Showing papers on "Symplectic vector space published in 1988"

The unregularized gradient flow of the symplectic action

Abstract: The symplectic action can be defined on the space of smooth paths in a symplectic manifold P which join two Lagrangian submanifolds of P. To pursue a new approach to the variational theory of this function, we define on a subset of the path space the flow whose trajectories are given by the solutions of the Cauchy-Riemann equation with respect to a suitable almost complex structure on P. In particular, we prove compactness and transversality results for the set of bounded trajectories.

Symplectic geometry of the convariant phase space

Abstract: Develops a formalism for a covariant treatment of the phase space of field theories. Within that formalism, an expression for the symplectic form of a general Lagrangian field theory is presented. As examples, the author derives the symplectic forms for general relativity and for the Green-Schwarz covariant superstring action. Finally, an extension of the covariant phase space formalism to theories in superspace is given.

On Lie groups with left-invariant symplectic or Kählerian structures

Abstract: Left-invariant symplectic structure on a group G; properties of the corresponding Lie algebra g. A unimodular symplectic Lie algebra has to be solvable (see [1]). Symplectic subgroups and left-invariant Poisson structures on a group. Affine Poisson structures: an affine Poisson structure associated to g and admitting g * as a unique leaf corresponds to a unimodular symplectic Lie algebra and the associate group is right-affine. If G is unimodular and endowed with a left-invariant metric g, harmonic theory for the left-invariant forms. Kahlerian group is metabelian and Riemannianly flat. Decomposition of a simply connected Kahlerian group. A symplectic group admitting a left-invariant metric with a nonnegative Ricci curvature is unimodular and admits a left-invariant flat Kahlerian structure.

Moments and Reduction for Symplectic Groupoids

Abstract: A hamiltonian action of a Lie group G on a symplectic manifold M is generated by a momentum map J:M-*Q* which is equivariant with respect to the coadjoint representation. The reduction procedure of Meyer [14] and Marsden and Weinstein [13] consists of forming the quotient M^ =J~ (//) /G^ where /£ is an element of Q* and G^ is its coadjoint isotropy group. In recent years (see [6], for example) a property of / already known to Lie [9] has been recognized as essential: J is a Poisson map from M to g* with its Lie-Poisson structure. This suggests the problem of replacing g* by an arbitrary Poisson manifold P3 but the question immediately arises as to what object will play the role of the group G. This object having just been identified by Karasev [7] and one of us [3] [20] as a symplectic groupoid^ the purpose of the present paper is to extend the reduction procedure to symplectic groupoid actions. An important stimulus for our work has been the reduction theory for Poisson Lie group actions developed in [16]. Using Drinfel'd's notion of Poisson Lie group [4], Semenov-Tian-Shansky explained the hamiltonian behavior of the dressing transformations which arise from the inverse-scattering approach to completely integrable systems. A new theory was necessary because dressing transformations do not preserve the Poisson structure of the spaces on which they act. In a sequel to this paper5 we hope to study how Poisson Lie groups and

Standard form of four-dimensional symplectic quadratic maps.

Abstract: It is shown that all four-dimensional symplectic quadratic maps generated by a generating function should be in a form that can be transformed into two coupled (two-dimensional) H\'enon maps. The latter is called the standard form of the four-dimensional symplectic quadratic map.

A characterization of the type of the Cauchy-Hua measure on real symmetric matrices

Abstract: The Cauchy-Hua measure is the probability on the set ofn×n real symmetric matrices of densitys ↦ Cn/det(I + s2)(n + 1)/2. For any probability measure μ on the set ofn×n real symmetric matrices, we define (if μ verifies an additional condition) the image of μ by a 2n×2n real symplectic matrix, and we introduce the type of μ. Then, the type of the Cauchy-Hua measure is characterized from its invariance by the symplectic group.

Homology of Symplectic

Abstract: In this paper we compute the homology of the infinite Lie algebra of orthogonal (resp. symplectic) matrices for an associative ring with involution over a characteristic zero field. It is shown (Theorem 5.5) to be the graded symmetric algebra over skew-dihedral homology. The surprising result is that the orthogonal and symplectic algebras have the same homology in the stable range. Skew-dihedral homology is a variation of cyclic homology, which is studied in detail in [L]. The proof of the main theorem is based on computations in invariant theory (cf. [Pl, P2, W]) for orthogonal and symplectic matrices. It follows the lines of proof of the analogue result for matrices given in [L-Q]. We also determine exactly the stable range (Theorem 7.1) and the first obstruction to stability (Theorem 7.3) in the symplectic case. In the first part we give a comprehensive and detailed review of invariant theory for orthogonal and symplectic matrices. In particular we make explicit the relations between the hyperoctahedral group, the trace iden- tities for matrices and the invariant space of the tensor algebra of matrices. The homology result in the stable range was also announced in [F-T]. Conventions. Vector spaces will be abbreviated as spaces. If the group G is acting on the space V, the space of invariants is VG = { VE V 1 VgE G, g. v = v} and the space of coinvariants is V, = V/{ g. v - v I g E G, v E V}. The signature of a permutation c is denoted E(O) = + 1. 93