Symplectic Pairs 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.

Symplectic Pairs and Intrinsically Harmonic Forms

Symplectic Pairs and Intrinsically Harmonic Forms

Weak dual pairs in Dirac–Jacobi geometry

Adopting the omni-Lie algebroid approach to Dirac–Jacobi structures, we propose and investigate a notion of weak dual pairs in Dirac–Jacobi geometry. Their main motivating examples arise from the theory of multiplicative precontact structures on Lie groupoids. Among other properties of weak dual pairs, we prove two main results. (1) We show that the property of fitting in a weak dual pair defines an equivalence relation for Dirac–Jacobi manifolds. So, in particular, we get the existence of self-dual pairs and this immediately leads to an alternative proof of the normal form theorem around Dirac–Jacobi transversals. (2) We prove the characteristic leaf correspondence theorem for weak dual pairs paralleling and extending analogous results for symplectic and contact dual pairs. Moreover, the same ideas of this proof apply to get a presymplectic leaf correspondence for weak dual pairs in Dirac geometry (not yet present in literature).

Contact Dual Pairs

Abstract We introduce and study the notion of contact dual pair adopting a line bundle approach to contact and Jacobi geometry. A contact dual pair is a pair of Jacobi morphisms defined on the same contact manifold and satisfying a certain orthogonality condition. Contact groupoids and contact reduction are the main sources of examples. Among other properties, we prove the characteristic leaf correspondence theorem for contact dual pairs that parallels the analogous result of Weinstein for symplectic dual pairs.

Symplectic symmetric pairs of contragredient Lie superalgebras

Let L be a contragredient Lie superalgebra. A symmetric pair of L is a pair $$({\mathfrak {g}},{\mathfrak {g}}^\sigma )$$ , where $${\mathfrak {g}}$$ is a real form of L, and $$\sigma $$ is a $${\mathfrak {g}}$$ -involution with invariant subalgebra $${\mathfrak {g}}^\sigma $$ . We show that a symmetric pair carries invariant symplectic forms if and only if $${\mathfrak {g}}^\sigma $$ has a 1-dimensional center. Furthermore, the symplectic form is pseudo-Kahler if and only if the center of $${\mathfrak {g}}^\sigma $$ is compact. As an application, we classify the symplectic symmetric pairs, as well as the pseudo-Kahler symmetric pairs.

The Marsden–Weinstein reduction theorem for contact–symplectic pairs

We show that the classical Marsden–Weinstein Reduction theorem for Hamiltonian systems with symmetries is still true for contact–symplectic pairs.

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.

Holomorphic Spheres and Four-Dimensional Symplectic Pairs

We classify four-dimensional manifolds endowed with symplectic pairs admitting embedded symplectic spheres with nonnegative self-intersection, following the strategy of McDuff’s classification of rational and ruled symplectic four-manifolds.

The classification of symplectic matrices and pairs

The classification of symplectic matrices and pairs

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.

Symplectic spaces and pairs of symmetric and nonsingular skew-symmetric matrices under congruence

Let F be a field of characteristic not 2, and let (A,B) be a pair of n×n matrices over F, in which A is symmetric and B is skew-symmetric. A canonical form of (A,B) with respect to congruence transformations (STAS,STBS) was given by Sergeichuk (1988) [25] up to classification of symmetric and Hermitian forms over finite extensions of F. We obtain a simpler canonical form of (A,B) if B is nonsingular. Such a pair (A,B) defines a quadratic form on a symplectic space, that is, on a vector space with scalar product given by a nonsingular skew-symmetric form. As an application, we obtain known canonical matrices of quadratic forms and Hamiltonian operators on real and complex symplectic spaces.

Affine Vogan diagrams and symmetric pairs

We introduce the affine Vogan diagrams of complex simple Lie algebras. These are generalizations of Vogan diagrams, and we study the involutions represented by them. We apply these diagrams to study the symmetric pairs, in particular the associated and symplectic symmetric pairs.

Canonical functions, differential graded symplectic pairs in supergeometry, and Alexandrov-Kontsevich-Schwartz-Zaboronsky sigma models with boundaries

Consistent boundary conditions for Alexandrov-Kontsevich-Schwartz-Zaboronsky (AKSZ) sigma models and the corresponding boundary theories are analyzed. As their mathematical structures, we introduce a generalization of differential graded symplectic manifolds, called twisted QP manifolds, in terms of graded symplectic geometry, canonical functions, and QP pairs. We generalize the AKSZ construction of topological sigma models to sigma models with Wess-Zumino terms and show that all the twisted Poisson-like structures known in the literature can actually be naturally realized as boundary conditions for AKSZ sigma models.

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.

A Structure-Preserving Curve for Symplectic Pairs and Its Applications

The main purpose of this paper is the study of numerical methods for the maximal solution of the matrix equation $X+A^*X^{-1}A = Q$, where $Q$ is Hermitian positive definite. We construct a smooth curve parameterized by $t\ge 1$ of symplectic pairs with a special structure, in which the curve passes through all iteration points generated by the known numerical methods, including the fixed-point iteration, the structure-preserving doubling algorithm (SDA), and Newton's method provided that $A^*Q^{-1}A=AQ^{-1}A^*$. In the theoretical section, we give a necessary and sufficient condition for the existence of this structure-preserving curve for each parameter $t\ge1$. We also study the monotonicity and boundedness properties of this curve. In the application section, we use this curve to measure the convergence rates of those numerical methods. Numerical results illustrating these solutions are also presented.

Local rigidity and nonrigidity of symplectic pairs

It is shown that indefinite strictly almost Kahler and opposite Kahler structures (J, J′) on a four-dimensional manifold with J-invariant Ricci operator are rigid, thus extending a previous result of Apostolov, Armstrong and Drăghici from the positive definite case to the indefinite one. In contrast to this, examples of nonhomogeneous four-dimensional manifolds which admit strictly almost paraKahler and opposite paraKahler structures \({(\mathfrak{J},\mathfrak{J}^{\prime})}\) with \({\mathfrak{J}}\) -invariant Ricci operator are shown.

Rank one reducibility for unitary groups

Let (G,G ' ) denote a dual reductive pair consisting of two unitary groups over a nonarchimedean local field of characteristic zero. We relate the reducibility of the parabolically induced representations of these two groups if the inducing data is cuspidal and related to each other by theta correspondence. We calculate theta lifts of the irreducible subquotients of these parabolically induced representations. To obtain these results, we explicitly calculate filtration of Jacquet modules of the appropriate Weil representation (as Kudla did for the orthogonal- symplectic dual pairs), but keeping in mind the explicit splittings of covers of these two unitary groups, also obtained by Kudla.

The Geometry of Recursion Operators

We study the fields of endomorphisms intertwining pairs of symplectic structures. Using these endomorphisms we prove an analogue of Moser’s theorem for simultaneous isotopies of two families of symplectic forms. We also consider the geometric structures defined by pairs and triples of symplectic forms for which the squares of the intertwining endomorphisms are plus or minus the identity. For pairs of forms we recover the notions of symplectic pairs and of holomorphic symplectic structures. For triples we recover the notion of a hypersymplectic structure, and we also find three new structures that have not been considered before. One of these is the symplectic formulation of hyper-Kahler geometry, which turns out to be a strict generalization of the usual definition in terms of differential or Kahler geometry.

The geometry of symplectic pairs

We study the geometry of manifolds carrying symplectic pairs consisting of two closed 2-forms of constant ranks, whose kernel foliations are complementary. Using a variation of the construction of Boothby and Wang we build contact-symplectic and contact pairs from symplectic pairs.

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.