spaces each equipped with a positive definite quadratic form | \2. Let u: uXMobea composition of these quadratic forms [3, p. 148] normalized in the sense that n(u0, v) = v for some u0 G u. Define = + 4|u'|2 + 2|u|2|t;f, (1) Received by the editors March 24, 1980 and, in revised form, September 15, 1980 and November 5, 1980. AMS (MOS) subject classifications (1970). Primary 43A80, 22E15; Secondary 35C05, 35H05.

Control systems on semi-simple Lie groups and their homogeneous spaces

Abstract: © Annales de l’institut Fourier, 1981, tous droits réservés. L’accès aux archives de la revue « Annales de l’institut Fourier » (http://annalif.ujf-grenoble.fr/) 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.

Integrable geodesic flows on homogeneous spaces

Abstract: A method is exposed which allows the construction of families of first integrals in involution for Hamiltonian systems which are invariant under the Hamiltonian action of a Lie group G. This is applied to invariant Hamiltonian systems on the tangent bundles of certain homogeneous spaces M = G/K. It is proved, for example, that every such invariant Hamiltonian system is completely integrable if M is a real or complex Grassmannian manifold or SU(n + 1)/SO(n + 1) or a distance sphere in ℂPn+1. In particular, the geodesic flows of these homogeneous spaces are integrable.

The Lie Algebra Structure of Nonlinear Evolution Equations Admitting Infinite Dimensional Abelian Symmetry Groups

Abstract: Hereditary operators in Lie algebras are investigated. These are operators which are characterized by a special algebraic equation and their main property is that they generate abelian subalgebras of the given Lie algebra. These abelian subalgebras are infinite dimensional if the hereditary operator is not cyclic. As a consequence hereditary operators generate on a systematic level nonlinear dynamical systems which possess infinite dimensional abelian groups of symmetry transformations. We show that hereditary operators can be understood as special Lie algebra deformations with a linear interpolation property. In order to construct new hereditary operators out of given ones we study the permanence properties of these operators; this study of permanence properties leads in a natural way to a notion of compatibility. For local hereditary operators it is shown that eigenvector decompositions are time invariant (such an eigenvector decomposition is known" to characterize pure multisoliton solutions). Apart from the well-known equations (KdV, sine-Gordon, etc.), we give-as examples-many new nonlinear equations with infinite dimensional groups of symmetry transformations. A detailed analysis of the celebrated Korteweg-de Vries equation reveals that this nonlinear evolution equation possesses an infinite dimensional abelian group of symmetry transformations. This group of symmetry transformations is given by the resolvents of the so-called generalized KdV equations. And this striking property is shared by many other nonlinear evolution equations; Only to name a few: Burgers equation, sine-Gordon equation, Zakharov-Shabat equations, Gardner equation etc. Furthermore one discovers that for these equations (except Burgers equation) the structure of this abelian symmetry group is intimately connected with the existence (and description) of multisoliton solutions, and in addition connected to the existence of infinitely many conservation laws (via Noether's theorem or rather a suitable generalization thereof).

On the symmetry of the Gibbs states in two-dimensional lattice systems

Abstract: Under fairly general conditions if a two dimensional classical lattice system has an internal symmetry groupG, which is a compact connected Lie group, then all Gibbs states areG-invariant.

Super Lie groups: global topology and local structure

Abstract: A general mathematical framework for the super Lie groups of supersymmetric theories is presented. The definition of super Lie group is given in terms of supermanifolds, and two theorems (analogous to theorems in classical Lie group theory) are proved. The relationship of the super Lie groups defined here to the formal groups of Berezin and Kac and the graded Lie groups of Kostant is analyzed.

Euler-Poisson equations on Lie algebras and the N-dimensional heavy rigid body

Abstract: The classical Euler-Poisson equations describing the motion of a heavy rigid body about a fixed point are generalized to arbitrary Lie algebras as Hamiltonian systems on coad-joint orbits of a tangent bundle Lie group. the N-dimensional Lagrange and symmetric heavy top are thereby shown to be completely integrable.

An exact invariant for a class of time‐dependent anharmonic oscillators with cubic anharmonicity

Abstract: An exact invariant is constructed for a class of time‐dependent anharmonic oscillators using the method of the Lie theory of extended groups. The presence of the anharmonic term imposes a constraint on the nature of the time dependence. For a subclass it is possible to obtain an energy‐like integral and a condition under which the motion is bounded.

Wave Front Sets of Representations of Lie Groups

Abstract: In the past few years the concept of wave front set [D] has proved fruitful for the theory of distributions and P.D.E. It seems it might also be of use in the representation theory of Lie groups. Its close relative, the singular spectrum of a hyperfunction, has already been discussed in a special context in [K-V] which served as the catalyst for this note. The purpose here is to define and discuss general properties of wave front sets of representations, and to give some examples.

Subgroups of Lie groups and separation of variables

Abstract: Separable systems of coordinates for the Helmholtz equation ΔdΨ =EΨ in pseudo‐Riemannian spaces of dimension d have previously been characterized algebraically in terms of sets of commuting second order symmetry operators for the operator Δd. They have also been characterized geometrically by the form that the metric ds2=gik(x)dxidxk can take. We complement these characterizations by a group theoretical one in which the second order operators are related to continuous and discrete subgroups of G, the symmetry group of Δd. For d=3 we study all separable coordinates that can be characterized in terms of the Lie algebra L of G and show that they are of eight types, seven of which are related to the subgroup structure of G. Our method clearly generalizes to the case d≳3. Although each separable system corresponds to a pair of commuting symmetry operators, there do exist pairs of commuting symmetries S1,S2 that are not associated with separable coordinates. For subgroup related operators we show in detail just...

The theory of Eisenstein systems

Abstract: It is called a Hecke operator. It commutes with ∆, and acts on its eigenspaces. The study of these operators and of those appearing in Hecke’s work promises to be of considerable importance for diophantine problems, in particular for the investigation of the Dirichlet series to which the names of Artin and Hasse-Weil are attached. However the spectral theory of ∆ on Γ-invariant functions is a purely analytic problem, of interest in its own right for any discrete subgroup Γ of SL(2,R) whose fundamental domain has finite volume. If the quotient of the upper half-plane by Γ is compact the spectrum is discrete, but otherwise there is a continuous spectrum and the corresponding eigenfunctions are called Eisenstein series. If the quotient is not compact there are cusps. By way of illustration we may assume that ∞ is a cusp. This means that Γ contains a subgroup of the form

Renormalizable Theories with Symmetry Breaking

Abstract: The description of symmetry breaking proposed by K. Symanzik within the framework of renormalizable theories is generalized from the geometrical point of view. For an arbitrary compact Lie group, a soft breaking of arbitrary covariance, and an arbitrary field multiplet, the expected integrated Ward identity are shown to hold to all orders of renormalized perturbation theory provided the Lagrangian is suitably chosen. The corresponding local Ward identity which provides the Lagrangian version of current algebra through the coupling to an external classical Yang-Mills field, is then proved to hold up to the classical Adler-Bardeen anomaly whose general form is written down. The BPHZ renormalization scheme is used throughout in such a way that the algebraic structure analyzed in the present context may serve as an introduction to the study of fully quantized gauge theories.

Harmonic Analysis in Euclidean Spaces

Abstract: Contains sections on Several complex variables, Pseudo differential operators and partial differential equations, Harmonic analysis in other settings: probability, martingales, local fields, and Lie groups and functional analysis.

Singularity theory : selected papers

Abstract: Intoduction C. T. C. Wall 1. Singularities of smooth mappings (Volume 23, 1968) 2. On matrices depending on parameters (Volume 26, 1971) 3. Remarks on the stationary phase method and Coxeter numbers (Volume 28, 1973) 4. Normal forms of functions in neighbourhoods of degenerate critical points (Volume 29, 1974) 5. Critical points of smooth functions and their normal forms (Volume 30, 1975) 6. Critical points of functions on a manifold with boundary, the simple Lie groups Bk, Ck and F4 and singularities of evolutes (Volume 33, 1978) 7. Indices of singular points of 1-forms on a manifold with boundary, convolution of invariants of reflection groups and singular projections of smooth surfaces (Volume 34, 1979).

A group-theoretic approach to discrete-time non-linear controllability

Abstract: The controllability of discrete-time non-linear systems is studied through the use of Ritt's formal differential groups which were introduced thirty years ago. A condition for weak local controllability is given in terms of Lie algebra. It is very similar to the one obtained in the continuous-time case. In the case of regular (or bilinear) systems this analysis leads to Lie group techniques. Various examples are discussed.