Volume 31, Issue 3, March 1990
Index of content:

Presentation of the constant r‐term Krichever–Novikov‐type algebras
View Description Hide DescriptionFor each constant r‐term Krichever–Novikov‐type algebra, a minimal set of defining generators is given from which all the operators of the algebra can be constructed by sequential definitions. A finite system of polynomial conditions on the defining generators is given, guaranteeing the commutation relations of the full algebra.

Fundamental representations of the m‐principal realization of g l _{∞}
View Description Hide DescriptionThe m‐principal realization of g l _{∞} is introduced and its fundamental representations are defined on the representation space associated with the homogeneous Heisenberg subalgebra of a ^{(1)} _{ m } introduced by Frenkel and Kac. The Hirota equations associated with the representations are briefly reviewed. The main results concern the reduction of the algebras to the finite rank affine Lie algebras g l ^{(1)} _{ n } and a ^{(1)} _{ n−1} where it is shown that a complete nonredundant set of realizations can be obtained. This means that the associated Hirota equations can be obtained directly from the reduction of the Hirota equations for the fundamental m‐principal representations of g l _{∞} without requiring the complete nonredundant set of fundamental representations of the Lie algebra a ^{(1)} _{ n−1} itself.

Representations of the braid group obtained from quantum sl(3) enveloping algebra
View Description Hide DescriptionThe quantum Clebsch–Gordan (CG) coefficients for the coproduct 6×6 of the quantum sl(3) enveloping algebra are computed. Based on the representation 6, the representation of the braid group and the corresponding link polynomial are obtained. The link polynomials based on the representations of the quantum sl(3) enveloping algebra with a one‐row Young tableau are discussed.

The potential group approach and hypergeometric differential equations
View Description Hide DescriptionThis paper proposes a generalized realization of the potential groups SO(2,1) and SO(2,2) to describe the confluent hypergeometric and the hypergeometric equations, respectively. It implies that the classes of Schrödinger equations with solvable potentials whose analytical solutions are related to the confluent hypergeometric and the hypergeometric functions can be realized in terms of the above group structure.

Parametrization of SU(n) with n−1 orthonormal vectors
View Description Hide DescriptionA generalization to SU(n) of a well‐known relation in SU(2) is proposed. It relies on the observation that an element of SU(n) has associated with it in a natural way n−1 orthonormal vectors in R ^{ n 2−1}. The meaning of these n−1 vectors is discussed as they relate to the geometry of the adjoint representation of SU(n).

Local realizations of kinematical groups with a constant electromagnetic field. I. The relativistic case
View Description Hide DescriptionThis paper is devoted to the study of the description of elementary physical systems interacting with an external constant electromagnetic field and the construction of their differential wave equations from a group‐theoretical point of view. In this context certain local realizations of the Poincaré group are studied. The linearization of this problem is carried out by building the associated representation group that turns out to be the well‐known Maxwell group. In this way the usual method (concerning local realizations) that has been employed in studying free systems to the interacting case is extended.

Highest weight irreducible unitary representations of Lie algebras of infinite matrices. I. The algebra gl(∞)
View Description Hide DescriptionTwo classes of irreducible highest weight modules of the general linear Lie algebra gl(∞), corresponding to two different Borel subalgebras, are constructed. Both classes contain all unitary representations. Within each module a basis is introduced. Expressions for the transformation of the basis under the action of the algebra are written down.

On induced scalar products and unitarization
View Description Hide DescriptionInfinitesimal unitarity of representations of the simple real Lie algebras su(2), su(1,1), and so(4) is discussed with respect to scalar products induced by sesquilinear forms on their universal enveloping algebras. Sesquilinear forms are explicitly calculated. The Verma modules, their irreducible quotients, their irreducible submodules, and their infinitesimal unitarizability are discussed.

Integral bounds for radar ambiguity functions and Wigner distributions
View Description Hide DescriptionAn upper bound is proved for the L ^{ p } norm of Woodward’s ambiguity function in radarsignal analysis and of the Wigner distribution in quantum mechanics when p>2. A lower bound is proved for 1≤p<2. In addition, a lower bound is proved for the entropy. These bounds set limits to the sharpness of the peaking of the ambiguity function or Wigner distribution. The bounds are best possible and equality is achieved in the L ^{ p } bounds if and only if the functions f and g that enter the definition are both Gaussians.

On Hamiltonian systems in two degrees of freedom with invariants quartic in the momenta of form p ^{2} _{1} p ^{2} _{2}⋅⋅⋅
View Description Hide DescriptionA search is made for autonomous Hamiltonian systems in two degrees of freedom which admit a second invariant quartic in the momenta with leading term p ^{2} _{1} p ^{2} _{2}/ 2. A sufficient condition for the resulting functional equation to possess solutions is deduced and a family of integrable systems is identified, which under the equivalence class of linear transformations reduce to a simpler integrable system found originally by Bozis. The method of Lax pairs is used to find further solutions to the functional equation and give new classes of integrable but nonseparable Hamiltonians.

On the Painlevé classification of partial differential equations
View Description Hide DescriptionThe classification of partial differential equations from the point of view of the singular point analysis is suggested. The general form of the equation for the Painlevé resonances is derived. The Painlevé‐type classification of semilinear second‐order polynomialpartial differential equations in two independent variables is performed.

The wetted solid—A generalization of the Plateau problem and its implications for sintered materials
View Description Hide DescriptionA new generalization of the Plateau problem that includes the constraint of enclosing a given region is introduced. Physically, the problem is important insofar as it bears on sintering processes and the structure of wetted porous media. Some primal and dual characterizations of the solutions are offered and aspects of the problem are illustrated in one and two dimensions in order to clarify the combinatorial elements and demonstrate the importance of numerous local minima.

The Duistermaat–Heckman integration formula on flag manifolds
View Description Hide DescriptionAn exposition is given of various geometrical properties of flag manifolds and of the Duistermaat–Heckman integration formula as applied to flag manifolds.

Stochastic action of dynamical systems on curved manifolds. The geodesic interpolation
View Description Hide DescriptionDynamical systems on curved manifolds, with a Lagrangian at most quadratic in the velocity, are considered. The classical action functional, on some finite time interval, is well defined for any smooth trial trajectory in configuration space. The action functional is at the basis of Lagrangianvariational principles, from which all dynamical properties of the system can be derived. Here the problem of extending the action functional from the smooth deterministic trajectories of classical mechanics to the very irregular random trajectories of diffusions in configuration space is considered. In this way the action becomes a functional of the trial diffusion processes and can be put at the basis of stochastic variational principles. Since the problem is beset with ultraviolet divergences, the general strategy of renormalization theory is followed, by regularizing the trial diffusion processes through piecewise smooth geodesic lines for a generic given connection on the manifold. After cutoff removal and infinite counterterm subtraction, the quadratic part of the action shows a residual dependence on the generic regularizing connection field. Therefore, in the frame of this geodesic interpolation strategy, it is shown that a change in the connection field is equivalent to a well‐defined renormalization of the scalar potential. These results apply to the problem of quantization of generic dynamical systems on curved manifolds, in particular to the definition of Feynman path integrals on curved configuration spaces.

Space‐times admitting special affine conformal vectors
View Description Hide DescriptionSpace‐times admitting a special affine conformal vector (SACV) are shown to be precisely the space‐times that admit a special conformal Killing vector. All possible SACV space‐times are listed together with the corresponding SACV’s and covariantly constant tensors.

The group theoretic analysis of hyperheavenly equations
View Description Hide DescriptionContact symmetries of generalized hyperheavenly and hyperheavenly equations are investigated. It is shown that the groups of contact transformations admitted by these equations are the first prolongations of appropriate point transformation groups.

Effects of the shear viscosity on the character of cosmological evolution
View Description Hide DescriptionBianchi type I cosmological models are studied that contain a stiff fluid with a shear viscosity that is a power function of the energy density, such as η=αε^{ n }. These models are analyzed by describing the cosmological evolutions as the trajectories in the phase plane of Hubble functions. The simple and exact equations that determine these flows are obtained when 2n is an integer. In particular, it is proved that there is no Einstein initial singularity in the models of 0≤n<1. Cosmologies are found to begin with zero energy density and in the course of evolution the gravitational field will create matter. At the final stage, cosmologies are driven to the isotropic Friedmann universe. It is also pointed out that although the anisotropy will always be smoothed out asymptotically, there are solutions that simultaneously possess nonpositive and non‐negative Hubble functions for all time. This means that the cosmological dimensional reduction can work even on matter fluid having shear viscosity. These characteristics can also be found in any‐dimensional models.

Interaction of null dust clouds fronted by impulsive plane gravitational waves
View Description Hide DescriptionThis paper discusses the energy‐momentum tensorT ^{μν} in the region of interaction of a space‐time in which two colliding plane impulsive gravitational waves, each followed by a null dust cloud, exist. It is shown that in the interaction region T ^{μν} is of three types: (i) that of two noninteracting null dusts; (ii) that of a scalar field (equivalently an irrotational perfect fluid with energy density equal to pressure), and (iii) that of the sum of two independent non‐interacting scalar fields [equivalently a complex scalar field or an anisotropic perfect fluid with energy density w and pressures (w,π,π)].

The graviton propagator in maximally symmetric spaces
View Description Hide DescriptionVacuum metric perturbation two‐point functions are found for maximally symmetric backgrounds of arbitrary dimension n, using tensor mode functions on the n‐sphere and a general gauge‐fixing term. The gauge‐invariant part of the resulting graviton propagator is isolated, and is fully evaluated for de Sitter spaces in terms of functions of the geodesic separation.

Phase‐spaces and dynamical descriptions of infinite mean‐field quantum systems
View Description Hide DescriptionFollowing the work of P. Bona [J. Math. Phys. 2 9, 2223 (1988)], a class of infinite mean‐field quantum lattice systems is considered within the framework of algebraic quantum mechanics, and the following question is discussed: Which are the possible dynamical descriptions of the system, i.e., in which representations π of the system’s quasilocal C ^{*}‐algebra A do the local time evolutions converge in the thermodynamic limit to an automorphism group of a certain enlarged C ^{*}‐algebra C_{π} containing π(A) and (some of) the classical observables of the system in π. To this end we associate a ‘‘classical state space’’ E _{π} to any representation π (which is a subset of the phase‐space E⊇R^{ L } defined by Bona for the largest possible representation), and show how it is characterized by the π‐normal states, thereby obtaining a way to determine E _{π} explicitly in many cases. The answer to the question posed then reads: The limiting (Heisenberg) dynamics exists in π on C_{π} if and only if the set E _{π} is invariant under the flow φ^{ Q } _{ t } on E which describes the time evolution of classical observables. Conversely, to any invariant, closed subset B of E, there exists a dynamical description of the quantum system with classical state space B.