Showing papers on "Symplectic vector space published in 1980"

On the structure of symplectic operators and hereditary symmetries

Abstract: In the last fifteen years, there has been a remarlmble development in the exact analysis of certain nonlinear evolution equations, like tbe Korteweg-de Vries equation. I t is weH known that among the surprising features of these so-called exactly solvable equations is the possession of infinitely many symmetries and conservation laws, of N-soliton solutions and Bäcklund transformations. It has turned out that considering an operator which maps symmetries (1) onto symmetries of a given equation yields a useful approach to all these features. 'l'his operator is called a strong synunetry (2 ) ( or recursion operator (3 ) ). It is particularly useful because its transpose generates conserved covariants from given ones and because its eigenfunctions are also symmetries (which actually characterize the N-soliton solutions). One way of finding strong symmetries is to use the fact that any translation-invariant operator fP(u), possessing the property defined by v below is a strong symmetry for the hierarchy of equations u, = ( fP(u) )\" Uz, n = 0, 1, 2, ... . These operators are called hereditary symmetries. The strong symmetries of all well-known exactly solvable equations are hereditary (2 ). Recently there has also been progress in understanding the Hamiltonian structure of these evolution equations (4). An evolution equations is said to be a Hamiltonian system if it can be written in the form u 1 = O(u)f(u), where O(u) is impletic (which is, roughly speaking, the same as saying that 0-1(u) is sympletic) and where f(u) is the gradient of a suitable potential. For these systems tbe operator-valued function O(u) is of particular interest because it is a N oether operator, i.e. it maps conserved covariants onto symmetries. Our paper is related to .1\\Iagri's work wbo considered bi-Hamiltonion systems u, = = 81(u)j1(u) = 82(u)/2(u) and who sbowed that these equations have fP(u) =81(u) 8;(u) as strong symmetries.

On compact homogeneous symplectic manifolds

Abstract: © Annales de l’institut Fourier, 1980, 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.

On six-dimensional canonical realizations of the so(4,2) algebra

Abstract: On each six‐dimensional symplectic manifold a coordinate‐free realization of the so(4,2) algebra can be constructed, the generators of which satisfy the polynomial relations fulfilled by the so(4,2) generators associated with the Kepler problem. This realization contains as a particular case several realizations of so(4,2) known in the literature. An expression of the symplectic form on a six‐dimensional symplectic manifold, in terms of the so(4,2) generators defined on this manifold, is obtained. In particular, on the six‐dimensional orbit of the SO(4,2) group in so(4,2) this symplectic form coincides with the symplectic form introduced by Kirillov Kostant and Souriau. The symplectic form is given a Darboux expression with the aid of three pairs of canonically conjugated variables, which are a generalization of the Delaunay elements defined in the Kepler problem.

The most degenerate irreducible representations of the symplectic group

Abstract: Matrix elements of the generators of the symplectic group Sp(2n) have been obtained for the most degenerate irreducible representations (m2n,?) and (?2n). These are the only two cases for n≳2 where the state labels according to the branching laws of Hegerfeldt give rise to an orthogonal set of basic vectors.

Homogeneous contact compact manifolds and homogeneous symplectic manifolds

Abstract: A homogeneous contact compact manifold can be considered as the total space of a principal circle bundle over a simply connected homogeneous symplectic manifold whose fundamental form determines an integral cohomology class. A similar result but assuming the contact manifold to be simply connected was given by Boothby and Wang. The proof we give here is independent of that of Boothby and Wang. We also prove that if a Lie group acts transitivelly on a simplectic manifold of integral class by diffeomorphism of the symplectic structure, there is a Lie group acting transitivelly on the total space of the bundle obtained from the symplectic form (Kobayashi's method) by diffeomorphisms of the contact structure. (The contact form is the correspondent connection form).

Genus minimizing in symplectic 4-manifolds**

Abstract: The authors show that a symplectically embedded surface in a symplectic 4-manifold with $b^+_2$ greater than one minimizes genus in its homology class.