Showing papers on "Symplectic vector space published in 1990"

A New Capacity for Symplectic Manifolds

Abstract: Publisher Summary This chapter discusses the axioms for a symplectic capacity. At present, very little is known about the nature of a symplectic map. The axioms for a symplectic capacity are useful for a more systematic study of the symplectic embedding problem; they led to a new rigidity result. The axioms such as monotonicity, conformality, local nontriviality, and nontriviality do not determine a capacity function uniquely. There are many ways to construct different capacity functions. The capacity of every symplectic manifold is positive or ∞. Every capacity singles out the subgroup of homeomorphisms of R2n preserving the capacity. The elements of this distinguished group of homeomorphisms have the additional property that they are symplectic or anti-symplectic in case they are differentiable. The associated pseudogroup can be used to define a topological symplectic manifold.

Elliptic methods in symplectic geometry

Abstract: The past few years have seen several exciting developments in the field of symplectic geometry, and a beginning has been made towards solving many important and hitherto inaccessible problems. The new techniques which have made this possible have come both from the calculus of variations and from the theory of elliptic partial differential operators. This paper describes some of the results that Gromov obtained using elliptic methods, and then shows how Floer applied these elliptic techniques to develop a new approach to Morse theory, which has important applications in the theory of 3and 4-manifolds as well as in symplectic geometry. To give some idea of the context of their results, we begin with a section on symplectic geometry, which concentrates on questions about symplectic diffeomorphisms. For more general recent surveys of the field, see for example [A2], [E2], [Gl], [G3], [H2], [VI], and [V2]. The contents of this paper are:

A symplectic geometry approach to generalized Casson's invariants of 3-manifolds

Abstract: 1. In 1985 lectures at MSRI, Andrew Casson introduced an integer valued invariant À(M) for any oriented integral homology 3-sphere M. This invariant has many remarkable properties; detailed discussions of some of these can be found in an exposé by S. Akbulut and J. McCarthy (see [AM]). Roughly, A(M) measures the 'oriented' number of irreducible representations of the fundamental group n{(M) in SU(2). In the preceding article of this journal, Kevin Walker [W] described results of his thesis which yield an invariant k{M ) of an arbitrary oriented rational homology 3-sphere (RHS: HX(M , Q) = 0) which extends Casson's invariant. His creative methods give generalizations of the properties which Casson had earlier shown for oriented integral homology 3-spheres (IHS: H{(M , Z) = 0). For homology lens spaces, Boyer and Lions [BL] have independently obtained an inductive definition of X{M ) . Earlier, a different extension of Casson's invariant to certain rational homology spheres, which does not equal Walker's invariant, had been studied by S. Boyer and A. Nicas [BN]. In all of the above works, one is considering only representations into SU(2). The present announcement solves the problem, which has been emphasized by Atiyah [A], of producing generalizations of Casson's invariant to invariants of M that would roughly measure the 'oriented' number of representations of nx (M) in G = SU(n), for each n > 2 . We introduce XG(M ) , an invariant which is defined for an arbitrary oriented rational homology 3-sphere (RHS).

Symplectic geometry and its applications

Abstract: Contents: Linear Symplectic Geometry: Symplectic Manifolds.- Symplectic Geometry and Mechanics. Contact Geometry. Lagrangian and Legendre Singularities. Lagrangian and Legendre Cobordism.- References.

The Jordan-Schwinger representations of Cayley-Klein groups. III. The symplectic groups

Abstract: The symplectic Cayley–Klein groups are defined as the groups that are obtained by the contractions and analytical continuations of the classical symplectic groups. The Jordan–Schwinger representations of the groups under consideration are discussed based on the mixed sets of creation and annihilation operators of boson or fermion type. The matrix elements of finite group transformations are obtained in the bases of coherent states.

Field extensions and isotropic subspaces in symplectic geometry

Abstract: Let L/k be a finite field extension and let (V, B) be a finite dimensional symplectic space over L. We examine the action of the symplectic group SpL(V) on the set of B′-isotropic k-subspaces of V, where B′=ψ°B is the k-symplectic form induced by a ‘trace’ map ψ:L→k. The orbits are completely classified in the case of a quadratic extension and for maximal B′-isotropic subspaces in the case of a cubic extension; the number of orbits of maximal B′-isotropic subspaces is shown to be infinite if the degree of the extension is at least 4.

Fourier-optics approach to the symplectic group

Abstract: The n-dimensional symplectic group is discussed and classified in terms of operators for the first time to my knowledge. The symplectic embedding theorem is proved by using Kronecker block multiplication. In addition, the fundamental relation of Fourier optics is proved. It follows that all operator relations have been proved for Fourier optics. In combination with a previous result, the decomposition theorem for nonsingular matrices [ J. Math. Phys.12, 1772 ( 1971)], which is extended here somewhat, it follows that Fourier optics (excluding the phase-conjugate operator) is isomorphic to the symplectic group, the group of linear canonical transforms, and the group of first-order systems. This implies that every mathematical result concerning the symplectic group can be understood simply as a property of ideal cylindrical lenses under the paraxial approximation.

Symplectic orbits in quantum state space

Abstract: All symplectic orbits of the action of an arbitrary compact connected Lie group on the space of density operators are found. It is shown that there is only one orbit that is Kahler (orbit of coherent states).

Representations of parabolic subgroups of the symplectic group

Abstract: It is proved that the topological space of irreducible unitary representations of a parabolic subgroup of the complex symplectic group contains an open everywhere dense set homeomorphic to the space of irreducible representations of some reductive group.

Nongaussian linear filtering, identification of linear systems and the symplectic group

Abstract: Consider stochastic linear dynamical systems, dx=Axdt+Bdw,dy=Cxdt+dv,y(0)=0, x(0) a given initial random variable independent of the standard independent Wiener noise processes w,v. The matrices A,B,C are supposed to be constant. In this paper I consider two problems. For the first one A,B and C are supposed known and the question is how to calculate the conditional probability density of x at time t given the observations y(s), 0≤s≤t in the case that x(0) is not necessarily gaussian. (In the gaussian case the answer is given by the Kalman-Bucy filter). The second problem concerns identification, i.e. the A,B,C are unknown (but assumed constant so that dA=0, dB=0, dC=0), and one wants to calculate the joint conditional probability density at time t of (x,A,B,C), again given the observations y(s), 0≤s≤t. The methods used rely on Wei-Norman theory, the Duncan-Mortensen-Zakai equation and a “real form” of the Segal-Shale-Weil representation of the symplectic group Spn(R).

Construction of a bilinear symmetric form in orthogonal vector space 1

Abstract: In this text we present unpublished results by Eugeniusz Kusak and Wojciech Leonczuk. They contain an axiomatic description of the class of all spaces 〈V ; ⊥ξ〉, where V is a vector space over a field F, ξ : V × V → F is a bilinear symmetric form i.e. ξ(x, y) = ξ(y, x) and x ⊥ξ y iff ξ(x, y) = 0 for x, y ∈ V . They also contain an effective construction of bilinear symmetric form ξ for given orthogonal space 〈V ; ⊥〉 such that ⊥=⊥ξ. The basic tool used in this method is the notion of orthogonal projection J(a, b, x) for a, b, x ∈ V . We should stress the fact that axioms of orthogonal and symplectic spaces differ only by one axiom, namely: x ⊥ y+ez&y ⊥ z+ex ⇒ z ⊥ x+ey. For e = −1 we get the axiom on three perpendiculars characterizing orthogonal geometry. For e = +1 we get the axiom characterizing symplectic geometry see [1].

Construction of a bilinear antisymmetric form in symplectic vector space 1

Abstract: In this text we will present unpublished results by Eugeniusz Kusak. It contains an axiomatic description of the class of all spaces 〈V; ⊥ξ〉, whereV is a vector space over a field F, ξ : V ×V → F is a bilinear antisymmetric form i.e. ξ(x,y) = −ξ(y,x) andx⊥ξ y iff ξ(x,y) = 0 for x, y ∈ V. It also contains an effective construction of bilinear antisymmetric formξ for given symplectic space 〈V;⊥〉 such that⊥=⊥ξ. The basic tool used in this method is the notion of orthogonal projection J (a,b,x) for a,b,x∈V. We should stress the fact that axioms of orthogonal and symplectic spaces differ only by one axiom, namely: x⊥ y+ εz& y⊥ z+ εx⇒ z⊥ x+ εy. For ε = +1 we get the axiom characterizing symplectic geometry. Forε = −1 we get the axiom on three perpendiculars characterizing orthogonal geometry see [5].

Symplectic structure on a grassmannian fibration

Abstract: The author describes a canonical structure on a Grassmannian fibration whose fiber is a Grassmann manifold of the tangent spaces of a smooth manifold. This structure generalizes the symplectic structure on the cotangent bundle. This symplectic form takes its values in a vector space or even in a vector bundle. This structure is canonical; it is uniquely defined by a smooth manifold. Bibliography: 5 titles.

Symplectic Structures, Complex, Structures, Symplectic Spinors and the Fourier Transform

Abstract: We have directed our attention to the problems raised by the Fourier transform on curved symplectic manifolds. The generalization of the remarkable tool offered by this transform in Rn to such manifolds is awkward and leads to the difficulty known as the ‘Maslov class’ or ‘index’. It appears that a geometrization of this transform is required before it can be turned into a global tool. In the process, the complex structure J adapted to the symplectic form and to an associated pseudo-riemannian form will prove to be closely linked with the Fourier transform, which can be identified, up to some constant factor, with the lifting J of J to a symplectic spin group. Then the geometrization can be carried out at once, if a few conditions, which hold in all usual cases, are satified; the geometric Fourier transform is the natural action of J on the symplectic spinors.

Algebra Deformations on Symplectic Manifolds

Abstract: To a (n = 2r)-dimensional paracompact symplectic manifold V, we associate the associative algebras C∞(V) and C∞(V, C) of differentiable real resp. complex-valued functions. F is the symplectic 2-form, dF = 0 and the rank of F is everywhere maximal.