Showing papers on "Symplectic group published in 2006"
TL;DR: The integrals of polynomial functions with respect to the Haar measure on the unitary group U(d) have been studied in this paper for all values of d.
Abstract: We revisit the work of the first named author and using simpler algebraic arguments we calculate integrals of polynomial functions with respect to the Haar measure on the unitary group U(d). The previous result provided exact formulas only for 2d bigger than the degree of the integrated polynomial and we show that these formulas remain valid for all values of d. Also, we consider the integrals of polynomial functions on the orthogonal group O(d) and the symplectic group Sp(d). We obtain an exact character expansion and the asymptotic behavior for large d. Thus we can show the asymptotic freeness of Haar-distributed orthogonal and symplectic random matrices, as well as the convergence of integrals of the Itzykson–Zuber type.
690 citations
Book•
18 May 2006
TL;DR: In this paper, a phase space Weyl Calculus is introduced and the uncertainty principle of the Density Operator is defined. But the complexity of the Weyl calculus is not discussed.
Abstract: Symplectic Geometry.- Symplectic Spaces and Lagrangian Planes.- The Symplectic Group.- Multi-Oriented Symplectic Geometry.- Intersection Indices in Lag(n) and Sp(n).- Heisenberg Group, Weyl Calculus, and Metaplectic Representation.- Lagrangian Manifolds and Quantization.- Heisenberg Group and Weyl Operators.- The Metaplectic Group.- Quantum Mechanics in Phase Space.- The Uncertainty Principle.- The Density Operator.- A Phase Space Weyl Calculus.
375 citations
01 Jan 2006
TL;DR: This article corrected an error in (1) pointed out to the author by Ritter and stated that this Erratum will not be published, which was later corrected by the author.
Abstract: We correct an error in (1), pointed out to the author by Ritter. This Erratum will not be published.
235 citations
TL;DR: In this article, the authors define an invariant of oriented links in S 3 using the symplectic geometry of certain spaces which arise naturally in Lie theory and present a knot as the closure of a braid, which in turn views as a loop in configuration space.
Abstract: We define an invariant of oriented links in S 3 using the symplectic geometry of certain spaces which arise naturally in Lie theory. More specifically, we present a knot as the closure of a braid, which in turn we view as a loop in configuration space. Fix an affine subspaceSm of the Lie algebra sl2m(C) which is a transverse slice to the adjoint action at a nilpotent matrix with two equal Jordan blocks. The adjoint quotient map restricted to Sm gives rise to a symplectic fibre bundle over configuration space. An inductive argument constructs a distinguished Lagrangian submani
170 citations
TL;DR: A general intrinsic tracking controller design for fully-actuated simple mechanical systems, when the configuration space is one of a general class of Lie groups, and it is shown that the full-state feedback law composed with an exponentially convergent velocity estimator converges globally for almost every initial tracking error.
Abstract: We present a general intrinsic tracking controller design for fully-actuated simple mechanical systems, when the configuration space is one of a general class of Lie groups. We first express a state-feedback controller in terms of a function-the "error function"-satisfying certain regularity conditions. If an error function can be found, then a general smooth and bounded reference trajectory may be tracked asymptotically from almost every initial condition, with locally exponential convergence. Asymptotic convergence from almost every initial condition is referred to as "almost-global" asymptotic stability. Error functions may be shown to exist on any compact Lie group, or any Lie group diffeomorphic to the product of a compact Lie group and R/sup n/. This covers many cases of practical interest, such as SO(n), SE(n), their subgroups, and direct products. We show here that for compact Lie groups the dynamic configuration-feedback controller obtained by composing the full state-feedback law with an exponentially convergent velocity observer is also almost-globally asymptotically stable with respect to the tracking error. We emphasize that no invariance is needed for these results. However, for the special case where the kinetic energy is left-invariant, we show that the explicit expression of these controllers does not require coordinates on the Lie group. The controller constructions are demonstrated on SO(3), and simulated for the axi-symmetric top. Results show excellent performance.
129 citations
Book•
26 Jul 2006
TL;DR: Symplectic spinors have been used in many applications, e.g., Lie Derivative and Quantization, Symplectic Connections, Symmlectic Dirac Operators, and Second Order Operators as discussed by the authors.
Abstract: Background on Symplectic Spinors.- Symplectic Connections.- Symplectic Spinor Fields.- Symplectic Dirac Operators.- An Associated Second Order Operator.- The Kahler Case.- Fourier Transform for Symplectic Spinors.- Lie Derivative and Quantization.
67 citations
TL;DR: In this paper, the standard notion of differential calculus in noncommutative geometry on algebras and quantum groups is considered, and the moduli space of bicovariant infinitesimal data for quasitriangular Poisson?Lie groups has a canonical reference point which is flat in the triangular case.
Abstract: We semiclassicalise the standard notion of differential calculus in noncommutative geometry on algebras and quantum groups. We show in the symplectic case that the infinitesimal data for a differential calculus is a symplectic connection, and interpret its curvature as lowest order nonassociativity of the exterior algebra. Semiclassicalisation of the noncommutative torus provides an example with zero curvature. In the Poisson?Lie group case we study left-covariant infinitesimal data in terms of partially defined preconnections. We show that the moduli space of bicovariant infinitesimal data for quasitriangular Poisson?Lie groups has a canonical reference point which is flat in the triangular case. Using a theorem of Kostant, we completely determine the moduli space when the Lie algebra is simple: the canonical preconnection is the unique point other than in the case of sln, n > 2, when the moduli space is 1-dimensional. We relate the canonical preconnection to Drinfeld twists and thereby quantise it to a super coquasi-Hopf exterior algebra. We also discuss links with Fedosov quantisation.
66 citations
TL;DR: In this paper, the l-exotic nilpotent cone is introduced for complex symplectic groups, and a character formula and multiplicity formula for simple H-modules are presented.
Abstract: Let G be a complex symplectic group. We introduce a G x (C ^x) ^{l + 1}-variety N_{l}, which we call the l-exotic nilpotent cone. Then, we realize the Hecke algebra H of type C_n ^(1) with three parameters via equivariant algebraic K-theory in terms of the geometry of N_2. This enables us to establish a Deligne-Langlands type classification of "non-critical" simple H-modules. As applications, we present a character formula and multiplicity formulas of H-modules.
63 citations
TL;DR: In this paper, it was shown that if ω is a Kahler form on a complex surface (M,J), then ω(M,ω) agrees with the usual holomorphic Kodaira dimension of (m,J).
Abstract: The Kodaira dimension of a non-minimal manifold is defined to be that of any of its minimal models. It is shown in [12] that, if ω is a Kahler form on a complex surface (M,J), then κ(M,ω) agrees with the usual holomorphic Kodaira dimension of (M,J). It is also shown in [12] that minimal symplectic 4−manifolds with κ = 0 are exactly those with torsion canonical class, thus can be viewed as symplectic Calabi-Yau surfaces. Known examples of symplectic 4−manifolds with torsion canonical class are either Kahler surfaces with (holomorphic) Kodaira dimension zero or T 2−bundles over T 2 ([10], [12]). They all have small Betti numbers and Euler numbers: b+ ≤ 3, b ≤ 19 and b1 ≤ 4; and the Euler number is between 0 and 24. It is speculated in [12] that these are the only ones. In this paper we prove that it is true up to rational homology.
62 citations
TL;DR: In this paper, Andre et al. studied super-characters of the unitriangular group and proved that every irreducible (complex) character occurs as a constituent of a unique super-character.
61 citations
TL;DR: In this article, Calabi quasi-morphisms on the universal cover of Ham were constructed for non-monotone manifolds and these quasimorphisms descend to non-trivial homomorphisms.
Abstract: In this work we construct Calabi quasi-morphisms on the universal cover e Ham.M/ of the group of Hamiltonian diffeomorphisms for some non-monotone symplectic manifolds. This complements a result by Entov and Polterovich which applies in the monotone case. Moreover, in contrast to their work, we show that these quasimorphisms descend to non-trivial homomorphisms on the fundamental group of Ham.M/.
TL;DR: The concept of pseudo symplectic capacities was introduced in this paper, which is a mild generalization of the Hofer-Zehnder capacity and is used to estimate pseudo-symplectic capacities of Grassmannians and product manifold.
Abstract: We introduce the concept of pseudo symplectic capacities which is a mild generalization of that of symplectic capacities. As a generalization of the Hofer-Zehnder capacity we construct a Hofer-Zehnder type pseudo symplectic capacity and estimate it in terms of Gromov-Witten invariants. The (pseudo) symplectic capacities of Grassmannians and some product symplectic manifolds are computed. As applications we first derive some general nonsqueezing theorems that generalize and unite many previous versions then prove the Weinstein conjecture for cotangent bundles over a large class of symplectic uniruled manifolds (including the uniruled manifolds in algebraic geometry) and also show that any closed symplectic submanifold of codimension two in any symplectic manifold has a small neighborhood whose Hofer-Zehnder capacity is less than a given positive number. Finally, we give two results on symplectic packings in Grassmannians and on Seshadri constants.
TL;DR: In this article, the theory of Abelian Routh reduction for discrete mechanical systems was developed and applied to the variational integration of mechanical systems with Abelian symmetry, and the reduction of variational Runge-Kutta discretizations was considered, as well as the extent to which symmetry reduction and discretization commute.
Abstract: This paper develops the theory of Abelian Routh reduction for discrete mechanical systems and applies it to the variational integration of mechanical systems with Abelian symmetry. The reduction of variational Runge–Kutta discretizations is considered, as well as the extent to which symmetry reduction and discretization commute. These reduced methods allow the direct simulation of dynamical features such as relative equilibria and relative periodic orbits that can be obscured or difficult to identify in the unreduced dynamics. The methods are demonstrated for the dynamics of an Earth orbiting satellite with a non-spherical J2 correction, as well as the double spherical pendulum. The J2 problem is interesting because in the unreduced picture, geometric phases inherent in the model and those due to numerical discretization can be hard to distinguish, but this issue does not appear in the reduced algorithm, where one can directly observe interesting dynamical structures in the reduced phase space (the cotangent bundle of shape space), in which the geometric phases have been removed. The main feature of the double spherical pendulum example is that it has a non-trivial magnetic term in its reduced symplectic form. Our method is still efficient as it can directly handle the essential non-canonical nature of the symplectic structure. In contrast, a traditional symplectic method for canonical systems could require repeated coordinate changes if one is evoking Darboux' theorem to transform the symplectic structure into canonical form, thereby incurring additional computational cost. Our method allows one to design reduced symplectic integrators in a natural way, despite the non-canonical nature of the symplectic structure.
TL;DR: In this paper, it was shown that Lie superalgebras can be constructed from symplectic triple systems (respectively orthogonal triple systems) with a different construction.
28 May 2006
TL;DR: This article shows that NTRU lattices are proportional to the so-called symplectic lattices, and shows that orthogonalization techniques which are at the heart of all reduction algorithms known, are all compatible with symplecticity, and that they can be significantly sped up for symplectic matrices.
Abstract: NTRU is a very efficient public-key cryptosystem based on polynomial arithmetic. Its security is related to the hardness of lattice problems in a very special class of lattices. This article is motivated by an interesting peculiar property of NTRU lattices. Namely, we show that NTRU lattices are proportional to the so-called symplectic lattices. This suggests to try to adapt the classical reduction theory to symplectic lattices, from both a mathematical and an algorithmic point of view. As a first step, we show that orthogonalization techniques (Cholesky, Gram-Schmidt, QR factorization, etc.) which are at the heart of all reduction algorithms known, are all compatible with symplecticity, and that they can be significantly sped up for symplectic matrices. Surprisingly, by doing so, we also discover a new integer Gram-Schmidt algorithm, which is faster than the usual algorithm for all matrices. Finally, we study symplectic variants of the celebrated LLL reduction algorithm, and obtain interesting speed ups.
TL;DR: In this article, the pseudo symplectic capacities of toric manifolds in combinatorial data are given. But they do not consider the impact of symplectic blow-up on the capacities of the polygon spaces.
Abstract: In this paper we give concrete estimations for the pseudo symplectic capacities of toric manifolds in combinatorial data. Some examples are given to show that our estimates can compute their pseudo symplectic capacities. As applications we also estimate the symplectic capacities of the polygon spaces. Other related results are impacts of symplectic blow-up on symplectic capacities, symplectic packings in symplectic toric manifolds, the Seshadri constant of an ample line bundle on toric manifolds, and symplectic capacities of symplectic manifolds with $S^{1}$-action.
TL;DR: In this paper, an analogue of the inflation technique of Lalonde-McDuff was introduced to obtain new symplectic forms from symplectic surfaces of negative self-intersection in symplectic 4-manifolds.
Abstract: We introduce an analogue of the inflation technique of Lalonde--McDuff, allowing us to obtain new symplectic forms from symplectic surfaces of negative self-intersection in symplectic 4-manifolds. We consider the implications of this construction for the symplectic cones of Kaahler surfaces, proving along the way a result which can be used to simplify the intersections of distinct pseudo-holomorphic curves via a perturbation.
TL;DR: In this paper, it was shown that any irreducible admissible representation of the symplectic similitude group GSp(V) decomposes with multiplicity one when restricted to the group Sp(V), where V is an orthogonal space overk.
Abstract: We prove several multiplicity one theorems in this paper. Fork a local field not of characteristic two, andV a symplectic space overk, any irreducible admissible representation of the symplectic similitude group GSp(V) decomposes with multiplicity one when restricted to the symplectic group Sp(V). We prove the analogous result for GO(V) and O(V), whereV is an orthogonal space overk. Whenk is non-archimedean, we prove the uniqueness of Fourier-Jacobi models for representations of GSp(4), and the existence of such models for supercuspidal representations of GSp(4).
TL;DR: In this paper, the determinant of any symplectic matrix is + 1 and the zero patterns compatible with the symplectic structure are also presented, together with some properties of Schur complements.
TL;DR: In this article, the authors determine which automorphic representations of the discrete spectrum of GL2n are distinguished by the symplectic group, and conclude a project initiated by Jacquet and Rallis [JR2].
Abstract: We determine which automorphic representations of the discrete spectrum ofGL2n are distinguished by the symplectic group. This concludes a project initiated by Jacquet and Rallis [JR2]
TL;DR: In particular, this article proved that the monodromy of the moduli space of trielliptic curves is the special unitary group (su, s) with signature σ(r, s).
Abstract: We compute the $\integ/\ell$ and $\integ_\ell$ monodromy of every irreducible component of the moduli spaces of hyperelliptic and trielliptic curves. In particular, we provide a proof that the $\integ/\ell$ monodromy of the moduli space of hyperelliptic curves of genus $g$ is the symplectic group $\sp_{2g}(\integ/\ell)$. We prove that the $\integ/\ell$ monodromy of the moduli space of trielliptic curves with signature $(r,s)$ is the special unitary group $\su_{(r,s)}(\integ/\ell\tensor\integ[\zeta_3])$.
TL;DR: The most general action, quadratic in the B fields as well as in the curvature F, having SO(3, 1) or SO(4) as the internal gauge group for a four-dimensional BF theory is presented and its symplectic geometry is displayed in this paper.
Abstract: The most general action, quadratic in the B fields as well as in the curvature F, having SO(3, 1) or SO(4) as the internal gauge group for a four-dimensional BF theory is presented and its symplectic geometry is displayed. It is shown that the space of solutions to the equations of motion for the BF theory can be endowed with symplectic structures alternative to the usual one. The analysis also includes topological terms and cosmological constant. The implications of this fact for gravity are briefly discussed.
TL;DR: In this article, it was shown that the SU(1, 1) single-mode squeezing operator F corresponds to the generalized Fresnel transform (GFT) in wave optics.
TL;DR: In this article, a generalization of a sympletic structure is an affine structure on a Lie algebra, and the cohomology space for certain Lie algebras is computed.
Abstract: We study symplectic structures on characteristically nilpotent Lie algebras (CNLAs) by computing the cohomology space $H^2(\Lg,k)$ for certain Lie algebras $\Lg$. Among these Lie algebras are filiform CNLAs of dimension $n\le 14$. It turns out that there are many examples of CNLAs which admit a symplectic structure. A generalization of a sympletic structure is an affine structure on a Lie algebra.
01 Jan 2006
TL;DR: The notion of admissible subgroups of a semidirect product G of the symplectic group and the Heisenberg group was introduced in this paper, based on the Wigner distribution.
Abstract: We consider the (extended) metaplectic representation of the semidirect product G of the symplectic group and the Heisenberg group. By looking at the standard resolution of the identity formula and inspired by previous work [5], [13], [4], we introduce the notion of admissible (reproducing) subgroup of G via the Wigner distribution. We prove some features of admissible groups and then exhibit an explicit example (d = 2) of such a group, in connection with wavelet theory.
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TL;DR: In this article, a simple characterization of Mackenzie's double Lie algebroids in terms of homological vector fields is given and an extension to the multiple case is suggested.
Abstract: We give a simple characterization of Mackenzie's double Lie algebroids in terms of homological vector fields. Application to the `Drinfeld double' of Lie bialgebroids is given and an extension to the multiple case is suggested.
TL;DR: The fundamental group and rational cohomology of the configuration spaces of the Skyrme and Faddeev-Hopf models are computed in this paper, where the topology of configuration space is taken to be a compact oriented 3-manifold, either with or without a marked point representing an end at infinity.
Abstract: The fundamental group and rational cohomology of the configuration spaces of the Skyrme and Faddeev-Hopf models are computed. Physical space is taken to be a compact oriented 3-manifold, either with or without a marked point representing an end at infinity. For the Skyrme model, the codomain is any Lie group, while for the Faddeev-Hopf model it is S
2. It is determined when the topology of configuration space permits fermionic and isospinorial quantization of the solitons of the model within generalizations of the frameworks of Finkelstein-Rubinstein and Sorkin. Fermionic quantization of Skyrmions is possible only if the target group contains a symplectic or special unitary factor, while fermionic quantization of Hopfions is always possible. Geometric interpretations of the results are given.
TL;DR: In this article, it was shown that closed symplectic four-manifolds do not admit any smooth free circle actions with contractible orbits, without assuming that the actions preserve the symplectic forms.
Abstract: We prove that closed symplectic four-manifolds do not admit any smooth free circle actions with contractible orbits, without assuming that the actions preserve the symplectic forms. In higher dimensions such actions by symplectomorphisms do exist, and we give explicit examples based on the constructions of FGM.