Showing papers on "Symplectic vector space published in 1991"
01 Jan 1991
TL;DR: In this article, the authors show that any symplectic vector space has even dimension and any isotropic subspace is contained in a Lagrangian subspace and Lagrangians have dimension equal to half the dimension of the total space.
Abstract: Proposition 1.4. (1) Any symplectic vector space has even dimension (2) Any isotropic subspace is contained in a Lagrangian subspace and Lagrangians have dimension equal to half the dimension of the total space. (3) If (V1,ω1), (V2,ω2) are symplectic vector spaces with L1, L2 Lagrangian subspaces, and if dim(V1) = dim(V2), then there is a linear isomorphism φ : V1 → V2 such that φω2 = ω1 and φ(L1) = L2.
605 citations
TL;DR: In this paper, it was shown that if the regularity assumptions are dropped, the space MO is a union of symplectic manifolds, i.e., it is a stratified symplectic space.
Abstract: Let (M, w) be a Hamiltonian G-space with proper momentum map J: M -> g*. It is well-known that if zero is a regular value of J and G acts freely on the level set J '(0), then the reduced space MO = J- '(0)/G is a symplectic manifold. We show that if the regularity assumptions are dropped, the space MO is a union of symplectic manifolds; i.e., it is a stratified symplectic space. Arms et al. [2] proved that MO possesses a natural Poisson bracket. Using their result, we study Hamiltonian dynamics on the reduced space. In particular we show that Hamiltonian flows are strata-preserving and give a recipe for lifting a reduced Hamiltonian flow to the level set J-'(0). Finally we give a detailed description of the stratification of MO and prove the existence of a connected
532 citations
TL;DR: In this article, a 4-dimensional symplectic manifold with disconnected boundary of contact type is constructed and a collection of other results about symplectic manifolds with contact-type boundaries are derived using the theory of J-holomorphic spheres.
Abstract: An example of a 4-dimensional symplectic manifold with disconnected boundary of contact type is constructed. A collection of other results about symplectic manifolds with contact-type boundaries are derived using the theory ofJ-holomorphic spheres. In particular, the following theorem of Eliashberg-Floer-McDuff is proved: if a neighbourhood of the boundary of (V, ω) is symplectomorphic to a neighbourhood ofS2n−1 in standard Euclidean space, and if ω vanishes on all 2-spheres inV, thenV is diffeomorphic to the ballB2n.
260 citations
100 citations
TL;DR: In this article, an approach to the computation of Lagrangian invariant subspaces of a Hamiltonian or a symplectic matrix was proposed, which is the missing link in the solution of the open problem of constructing a stable structure-preserving QR -like method of complexity O(n 3 ) for the computations of invariant subsets of Hamiltonian and symplectic matrices.
68 citations
TL;DR: In this article, a Hamiltonian structure for the first variation equation of the Hamiltonian along a given dynamic solution is given, which is different from the well-known and elementary tangent space construction.
Abstract: This paper uses symplectic connections to give a Hamiltonian structure to the first variation equation for a Hamiltonian system along a given dynamic solution. This structure generalises that at an equilibrium solution obtained by restricting the symplectic structure to that point and using the quadratic form associated with the second variation of the Hamiltonian (plus Casimir) as energy. This structure is different from the well-known and elementary tangent space construction. Our results are applied to systems with symmetry and to Lie-Poisson systems in particular.
54 citations
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01 Jan 1991
TL;DR: In this article, the Hamilton dynamics constraints on dynamics contact spaces were introduced and the quantum-classical correspondence between affinely-rigid body and ellipsoidal figures of equilibrium was studied.
Abstract: Philosophical preliminaries geometry of bilinear forms and affine spaces symplectic spaces and symplectic Pfaff problem symplectic manifolds Newton mechanics, Galilean symmetry and the origin of the Hamilton theory basic concepts of the Hamilton dynamics constraints on dynamics contact spaces statistical concepts and the quantum-classical correspondence affinely-rigid body and ellipsoidal figures of equilibrium.
51 citations
43 citations
TL;DR: In this paper, the authors used the symplectic structure of the Hilbert space of a quantum system to derive a natural expression for the geometric phase present when the system performs a cyclic evolution.
36 citations
01 Jan 1991
TL;DR: In this paper, the authors studied the stochasticity of singular set of symplectic mtrices and derived a Maslov-type index theory for (degenerate) paths in symplectic groups, and therefore established the existence of periodic solutions of asymptotically linear Hamiltonian systems.
Abstract: A symplectic matrix M is singular, if det(M-I)=0. In this paper we study the struc-ture of the singular set of symplectic mtrices. We discuss the changes of the dimension ofthe null space and the determinant of the difference between a singular symplectic matrixand the identity matrix under rotational perturbations. The results obtained will be used todefine a Maslov-type index theory for (degenerate) paths in symplectic groups, and thereforeto establish the existence of periodic solutions of asymptotically linear Hamiltonian systems.
29 citations
TL;DR: In this article, a new interpretation of Poincare supersymmetric theories in terms of loop space symplectic geometry is presented, which opens a new point of view to a large class of problems, including the mechanism of supersymmetry breaking, structure of topological field theories and even aspects of quantum integrability.
TL;DR: Symplectic lattice maps as mentioned in this paper approximate the flow map of a Hamiltonian system to arbitrarily high order, and are a viable tool in the study of Hamiltonian systems, which can be thought of as the restriction of a symplectic map to an invariant lattice.
TL;DR: In this paper, the phase space of an evolution system with a pre-symplectic structure is considered and links with integrability, geometry of loop spaces, and Backlund transformations are traces.
Abstract: Matrices with entries being differential operators, that endow the phase space of an evolution system with a (pre)symplectic structure are considered. Special types of such structures are explicitly described. Links with integrability, geometry of loop spaces, and Backlund transformations are traces.
TL;DR: In this article, the authors present an account of this labelling which is based on the theory of bideterminants and is valid for symplectic groups in arbitrary characteristic, in which the authors show that it can be used to calculate the dimensions of the weight spaces of an irreducible module for the syrnplectic group.
Abstract: It was shown by R.C. King that symplectic standard tableaux can be used to calculate the dimensions of the weight spaces of an irreducible module for the syrnplectic group . In a recent paper. A. Berele gave a basis of this module, labelled by the symplectic standard tableaux of R. C King. The present paper gives an account of this labelling which is based on the theory of bideterminants and is valid for symplectic groups in arbitrary characteristic.
01 Jan 1991
TL;DR: In this article, it was shown that by taking stationary flows of integrable evolution equations on lattices, one obtains integrably symplectic maps and an alternative method based on the so-called nonlinearization of a scattering problem, and elucidate its intimate connections with the previous one.
Abstract: We show that by taking stationary flows of integrable evolution equations on lattices one obtains integrable symplectic maps. We also tersely discuss an alternative method based on the so-called nonlinearization of a scattering problem, and elucidate its intimate connections with the previous one. A few examples of possibly interesting integrable maps are presented.
TL;DR: In this paper, a new elementary geometric proof exploiting the positive curvature of complex projective space is presented for a basic lemma in the theory of lagrangian pairs in hermitian vector space.
01 Jan 1991
TL;DR: In this article, it was shown that given a Hamiltonian action of a compact and connected Lie group G on a symplectic manifold (M, ω) of finite type, there exists a linear symplectic action of G on some R 2n equipped with its standard symplectic structure such that the manifold can be realized as a reduction of this manifold with G.
Abstract: We show that given a Hamiltonian action of a compact and connected Lie group G on a symplectic manifold (M, ω) of finite type, there exists a linear symplectic action of G on some R 2n equipped with its standard symplectic structure such that (M, ω, G) can be realized as a reduction of this R 2n with the induced action of G.
01 Jul 1991
TL;DR: In this article, the classification theory of Arnold and Zakalyukin for singularities of Lagrange projections was generalized to projections that commute with a symplectic action of a compact Lie group and applied to the classification of infinitesimally stable corank 1 projections with ℤ2 symmetry.
Abstract: We generalise the classification theory of Arnold and Zakalyukin for singularities of Lagrange projections to projections that commute with a symplectic action of a compact Lie group. The theory is applied to the classification of infinitesimally stable corank 1 projections with ℤ2 symmetry. However examples show that even in very low dimensions there exist generic projections which are not infinitesimally stable.
TL;DR: In this article, the authors describe new connections between the characteristic-free representation theories of the symplectic group and the corresponding general linear group (Theorem 2.2 and 2.6).
Abstract: The purpose of this work is to describe some new connections between the characteristic-free representation theories of the symplectic group and the corresponding general linear group (Theorem 2.2 and Theorem 2.6).
TL;DR: In this paper, the q-deformation of SU(2) algebra at classical level, SUq,→0(2), in a Hamiltonian approach was obtained by deforming the symplectic structure on S2.
Abstract: By deforming the symplectic structure on S2, we get the q-deformation of SU(2) algebra at classical level, SUq,→0(2), in a Hamiltonian approach. Furthermore, we construct a set of operators on the line bundle over the deformed symplectic manifold such that they form SUq,→0(2) in Lie brackets and set up a nontrivial Hopf algebra with a parameter q only in such a classical Hamiltonian system. We also show that the deformations from S2 to are a set of quasiconformal transformations. The quantization via geometric approach of the system gives rise to the quantum q-deformed algebra SUq,(2), which has a Hopf algebraic structure with two independent parameters q and .
TL;DR: In this paper, it was shown that a Hermitian manifold is a complex space form if and only if the local reflections with respect to any holomorphic surface are symplectic, i.e., preserve the Kahler form.
Abstract: We prove that a Hermitian manifold is a complex space form if and only if the local reflections with respect to any holomorphic surface are symplectic, i.e., preserve the Kahler form.
TL;DR: A diffeomorphism of a finite-dimensional flat manifold which is canonoid with respect to all linear and quadratic Hamiltonians preserves the manifold's structure up to a factor as discussed by the authors.
Abstract: A diffeomorphism of a finite-dimensional flat symplectic manifold which is canonoid with respect to all linear and quadratic Hamiltonians, preserves the symplectic structure up to a factor: so runs the ‘quadratic Hamiltonian theorem’. Here we show that the same conclusion holds for much smaller ‘sufficiency subsets’ of quadratic Hamiltonians, and the theorem may thus be extended to homogeneous infinite-dimensional symplectic manifolds. In this way, we identify the distinguished Hamiltonians for the Kahler manifold of equivalent quantizations of a Hilbertizable symplectic space.
TL;DR: In this article, an exact sequence over the symplectic group whose Euler-Poincare characteristic gives the branching rule for this group was constructed and a representation-theoretic proof of the restriction rule for the group was given.
Abstract: In characteristic zero we construct an exact sequence over the symplectic group whose Euler-Poincare characteristic gives the branching rule for this group.This exact sequence is one way of realizing the Giambelli formula for the symplectic group.Also we give a representation-theoretic proof of the restriction rule for the symplectic group.
TL;DR: In this paper, the authors considered a conjugacy class of elementary abelian subgroups of order q =pa (p prime) in the finite group G. The class RG is called a class of p-transvection subgroups and is called PSp if any two members of the class generate either their direct product or a subgroup isomorphic to SL, (q) or PSL,(q).
TL;DR: In this article, it is shown how to construct an infinite class of nonlinear sympletic maps of the space that are stable, i.e., have invariant closed balls.
Abstract: For a symplectic vector space of arbitrary even dimension it is shown how to construct an infinite class of nonlinear sympletic maps of the space that are absolutely stable, i.e., have invariant closed balls. Explicit examples are presented.
01 Jan 1991
TL;DR: The group of degree 2v over F is a group with respect to matrix multiplication and is called the symplectic group over F, see as mentioned in this paper for a discussion of matrix multiplication groups.
Abstract: （F） is a group with respect to the matrix multiplication and is called the symplectic group of degree 2v over F