Symplectic Pair Research Articles

Classical limits of Hilbert bimodules as symplectic dual pairs

Hilbert bimodules are morphisms between C*-algebraic models of quantum systems, while symplectic dual pairs are morphisms between Poisson geometric models of classical systems. Both of these morphisms preserve representation-theoretic structures of the relevant types of models. Previously, it has been shown that one can functorially associate certain symplectic dual pairs to Hilbert bimodules through strict deformation quantization. We show that, in the inverse direction, strict deformation quantization also allows one to functorially take the classical limit of a Hilbert bimodule to reconstruct a symplectic dual pair.

Geodesics and magnetic curves in the 4-dim almost Kähler model space F4

Abstract We study geodesics and magnetic trajectories in the model space F 4 {{\rm{F}}}^{4} . The space F 4 {{\rm{F}}}^{4} is isometric to the 4-dim simply connected Riemannian 3-symmetric space due to Kowalski. We describe the solvable Lie group model of F 4 {{\rm{F}}}^{4} and investigate its curvature properties. We introduce the symplectic pair of two Kähler forms on F 4 {{\rm{F}}}^{4} . Those symplectic forms induce invariant Kähler structure and invariant strictly almost Kähler structure on F 4 {{\rm{F}}}^{4} . We explore some typical submanifolds of F 4 {{\rm{F}}}^{4} . Next, we explore the general properties of magnetic trajectories in an almost Kähler 4-manifold and characterize Kähler magnetic curves with respect to the symplectic pair of Kähler forms. Finally, we study homogeneous geodesics and homogeneous magnetic curves in F 4 {{\rm{F}}}^{4} .

Manifestly covariant worldline actions from coadjoint orbits. Part I. Generalities and vectorial descriptions

We derive manifestly covariant actions of spinning particles starting from coadjoint orbits of isometry groups, by using Hamiltonian reductions. We show that the defining conditions of a classical Lie group can be treated as Hamiltonian constraints which generate the coadjoint orbits of another, dual, Lie group. In case of (inhomogeneous) orthogonal groups, the dual groups are (centrally-extended inhomogeneous) symplectic groups. This defines a symplectic dual pair correspondence between the coadjoint orbits of the isometry group and those of the dual Lie group, whose quantum version is the reductive dual pair correspondence à la Howe. We show explicitly how various particle species arise from the classification of coadjoint orbits of Poincaré and (A)dS symmetry. In the Poincaré case, we recover the data of the Wigner classification, which includes continuous spin particles, (spinning) tachyons and null particles with vanishing momenta, besides the usual massive and massless spinning particles. In (A)dS case, our classification results are not only consistent with the pattern of the corresponding unitary irreducible representations observed in the literature, but also contain novel information. In dS, we find the presence of partially massless spinning particles, but continuous spin particles, spinning tachyons and null particles are absent. The AdS case shows the largest diversity of particle species. It has all particles species of Poincaré symmetry except for the null particle, but allows in addition various exotic entities such as one parameter extension of continuous particles and conformal particles living on the boundary of AdS. Notably, we also find a large class of particles living in “bitemporal” AdS space, including ones where mass and spin play an interchanged role. We also discuss the relative inclusion structure of the corresponding orbits.

Symplectic P-stable additive Runge—Kutta methods

<p style='text-indent:20px;'>Classical symplectic partitioned Runge–Kutta methods can be obtained from a variational formulation where all the terms in the discrete Lagrangian are treated with the same quadrature formula. We construct a family of symplectic methods allowing the use of different quadrature formulas (primary and secondary) for different terms of the Lagrangian. In particular, we study a family of methods using Lobatto quadrature (with corresponding Lobatto IIIA-B symplectic pair) as a primary method and Gauss–Legendre quadrature as a secondary method. The methods have the same implicitness as the underlying Lobatto IIIA-B pair, and, in addition, they are <i>P-stable</i>, therefore suitable for application to highly oscillatory problems.</p>

A structure preserving flow for computing Hamiltonian matrix exponential

This article focuses on computing Hamiltonian matrix exponential. Given a Hamiltonian matrix $$\mathcal {H}$$ , it is well-known that the matrix exponential $$e^{\mathcal {H}}$$ is a symplectic matrix and its eigenvalues form reciprocal $$(\lambda ,1/\bar{\lambda })$$ . It is important to take care of the symplectic structure for computing $$e^{\mathcal {H}}$$ . Based on the structure-preserving flow proposed by Kuo et al. (SIAM J Matrix Anal Appl 37:976–1001, 2016), we develop a numerical method for computing the symplectic matrix pair $$(\mathcal {M},\mathcal {L})$$ which represents $$e^{\mathcal {H}}$$ .

Contact-pair neighborhood theorem for submanifolds in symplectic pairs

Let M be a 2n-dimensional smooth manifold associated with the structure of symplectic pair which is a pair of closed 2-forms of constant ranks with complementary kernel foliations. Let Q ⊂ M be a codimension 2 compact submanifold. We show some sufficient and necessary conditions on the existence of the structure of contact pair (α, β) on Q, which is a pair of 1-forms of constant classes whose characteristic foliations are transverse and complementary such that α and β restrict to contact forms on the leaves of the characteristic foliations of β and α, respectively. This is a generalization of the neighborhood theorem for contact-type hypersurfaces in symplectic topology.

Smoothness of isotopy for symplectic pairs

Let M be a 2n-dimensional smooth manifold with a symplectic pair which is a pair of closed 2-forms of constant ranks with complementary kernel foliations. Similar to Moser's stability theorem for symplectic forms, one desires to establish a stability theorem for symplectic pairs. Some sufficient and necessary conditions are obtained by Bande, Ghiggini and Kotschick. In this article, we consider a technical problem relating to the stability theorem. To complete the proof of the stability theorem for symplectic pairs, we verify the smoothness of the isotopy which is ignored in the literature. The Hodge theory for Riemannian foliation is crucial to our discussion.

Almost complex structures for symplectic pairs

Let M be a 2n-dimensional smooth manifold with a linear symplectic pair (ω,τ) which is a pair of skew-symmetric 2-forms of constant ranks with complementary kernel foliations. We study properties of almost complex structures compatible with the symplectic pair on M. We show that the space of compatible almost complex structures is contractible. This generalizes a result on compatible almost complex structures for symplectic manifolds.

Taut contact circles and bi-contact metric structures on three-manifolds

Geiges and Gonzalo (Invent. Math. 121:147–209 1995, J. Differ. Geom. 46:236–286 1997, Acta. Math. Vietnam 38:145–164 2013) introduced and studied the notion of taut contact circle on a three-manifold. In this paper, we introduce a Riemannian approach to the study of taut contact circles on three-manifolds. We characterize the existence of a taut contact metric circle and of a bi-contact metric structure. Then, we give a complete classification of simply connected three-manifolds which admit a bi-H-contact metric structure. In particular, a simply connected three-manifold admits a homogeneous bi-contact metric structure if and only if it is diffeomorphic to one of the following Lie groups: SU(2), \({\widetilde{SL}}(2,{\mathbb {R}})\), \({\widetilde{E}}(2)\), E(1, 1). Moreover, we obtain a classification of three-manifolds which admit a Cartan structure \((\eta _1,\eta _2)\) with the so-called Webster function \({\mathcal {W}}\) constant along the flow of \(\xi _1\) (equivalently \(\xi _2\)). Finally, we study the metric cone, i.e., the symplectization, of a bi-contact metric three-manifold. In particular, the notion of bi-contact metric structure is related to the notions of conformal symplectic couple (in the sense of Geiges (Duke Math. J. 85:701–711 1996)) and symplectic pair (in the sense of Bande and Kotschick (Trans. Am. Math. Soc. 358(4):1643–1655 2005)).

H∞Controller Design for Asynchronous Hybrid Systems with Multiple Delays

Solutions for theH∞synthesis problems of asynchronous hybrid systems with input-output delays are proposed. The continuous-time lifting approach of sampled-data systems is extended to a hybrid system with multiple delays, and some feasible formulas to calculate the operators of the equivalent discrete-time (DT) system are given. Different from the existing methods derived from symplectic pair theory or by state augmentation, a Lyapunov-Krasovskii functional to solve the synthesis problem is explicitly constructed. The delay-dependent stability conditions we obtained can be described in terms of nonstrict linear matrix inequalities (LMIs), which are much more convenient to be solved by LMI tools.

Rozansky–Witten-Type Invariants from Symplectic Lie Pairs

We introduce symplectic structures on "Lie pairs" of (real or complex) algebroids as studied by Chen, Stienon and the second author (From Atiyah classes to homotopy Leibniz algebras, arXiv:1204.1075), encompassing homogeneous symplectic spaces, symplectic manifolds with a $\mathfrak g$-action and holomorphic symplectic manifolds. We show that to each such symplectic Lie pair are associated Rozansky-Witten-type invariants of three-manifolds and knots, given respectively by weight systems on trivalent and chord diagrams.

Symplectic dual pair related to soa1,...,an-1(n)

We investigate a symplectic dual pair related to a Lie algebra soa1,...,an-1 (n) of deformed skew-symmetric n × n matrices. The deformations are parametrized by a sequence of real numbers a1,...,an-1.

On neighborhood theorems for symplectic pairs

Let M be a 2n-dimensional smooth manifold. We study local properties for a symplectic pair on M which is a pair of closed 2-forms of constant ranks with complementary kernel foliations. We show that some neighborhood theorems for submanifolds of M and Darboux type theorem hold. For more general case of recursion operators intertwining a pair of symplectic forms, we also prove that some neighborhood theorem holds.

Symmetry in the geometry of metric contact pairs

We prove that the universal covering of a complete locally symmetric normal metric contact pair manifold with decomposable ϕ is a Calabi‐Eckmann manifold or the Riemannian product of a sphere and . We show that a complete, simply connected, normal metric contact pair manifold with decomposable ϕ, such that the foliation induced by the vertical subbundle is regular and reflections in the integral submanifolds of the vertical subbundle are isometries, is the product of globally ϕ‐symmetric spaces or the product of a globally ϕ‐symmetric space and . Moreover in the first case the manifold fibers over a locally symmetric space endowed with a symplectic pair.

Bijective Proof of a Symplectic Dual Pair Identity

We provide a combinatorial proof of a symplectic character identity relating the sum of a product of symplectic Schur functions to the product . This formula owes its origin to the existence of a dual pair of symplectic groups acting on spinors, as pointed out by Hasegawa. The first combinatorial proof, based on symplectic tableaux and a variation of the Robinson–Schensted–Knuth correspondence, was due to Terada. Here we use Schützenberger’s jeu de taquin, augmented by two simple zero weight transformations. The identity itself generalizes a well-known identity expressing as a sum of products of Schur functions that was due to Littlewood and proved combinatorially by Remmel. We offer an alternative combinatorial proof of this identity by means of the jeu de taquin, as a precursor to the proof of the symplectic identity.

Structure-Preserving Algorithms for Periodic Discrete-Time Algebraic Riccati Equations

In this paper we investigate structure-preserving algorithms for computing the symmetric positive semi-definite solutions to the periodic discrete-time algebraic Riccati equations (P-DAREs). Using a structure-preserving swap and collapse procedure, a single symplectic matrix pair in standard symplectic form is obtained. The P-DAREs can then be solved via a single DARE, using a structure-preserving doubling algorithm. We develop the structure-preserving doubling algorithm from a new point of view and show its quadratic convergence under assumptions which are weaker than stabilizability and detectability. With several numerical results, the algorithm is shown to be efficient, out-performing other algorithms on a large set of benchmark problems.

POISSON GEOMETRY IN CONSTRAINED SYSTEMS

Associated to a constrained system with closed constraint algebra there are two Poisson manifolds P and Q forming a symplectic dual pair with respect to the original, unconstrained phase space: P is the image of the constraint map (equipped with the algebra of constraints) and Q the Poisson quotient with respect to the orbits generated by the constraints (the orbit space is assumed to be a manifold). We provide sufficient conditions so that the reduced phase space of the constrained system may be identified with a symplectic leaf of Q. By these methods, a second class constrained system with closed algebra is reformulated as an abelian first class system in an extended phase space.While any Poisson manifold (P,Π) has a symplectic realization (Karasev, Weinstein 87), it does not always permit a leafwise symplectic embedding into a symplectic manifold (M,ω). For regular P, it is seen that such an embedding exists, iff the characteristic form-class of Π, a certain element of the third relative cohomology of P, vanishes. A tubular neighborhood of the constraint surface of a general second class constrained system equipped with the Dirac bracket provides a physical example for such an embedding into the original symplectic manifold. In contrast, a leafwise symplectic embedding of e.g. (the maximal regular part of) a Poisson Lie manifold associated to a compact, semisimple Lie algebra does not exist.

Full sign-invertibility and symplectic matrices

An n × n sign pattern H is said to be sign-invertible if there exists a sign pattern H −1 (called the sign inverse of H) such that, for all matrices A ∈ Q( H), A −1 exists and A −1 ∈ Q( H −1). If, in addition, H −1 is sign-invertible [implying ( H −1) −1 = H], H is said to be fully sign-invertible and ( H, H −1) is called a sign-invertible pair. Given an n × n sign pattern H, a symplectic pair in Q( H) is a pair of matrices ( A, D) such that A ∈ Q( H), D ∈ Q( H), and A T D = I. (Symplectic pairs are a pattern generalization of orthogonal matrices which arise from a special symplectic matrix found in n-body problems in celestial mechanics [1].) We discuss the digraphical relationship between a sign-invertible pattern H and its sign inverse H −1, and use this to cast a necessary condition for full sign-invertibility of H. We proceed to develop sufficient conditions for H's full sign-invertibility in terms of allowed paths and cycles in the digraph of H, and conclude with a complete characterization of those sign patterns that require symplectic pairs.