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Showing papers on "Lie group published in 2005"


Journal ArticleDOI
TL;DR: In this paper, a Lagrangian submanifold of a symplectic Lie algebroid is introduced and the Lagrange-Poincare equations are the local equations defining certain Lagrangians of Atiyah algebroids.
Abstract: In some previous papers, a geometric description of Lagrangian mechanics on Lie algebroids has been developed. In this topical review, we give a Hamiltonian description of mechanics on Lie algebroids. In addition, we introduce the notion of a Lagrangian submanifold of a symplectic Lie algebroid and we prove that the Lagrangian (Hamiltonian) dynamics on Lie algebroids may be described in terms of Lagrangian submanifolds of symplectic Lie algebroids. The Lagrangian (Hamiltonian) formalism on Lie algebroids permits us to deal with Lagrangian (Hamiltonian) functions not defined necessarily on tangent (cotangent) bundles. Thus, we may apply our results to the projection of Lagrangian (Hamiltonian) functions which are invariant under the action of a symmetry Lie group. As a consequence, we obtain that Lagrange–Poincare (Hamilton–Poincare) equations are the Euler–Lagrange (Hamilton) equations associated with the corresponding Atiyah algebroid. Moreover, we prove that Lagrange–Poincare (Hamilton–Poincare) equations are the local equations defining certain Lagrangian submanifolds of symplectic Atiyah algebroids.

214 citations


Posted Content
TL;DR: In this paper, a necessary and sufficient condition for a 2-dimensional Riemannian manifold to be locally isometrically immersed into a 3-dimensional homogeneous manifold with a 4-dimensional isometry group is given.
Abstract: We give a necessary and sufficient condition for a 2-dimensional Riemannian manifold to be locally isometrically immersed into a 3-dimensional homogeneous manifold with a 4-dimensional isometry group. The condition is expressed in terms of the metric, the second fundamental form, and data arising from an ambient Killing field. This class of 3-manifolds includes in particular the Berger spheres, the Heisenberg space Nil(3), the universal cover of the Lie group PSL(2,R) and the product spaces S^2 x R and H^2 x R. We give some applications to constant mean curvature (CMC) surfaces in these manifolds; in particular we prove the existence of a generalized Lawson correspondence, i.e., a local isometric correspondence between CMC surfaces in homogeneous 3-manifolds.

207 citations


Book
01 Jan 2005
TL;DR: Multiscale representations for data observed on equispaced grids and taking values in manifolds such as the sphere, the special orthogonal group, the positive definite matrices, and the Grassmann manifolds, using theExp and Log maps of those manifolds are described.
Abstract: We describe multiscale representations for data observed on equispaced grids and taking values in manifolds such as the sphere $S^2$, the special orthogonal group $SO(3)$, the positive definite matrices $SPD(n)$, and the Grassmann manifolds $G(n,k)$. The representations are based on the deployment of Deslauriers--Dubuc and average-interpolating pyramids "in the tangent plane" of such manifolds, using the $Exp$ and $Log$ maps of those manifolds. The representations provide "wavelet coefficients" which can be thresholded, quantized, and scaled in much the same way as traditional wavelet coefficients. Tasks such as compression, noise removal, contrast enhancement, and stochastic simulation are facilitated by this representation. The approach applies to general manifolds but is particularly suited to the manifolds we consider, i.e., Riemannian symmetric spaces, such as $S^{n-1}$, $SO(n)$, $G(n,k)$, where the $Exp$ and $Log$ maps are effectively computable. Applications to manifold-valued data sources of a geometric nature (motion, orientation, diffusion) seem particularly immediate. A software toolbox, SymmLab, can reproduce the results discussed in this paper.

193 citations


Journal ArticleDOI
TL;DR: In this article, a generalized Gaudin Lie algebra and a complete set of mutually commuting quantum invariants allowing the derivation of several families of exactly solvable Hamiltonians are introduced.

175 citations


Journal ArticleDOI
TL;DR: This work verifies that a linear scheme S and its analogous nonlinear scheme T satisfy a proximity condition, and shows that the proximity condition implies the convergence of T and continuity of its limit curves, if S has the same property.

154 citations


Journal ArticleDOI
TL;DR: Three Lie group integrators, the Crouch-Grossman method, commutator-free method, and Munthe-Kaas method, are formulated for the equations of motion of articulated multibody systems, providing singularity-free integration and approximated solutions always evolve on the underlying manifold structure, unlike the quaternion method.
Abstract: Numerical integration methods based on the Lie group theoretic geometrical approach are applied to articulated multibody systems with rigid body displacements, belonging to the special Euclidean group SE(3), as a part of generalized coordinates. Three Lie group integrators, the Crouch-Grossman method, commutator-free method, and Munthe-Kaas method, are formulated for the equations of motion of articulated multibody systems. The proposed methods provide singularity-free integration, unlike the Euler-angle method, while approximated solutions always evolve on the underlying manifold structure, unlike the quaternion method. In implementing the methods, the exact closed-form expression of the differential of the exponential map and its inverse on SE(3) are formulated in order to save computations for its approximation up to finite terms. Numerical simulation results validate and compare the methods by checking energy and momentum conservation at every integrated system state.

139 citations


Proceedings ArticleDOI
04 Jul 2005
TL;DR: A new analytic method based on singular value decompositions that yields a closed-form solution for simultaneous multiview registration in the noise-free scenario and an iterative scheme based on Newton's method on SO3 that has locally quadratic convergence is presented.
Abstract: We propose a novel algorithm to register multiple 3D point sets within a common reference frame using a manifold optimization approach. The point sets are obtained with multiple laser scanners or a mobile scanner. Unlike most prior algorithms, our approach performs an explicit optimization on the manifold of rotations, allowing us to formulate the registration problem as an unconstrained minimization on a constrained manifold. This approach exploits the Lie group structure of SO3 and the simple representation of its associated Lie algebra so3 in terms of R3.Our contributions are threefold. We present a new analytic method based on singular value decompositions that yields a closed-form solution for simultaneous multiview registration in the noise-free scenario. Secondly, we use this method to derive a good initial estimate of a solution in the noise-free case. This initialization step may be of use in any general iterative scheme. Finally, we present an iterative scheme based on Newton's method on SO3 that has locally quadratic convergence. We demonstrate the efficacy of our scheme on scan data taken both from the Digital Michelangelo project and from scans extracted from models, and compare it to some of the other well known schemes for multiview registration. In all cases, our algorithm converges much faster than the other approaches, (in some cases orders of magnitude faster), and generates consistently higher quality registrations.

122 citations


Journal ArticleDOI
TL;DR: In this paper, a direct method was developed to find finite symmetry groups of nonlinear mathematical physics systems using the direct method for the well-known (2+1)-dimensional Kadomtsev-Petviashvili equation and the Ablowitz-Kaup-Newell-Segur system, both the Lie point symmetry groups and the non-Lie symmetry groups are obtained.
Abstract: A new direct method is developed to find finite symmetry groups of nonlinear mathematical physics systems. Using the direct method for the well-known (2+1)-dimensional Kadomtsev–Petviashvili equation and the Ablowitz–Kaup–Newell–Segur system, both the Lie point symmetry groups and the non-Lie symmetry groups are obtained. The Lie symmetry groups obtained via traditional Lie approaches are only special cases. Furthermore, the expressions of the exact finite transformations of the Lie groups are much simpler than those obtained via the standard approaches.

121 citations


Proceedings ArticleDOI
18 Mar 2005
TL;DR: In this paper, a Lie group formulation is proposed for determining relative equilibria for arbitrary G-invariant control (where G = SE(3) is a symmetry group for the control law), and global convergence and non-collision results for specific two-vehicle interaction laws in three dimensions are presented.
Abstract: Motivated by the problem of formation control for vehicles moving at unit speed in three-dimensional space, we are led to models of gyroscopically interacting particles, which require the machinery of curves and frames to describe and analyze. A Lie group formulation arises naturally, and we discuss the general problem of determining (relative) equilibria for arbitrary G-invariant controls (where G = SE(3) is a symmetry group for the control law). We then present global convergence (and non-collision) results for specific two-vehicle interaction laws in three dimensions, which lead to specific formations (i.e., relative equilibria). Generalizations of the interaction laws to n vehicles is also discussed, and simulation results presented.

120 citations


Posted Content
TL;DR: In this paper, the equations of motion for full body models that describe the dynamics of rigid bodies, acting under their mutual gravity are derived using a variational approach where variations are defined on the Lie group of rigid body configurations.
Abstract: We develop the equations of motion for full body models that describe the dynamics of rigid bodies, acting under their mutual gravity. The equations are derived using a variational approach where variations are defined on the Lie group of rigid body configurations. Both continuous equations of motion and variational integrators are developed in Lagrangian and Hamiltonian forms, and the reduction from the inertial frame to a relative coordinate system is also carried out. The Lie group variational integrators are shown to be symplectic, to preserve conserved quantities, and to guarantee exact evolution on the configuration space. One of these variational integrators is used to simulate the dynamics of two rigid dumbbell bodies.

118 citations


Proceedings ArticleDOI
17 Oct 2005
TL;DR: A new method to estimate multiple rigid motions from noisy 3D point correspondences in the presence of outliers is proposed and a mean shift algorithm which estimates modes of the sampled distribution using the Lie group structure of the rigid motions is developed.
Abstract: We propose a new method to estimate multiple rigid motions from noisy 3D point correspondences in the presence of outliers. The method does not require prior specification of number of motion groups and estimates all the motion parameters simultaneously. We start with generating samples from the rigid motion distribution. The motion parameters are then estimated via mode finding operations on the sampled distribution. Since rigid motions do not lie on a vector space, classical statistical methods can not be used for mode finding. We develop a mean shift algorithm which estimates modes of the sampled distribution using the Lie group structure of the rigid motions. We also show that proposed mean shift algorithm is general and can be applied to any distribution having a matrix Lie group structure. Experimental results on synthetic and real image data demonstrate the superior performance of the algorithm.

Journal ArticleDOI
TL;DR: In this paper, the authors studied the existence of L p -type gradient estimates for the heat kernel of the natural hypoelliptic Laplacian on the real three-dimensional Heisenberg Lie group.

Proceedings ArticleDOI
08 Jun 2005
TL;DR: In this paper, 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, is presented, and it is shown that if a suitable error function can he found, then a general smooth and hounded reference trajectory may be tracked asymptotically from almost every initial condition with locally exponential convergence.
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 show that if a suitable error function can he found, then a general smooth and hounded reference trajectory may be tracked asymptotically from almost every initial condition, with locally exponential convergence. Such functions may be shown to exist on any compact Lie group, or on any product of a compact Lie group and R/sup n/. In the case of compact Lie groups, we show that the full-state feedback law composed with an exponentially convergent velocity estimator, also converges globally for almost every initial tracking error. We explicitly compute these controllers on SO(3), and simulate their performance for the axisymmetric top problem.

Posted Content
TL;DR: The relation between Lie 2-algebras, the Kac-Moody central extensions of loop groups, and the group String(n) was discussed in this paper.
Abstract: We describe an interesting relation between Lie 2-algebras, the Kac-Moody central extensions of loop groups, and the group String(n). A Lie 2-algebra is a categorified version of a Lie algebra where the Jacobi identity holds up to a natural isomorphism called the "Jacobiator". Similarly, a Lie 2-group is a categorified version of a Lie group. If G is a simply-connected compact simple Lie group, there is a 1-parameter family of Lie 2-algebras g_k each having Lie(G) as its Lie algebra of objects, but with a Jacobiator built from the canonical 3-form on G. There appears to be no Lie 2-group having g_k as its Lie 2-algebra, except when k = 0. Here, however, we construct for integral k an infinite-dimensional Lie 2-group whose Lie 2-algebra is equivalent to g_k. The objects of this 2-group are based paths in G, while the automorphisms of any object form the level-k Kac-Moody central extension of the loop group of G. This 2-group is closely related to the kth power of the canonical gerbe over G. Its nerve gives a topological group that is an extension of G by K(Z,2). When k = +-1, this topological group can also be obtained by killing the third homotopy group of G. Thus, when G = Spin(n), it is none other than String(n).

Journal ArticleDOI
TL;DR: In this paper, the authors apply techniques of subriemannian geometry on Lie groups and of optimal synthesis on 2-D manifolds to the population transfer problem in a three-level quantum system driven by two laser pulses, of arbitrary shape and frequency.
Abstract: We apply techniques of subriemannian geometry on Lie groups and of optimal synthesis on 2-D manifolds to the population transfer problem in a three-level quantum system driven by two laser pulses, of arbitrary shape and frequency. In the rotating wave approximation, we consider a nonisotropic model, i.e., a model in which the two coupling constants of the lasers are different. The aim is to induce transitions from the first to the third level, minimizing 1) the time of the transition (with bounded laser amplitudes), 2) the energy transferred by lasers to the system (with fixed final time). After reducing the problem to real variables, for the purpose 1) we develop a theory of time optimal syntheses for distributional problem on 2-D manifolds, while for the purpose 2) we use techniques of subriemannian geometry on 3-D Lie groups. The complete optimal syntheses are computed.

Journal ArticleDOI
TL;DR: The use of geometrical methods to tackle the non-negative independent component analysis (non-negative ICA) problem, without assuming the reader has an existing background in differential geometry, is explored.

Journal ArticleDOI
TL;DR: In this article, simple methods to establish the property of Rapid Decay for a number of groups arising geometrically were presented. But these methods are not suitable for non-cocompact lattices in rank one Lie groups.
Abstract: We explain simple methods to establish the property of Rapid Decay for a number of groups arising geometrically. Those lead to new examples of groups with the property of Rapid Decay, notably including non-cocompact lattices in rank one Lie groups.

Book
18 Nov 2005
TL;DR: A Fast Trip Through the Classical Theory as discussed by the authors shows that real singularities with a Milnor Fibration can be found in the 3-dimensional Brieskorn Manifolds.
Abstract: A Fast Trip Through the Classical Theory.- Motions in Plane Geometry and the 3-dimensional Brieskorn Manifolds.- 3-dimensional Lie Groups and Surface Singularities.- Within the Realm of the General Index Theorem.- On the Geometry and Topology of Quadrics in ??n.- Real Singularities and Complex Geometry.- Real Singularities with a Milnor Fibration.- Real Singularities and Open Book Decompositions of the 3-sphere.

Journal ArticleDOI
TL;DR: In this paper, it was shown that any Riemannian isometric action of a discrete group with property T of Kazhdan is locally rigid on a compact manifold X and a foliated version of this result was used in their proof of local rigidity for standard actions of higher rank semisimple Lie groups and their lattices.
Abstract: Let Γ be a discrete group with property (T) of Kazhdan. We prove that any Riemannian isometric action of Γ on a compact manifold X is locally rigid. We also prove a more general foliated version of this result. The foliated result is used in our proof of local rigidity for standard actions of higher rank semisimple Lie groups and their lattices in [FM2].

Journal ArticleDOI
TL;DR: In this article, the equivalence of conservation laws with respect to Lie symmetry groups for fixed systems of differential equations was introduced and the notion of local dependence of potentials was defined.
Abstract: We introduce notions of equivalence of conservation laws with respect to Lie symmetry groups for fixed systems of differential equations and with respect to equivalence groups or sets of admissible transformations for classes of such systems. We also revise the notion of linear dependence of conservation laws and define the notion of local dependence of potentials. To construct conservation laws, we develop and apply the most direct method which is effective to use in the case of two independent variables. Admitting possibility of dependence of conserved vectors on a number of potentials, we generalize the iteration procedure proposed by Bluman and Doran-Wu for finding nonlocal (potential) conservation laws. As an example, we completely classify potential conservation laws (including arbitrary order local ones) of diffusion-convection equations with respect to the equivalence group and construct an exhaustive list of locally inequivalent potential systems corresponding to these equations.

Journal ArticleDOI
TL;DR: In this paper, the authors reformulate the traditional velocity based vector space Newmark algorithm for the rotational dynamics of rigid bodies, that is for the setting of the SO(3) Lie group.
Abstract: We reformulate the traditional velocity based vector-space Newmark algorithm for the rotational dynamics of rigid bodies, that is for the setting of the SO(3) Lie group. We show that the most naive re-write of the vector space algorithm possesses the properties of symplecticity and (almost) momentum conservation. Thus, we obtain an explicit algorithm for rigid body dynamics that matches or exceeds performance of existing algorithms, but which curiously does not seem to have been considered in the open literature so far. Copyright © 2005 John Wiley & Sons, Ltd.

Journal ArticleDOI
TL;DR: In this article, a new class of nonfactorising D-branes in the product group G × G where the fluxes and metrics on the two factors do not necessarily coincide was proposed.
Abstract: We propose a new class of non-factorising D-branes in the product group G × G where the fluxes and metrics on the two factors do not necessarily coincide. They generalise the maximally symmetric permutation branes which are known to exist when the fluxes agree, but break the symmetry down to the diagonal current algebra in the generic case. Evidence for the existence of these branes comes from a lagrangian description for the open string world-sheet and from effective Dirac-Born-Infeld theory. We state the geometry, gauge fields and, in the case of SU(2) × SU(2), tensions and partial results on the open string spectrum. In the latter case the generalised permutation branes provide a natural and complete explanation for the charges predicted by K-theory including their torsion.

Journal ArticleDOI
TL;DR: In this paper, the discrete Euler-Lagrange equations are reduced to the discrete euler-Poincare-Suslov equations and the dynamics associated with these equations are shown to evolve on a subvariety of the Lie group G. The theory is illustrated with two classical nonholonomic systems, the Suslov top and the Chaplygin sleigh.
Abstract: This paper studies discrete nonholonomic mechanical systems whose configuration space is a Lie group G. Assuming that the discrete Lagrangian and constraints are left-invariant, the discrete Euler–Lagrange equations are reduced to the discrete Euler–Poincare–Suslov equations. The dynamics associated with the discrete Euler–Poincare–Suslov equations is shown to evolve on a subvariety of the Lie group G. The theory is illustrated with the discrete versions of two classical nonholonomic systems, the Suslov top and the Chaplygin sleigh. The preservation of the reduced energy by the discrete flow is observed and discrete momentum conservation is discussed.

Posted Content
TL;DR: In this paper, the authors lay the foundations of differential geometry and Lie theory over the general class of topological base fields and -rings for which a differential calculus has been developed in recent work (collaboration with H. Gloeckner and K.-H. Neeb).
Abstract: The aim of this work is to lay the foundations of differential geometry and Lie theory over the general class of topological base fields and -rings for which a differential calculus has been developed in recent work (collaboration with H. Gloeckner and K.-H. Neeb), without any restriction on the dimension or on the characteristic. Two basic features distinguish our approach from the classical real (finite or infinite dimensional) theory, namely the interpretation of tangent- and jet functors as functors of scalar extensions and the introduction of multilinear bundles and multilinear connections which generalize the concept of vector bundles and linear connections.

Journal ArticleDOI
TL;DR: It is proved that if G is a group definable in a saturated o-minimal structure, then G has no infinite descending chain of type-definable subgroups of bounded index and G/G 00 equipped with the “logic topology” is a compact Lie group.

Journal ArticleDOI
TL;DR: In this paper, the authors generalize Kirichenko's Theorem and describe almost hermitian manifolds with parallel torsion form for dimension six, and they classify all naturally reductive hermitians with small isotropy group of the characteristic torsions.

Journal ArticleDOI
TL;DR: Grothendieck's torsion index of the compact Lie group Spin(n) for any n is computed in this paper, where the authors explain the applications of this index to topology and to the study of splitting fields for quadratic forms.
Abstract: We compute Grothendieck's torsion index of the compact Lie group Spin(n) for any n. We explain the applications of the torsion index to topology and to the study of splitting fields for quadratic forms.

29 Aug 2005
TL;DR: HAL as mentioned in this paper is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not, which may come from teaching and research institutions in France or abroad, or from public or private research centers.
Abstract: HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Infinite-Dimensional Lie Groups Karl-Hermann Neeb

Journal ArticleDOI
TL;DR: In this article, it was shown that every countable direct system of finite-dimensional real or complex Lie groups has a direct limit in the category of Lie groups modelled on locally convex spaces.
Abstract: We show that every countable direct system of finite-dimensional real or complex Lie groups has a direct limit in the category of Lie groups modelled on locally convex spaces. This enables us to push all basic constructions of finite-dimensional Lie theory to the case of direct limit groups. In particular, we obtain an analogue of Lie's third theorem: Every countable-dimensional real or complex locally finite Lie algebra is enlargible, i.e., it is the Lie algebra of some regular Lie group (a suitable direct limit group).

Journal ArticleDOI
TL;DR: In this article, the authors define what it means for a state in a convex cone of states on a space of observables to be generalized-entangled relative to a subspace of the observables, in a general ordered linear spaces framework for operational theories.
Abstract: We define what it means for a state in a convex cone of states on a space of observables to be generalized-entangled relative to a subspace of the observables, in a general ordered linear spaces framework for operational theories. This extends the notion of ordinary entanglement in quantum information theory to a much more general framework. Some important special cases are described, in which the distinguished observables are subspaces of the observables of a quantum system, leading to results like the identification of generalized unentangled states with Lie-group-theoretic coherent states when the special observables form an irreducibly represented Lie algebra. Some open problems, including that of generalizing the semigroup of local operations with classical communication to the convex cones case, are discussed.