Showing papers on "Symplectic vector space published in 2017"

Construction of Symplectic Runge-Kutta Methods for Stochastic Hamiltonian Systems

Abstract: We study the construction of symplectic Runge-Kutta methods for stochastic Hamiltonian systems (SHS). Three types of systems, SHS with multiplicative noise, special separable Hamiltonians and multiple additive noise, respectively, are considered in this paper. Stochastic Runge-Kutta (SRK) methods for these systems are investigated, and the corresponding conditions for SRK methods to preserve the symplectic property are given. Based on the weak/strong order and symplectic conditions, some effective schemes are derived. In particular, using the algebraic computation, we obtained two classes of high weak order symplectic Runge-Kutta methods for SHS with a single multiplicative noise, and two classes of high strong order symplectic Runge-Kutta methods for SHS with multiple multiplicative and additive noise, respectively. The numerical case studies confirm that the symplectic methods are efficient computational tools for long-term simulations.

Hamiltonian and symplectic symmetries: An introduction

Abstract: Classical mechanical systems are modeled by a symplectic manifold $(M,\omega)$, and their symmetries, encoded in the action of a Lie group $G$ on $M$ by diffeomorphisms that preserves $\omega$ These actions, which are called "symplectic", have been studied in the past forty years, following the works of Atiyah, Delzant, Duistermaat, Guillemin, Heckman, Kostant, Souriau, and Sternberg in the 1970s and 1980s on symplectic actions of compact abelian Lie groups that are, in addition, of "Hamiltonian" type, ie they also satisfy Hamilton's equations Since then a number of connections with combinatorics, finite dimensional integrable Hamiltonian systems, more general symplectic actions, and topology, have flourished In this paper we review classical and recent results on Hamiltonian and non Hamiltonian symplectic group actions roughly starting from the results of these authors The paper also serves as a quick introduction to the basics of symplectic geometry

Donaldson’s Q -operators for symplectic manifolds

Abstract: We prove an estimate for Donaldson’s Q-operator on a prequantized compact symplectic manifold. This estimate is an ingredient in the recent result of Keller and Lejmi (2017) about a symplectic generalization of Donaldson’s lower bound for the L 2-norm of the Hermitian scalar curvature.

Parabolic conformally symplectic structures III; Invariant differential operators and complexes

Abstract: This is the last part of a series of articles on a family of geometric structures (PACS-structures) which all have an underlying almost conformally symplectic structure. While the first part of the series was devoted to the general study of these structures, the second part focused on the case that the underlying structure is conformally symplectic (PCS-structures). In that case, we obtained a close relation to parabolic contact structures via a concept of parabolic contactification. It was also shown that special symplectic connections (and thus all connections of exotic symplectic holonomy) arise as the canonical connection of such a structure. In this last part, we use parabolic contactifications and constructions related to Bernstein-Gelfand-Gelfand (BGG) sequences for parabolic contact structures, to construct sequences of differential operators naturally associated to a PCS-structure. In particular, this gives rise to a large family of complexes of differential operators associated to a special symplectic connection. In some cases, large families of complexes for more general instances of PCS-structures are obtained.

Quantum quaternion spheres

Abstract: For the quantum symplectic group SP q (2n), we describe the C ∗-algebra of continuous functions on the quotient space S P q (2n)/S P q (2n−2) as an universal C ∗-algebra given by a finite set of generators and relations. The proof involves a careful analysis of the relations, and use of the branching rules for representations of the symplectic group due to Zhelobenko. We then exhibit a set of generators of the K-groups of this C ∗-algebra in terms of generators of the C ∗-algebra.

On asymptotic expansions of generalized Bergman kernels on symplectic manifolds

Abstract: A full off-diagonal asymptotic expansion is established for the generalized Bergman kernels of the renormalized Bochner Laplacians associated with high tensor powers of a positive line bundle over a compact symplectic manifold. As an application, the algebra of Toeplitz operators on the symplectic manifold associated with the renormalized Bochner Laplacian is constructed.

Structure of locally conformally symplectic Lie algebras and solvmanifolds

Abstract: We obtain structure results for locally conformally symplectic Lie algebras. We classify locally conformally symplectic structures on four-dimensional Lie algebras and construct locally conformally symplectic structures on compact quotients of all four-dimensional connected and simply connected solvable Lie groups.

Exact low-order polynomial expressions to compute the Kolmogoroff–Nagumo mean in the affine symplectic group of optical transference matrices

Abstract: The current contribution presents exact third-order polynomial expressions of matrix functions that arise in the computation of the Kolmogoroff–Nagumo mean of a set of optical transference matrices, that belong to the affine symplectic group .

On quasi-heredity and cell module homomorphisms in the symplectic blob algebra

Abstract: This paper reports key advances in the study of the representation theory of the symplectic blob algebra. For suitable specialisations of the parameters we construct four large families of homomorphisms between cell modules. We hence find a large family of non-semisimple specialisations. We find a minimal poset (i.e. least number of relations) for the symplectic blob as a quasi-hereditary algebra.

Gauging the Relativistic Particle Model on the Noncommutative Plane

Abstract: We construct a new model for relativistic particle on the noncommutative plane, using the symplectic formalism of constrained systems. We suggest a shortcut approach to construct the gauged Lagrangian, using the Poisson algebra of constraints, without calculating the whole procedure of symplectic formalism. We also propose an approach for the systems, in which the symplectic formalism is not applicable, due to the truncation of production of secondary constraints at the first level. After gauging the model, we obtain the corresponding generators of gauge transformations of the physical system. Finally, by extracting the Poisson structure of all constraints, we show the effect of gauging on the canonical structure of the phase spaces of both primary and gauged models.

Odd sphere bundles, symplectic manifolds, and their intersection theory

Abstract: Recently, Tsai-Tseng-Yau constructed new invariants of symplectic manifolds: a sequence of Aoo-algebras built of differential forms on the symplectic manifold. We show that these symplectic Aoo-algebras have a simple topological interpretation. Namely, when the cohomology class of the symplectic form is integral, these Aoo-algebras are equivalent to the standard de Rham differential graded algebra on certain odd-dimensional sphere bundles over the symplectic manifold. From this equivalence, we deduce for a closed symplectic manifold that Tsai-Tseng-Yau's symplectic Aoo-algebras satisfy the Calabi-Yau property, and importantly, that they can be used to define an intersection theory for coisotropic/isotropic chains. We further demonstrate that these symplectic Aoo-algebras satisfy several functorial properties and lay the groundwork for addressing Weinstein functoriality and invariance in the smooth category.

Second-order evaluations of orthogonal and symplectic Yangians

Abstract: Orthogonal or symplectic Yangians are defined by the Yang–Baxter RLL relation involving the fundamental R-matrix with the corresponding so(n) or sp(2m) symmetry. We investigate the second-order solution conditions, where the expansion of L(u) in u −1 is truncated at the second power, and we derive the relations for the two nontrivial terms in L(u).

Global description of action-angle duality for a Poisson-Lie deformation of the trigonometric $\mathrm{BC}_n$ Sutherland system

Abstract: Integrable many-body systems of Ruijsenaars--Schneider--van Diejen type displaying action-angle duality are derived by Hamiltonian reduction of the Heisenberg double of the Poisson-Lie group $\mathrm{SU}(2n)$. New global models of the reduced phase space are described, revealing non-trivial features of the two systems in duality with one another. For example, after establishing that the symplectic vector space $\mathbb{C}^n\simeq\mathbb{R}^{2n}$ underlies both global models, it is seen that for both systems the action variables generate the standard torus action on $\mathbb{C}^n$, and the fixed point of this action corresponds to the unique equilibrium positions of the pertinent systems. The systems in duality are found to be non-degenerate in the sense that the functional dimension of the Poisson algebra of their conserved quantities is equal to half the dimension of the phase space. The dual of the deformed Sutherland system is shown to be a limiting case of a van Diejen system.

Special homogeneous almost complex structures on symplectic manifolds

Abstract: Homogeneous compatible almost complex structures on symplectic manifolds are studied, focusing on those which are special, meaning that their Chern-Ricci form is a multiple of the symplectic form. Non Chern-Ricci flat ones are proven to be covered by co-adjoint orbits. Conversely, compact isotropy co-adjoint orbits of semi-simple Lie groups are shown to admit special compatible almost complex structures whenever they satisfy a necessary topological condition. Some classes of examples including twistor spaces of hyperbolic manifolds and discrete quotients of Griffiths period domains of weight two are discussed.

Basic aspects of symplectic Clifford analysis for the symplectic Dirac operator

Abstract: In the present article we study basic aspects of the symplectic version of Clifford analysis associated to the symplectic Dirac operator. Focusing mostly on the symplectic vector space of real dimension 2, this involves the analysis of first order symmetry operators, symplectic Clifford-Fourier transform, reproducing kernel for the symplectic Fischer product and the construction of bases of symplectic monogenics for the symplectic Dirac operator.

On the composition of Berezin-Toeplitz operators on symplectic manifolds

Abstract: We compute the second coefficient of the composition of two Berezin-Toeplitz operators associated with the $\text{spin}^c$ Dirac operator on a symplectic manifold, making use of the full-off diagonal expansion of the Bergman kernel.

The geometric structure of symplectic contraction

Abstract: We show that the symplectic contraction map of Hilgert-Manon-Martens -- a symplectic version of Popov's horospherical contraction -- is simply the quotient of a Hamiltonian manifold $M$ by a "stratified null foliation" that is determined by the group action and moment map. We also show that the quotient differential structure on the symplectic contraction of $M$ supports a Poisson bracket. We end by proving a very general description of the topology of fibers of Gelfand-Zeitlin systems on multiplicity free Hamiltonian $U(n)$ and $SO(n)$ manifolds.

Injectivity of Generalized Wronski Maps

Abstract: We study linear projections on Pluecker space whose restriction to the Grassmannian is a non-trivial branched cover. When an automorphism of the Grassmannian preserves the fibers, we show that the Grassmannian is necessarily of m-dimensional linear subspaces in a symplectic vector space of dimension 2m, and the linear map is the Lagrangian involution. The Wronski map for a self-adjoint linear differential operator and pole placement map for symmetric linear systems are natural examples.

First Betti number of weighted homology group of Hamiltonian vector fields on symplectic tori

Abstract: Like (co)homology group theory of formal Hamiltonian vector fields on symplectic vector spaces, we try studying homology group theory on symplecit tori introducing the notion of weight

Constant Symplectic 2-groupoids

Abstract: We propose a definition of symplectic 2-groupoid which includes integrations of Courant algebroids that have been recently constructed. We study in detail the simple but illustrative case of constant symplectic 2-groupoids. We show that the constant symplectic 2-groupoids are, up to equivalence, in one-to-one correspondence with a simple class of Courant algebroids that we call constant Courant algebroids. Furthermore, we find a correspondence between certain Dirac structures and Lagrangian sub-2-groupoids.