Showing papers on "Lie group published in 2006"

Covariance Tracking using Model Update Based on Lie Algebra

Abstract: We propose a simple and elegant algorithm to track nonrigid objects using a covariance based object description and a Lie algebra based update mechanism. We represent an object window as the covariance matrix of features, therefore we manage to capture the spatial and statistical properties as well as their correlation within the same representation. The covariance matrix enables efficient fusion of different types of features and modalities, and its dimensionality is small. We incorporated a model update algorithm using the Lie group structure of the positive definite matrices. The update mechanism effectively adapts to the undergoing object deformations and appearance changes. The covariance tracking method does not make any assumption on the measurement noise and the motion of the tracked objects, and provides the global optimal solution. We show that it is capable of accurately detecting the nonrigid, moving objects in non-stationary camera sequences while achieving a promising detection rate of 97.4 percent.

Towards a Lie theory of locally convex groups

Abstract: In this survey, we report on the state of the art of some of the fundamental problems in the Lie theory of Lie groups modeled on locally convex spaces, such as integrability of Lie algebras, integrability of Lie subalgebras to Lie subgroups, and integrability of Lie algebra extensions to Lie group extensions. We further describe how regularity or local exponentiality of a Lie group can be used to obtain quite satisfactory answers to some of the fundamental problems. These results are illustrated by specialization to some specific classes of Lie groups, such as direct limit groups, linear Lie groups, groups of smooth maps and groups of diffeomorphisms.

Nonassociative Tori and Applications to T-duality

Abstract: In this paper, we initiate the study of C *-algebras endowed with a twisted action of a locally compact abelian Lie group , and we construct a twisted crossed product , which is in general a nonassociative, noncommutative, algebra. The duality properties of this twisted crossed product algebra are studied in detail, and are applied to T-duality in Type II string theory to obtain the T-dual of a general principal torus bundle with general H-flux, which we will argue to be a bundle of noncommutative, nonassociative tori. Nonassociativity is interpreted in the context of monoidal categories of modules. We also show that this construction of the T-dual includes the other special cases already analysed in a series of papers.

Continuous symmetries of difference equations

Abstract: Lie group theory was originally created more than 100 years ago as a tool for solving ordinary and partial differential equations. In this article we review the results of a much more recent program: the use of Lie groups to study difference equations. We show that the mismatch between continuous symmetries and discrete equations can be resolved in at least two manners. One is to use generalized symmetries acting on solutions of difference equations, but leaving the lattice invariant. The other is to restrict them to point symmetries, but to allow them to also transform the lattice.

Time-optimal Control of Spin Systems

Abstract: The paper discusses various aspects of time-optimal control of quantum spin systems, modelled as right-invariant systems on a compact Lie group G. The main results are the reduction of such a system to an equivalent system on a homogeneous space G/H, and the explicit determination of optimal trajectories on G/H in the case where G/H is a Riemannian symmetric space. These results are mainly obtained by using methods from Lie theory and geometric control.

Differential Geometry and Lie Groups for Physicists

Abstract: Introduction 1 The concept of a manifold 2 Vector and tensor fields 3 Mappings of tensors induced by mappings of manifolds 4 Lie derivative 5 Exterior algebra 6 Differential calculus of forms 7 Integral calculus of forms 8 Particular cases and applications of Stoke's Theorem 9 Poincare Lemma and cohomologies 10 Lie Groups - basic facts 11 Differential geometry of Lie Groups 12 Representations of Lie Groups and Lie Algebras 13 Actions of Lie Groups and Lie Algebras on manifolds 14 Hamiltonian mechanics and symplectic manifolds 15 Parallel transport and linear connection on M 16 Field theory and the language of forms 17 Differential geometry on TM and T*M 18 Hamiltonian and Lagrangian equations 19 Linear connection and the frame bundle 20 Connection on a principal G-bundle 21 Gauge theories and connections 22 Spinor fields and Dirac operator Appendices Bibliography Index

Automorphic distributions, L-functions, and Voronoi summation for GL(3)

Abstract: This paper is third in a series of three, following "Summation Formulas, from Poisson and Voronoi to the Present" (math.NT/0304187) and "Distributions and Analytic Continuation of Dirichlet Series" (math.FA/0403030). The first is primarily an expository paper explaining the present one, whereas the second contains some distributional machinery used here as well. These papers concern the boundary distributions of automorphic forms, and how they can be applied to study questions about cusp forms on semisimple Lie groups. The main result of this paper is a Voronoi-style summation formula for the Fourier coefficients of a cusp form on GL(3,Z)\GL(3,R). We also give a treatment of the standard L-function on GL(3), focusing on the archimedean analysis as performed using distributions. Finally a new proof is given of the GL(3)xGL(1) converse theorem of Jacquet, Piatetski-Shapiro, and Shalika. This paper is also related to the later papers math.NT/0402382 and math.NT/0404521.

Almost-global tracking of simple mechanical systems on a general class of Lie Groups

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.

Nonlinear Mean Shift for Clustering over Analytic Manifolds

Abstract: The mean shift algorithm is widely applied for nonparametric clustering in Euclidean spaces. Recently, mean shift was generalized for clustering on matrix Lie groups. We further extend the algorithm to a more general class of nonlinear spaces, the set of analytic manifolds. As examples, two specific classes of frequently occurring parameter spaces, Grassmann manifolds and Lie groups, are considered. When the algorithm proposed here is restricted to matrix Lie groups the previously proposed method is obtained. The algorithm is applied to a variety of robust motion segmentation problems and multibody factorization. The motion segmentation method is robust to outliers, does not require any prior specification of the number of independent motions and simultaneously estimates all the motions present.

Applications of equivariant cohomology

Abstract: We will discuss the equivariant cohomology of a manifold endowed with the action of a Lie group. Localization formulae for equivariant integrals are explained by a vanishing theorem for equivariant cohomology with generalized coefficients. We then give applications to integration of characteristic classes on symplectic quotients and to indices of transversally elliptic operators. In particular, we state a conjecture for the index of a transversally elliptic operator linked to a Hamiltonian action. In the last part, we describe algorithms for numerical computations of values of multivariate spline functions and of vector-partition functions of classical root systems.

Cluster χ-varieties, amalgamation, and Poisson—Lie groups

Abstract: In this paper, starting from a split semisimple real Lie group G with trivial center, we define a family of varieties with additional structures. We describe them as cluster χ-varieties, as defined in [FG2]. In particular they are Poisson varieties. We define canonical Poisson maps of these varieties to the group G equipped with the standard Poisson—Lie structure defined by V. Drinfeld in [D, D1]. One of them maps to the group birationally and thus provides G with canonical rational coordinates.

Contractions of low-dimensional Lie algebras

Abstract: Theoretical background of continuous contractions of finite-dimensional Lie algebras is rigorously formulated and developed. In particular, known necessary criteria of contractions are collected and new criteria are proposed. A number of requisite invariant and semi-invariant quantities are calculated for wide classes of Lie algebras including all low-dimensional Lie algebras. An algorithm that allows one to handle one-parametric contractions is presented and applied to low-dimensional Lie algebras. As a result, all one-parametric continuous contractions for both the complex and real Lie algebras of dimensions not greater than 4 are constructed with intensive usage of necessary criteria of contractions and with studying correspondence between real and complex cases. Levels and colevels of low-dimensional Lie algebras are discussed in detail. Properties of multiparametric and repeated contractions are also investigated.

Proper groupoids and momentum maps : Linearization, affinity, and convexity

Abstract: We show that proper Lie groupoids are locally linearizable. As a consequence, the orbit space of a proper Lie groupoid is a smooth orbispace (a Hausdorff space which locally looks like the quotient of a vector space by a linear compact Lie group action). In the case of proper (quasi-)symplectic groupoids, the orbit space admits a natural integral affine structure, which makes it into an affine orbifold with locally convex polyhedral boundary, and the local structure near each boundary point is isomorphic to that of a Weyl chamber of a compact Lie group. We then apply these results to the study of momentum maps of Hamiltonian actions of proper (quasi-)symplectic groupoids, and show that these momentum maps preserve natural transverse affine structures with local convexity properties. Many convexity theorems in the literature can be recovered from this last statement and some elementary results about affine maps.

Littlewood-Paley decompositions and Besov spaces on Lie groups of polynomial growth

Abstract: We introduce a Littlewood--Paley decomposition related to any sub-Laplacian on a Lie group G of polynomial volume growth; this allows us to prove a Littlewood--Paley theorem in this general setting and to provide a dyadic characterization of Besov spaces B^{s,q}_p(G), s in R, equivalent to the classical definition through the heat kernel.

Yang-Baxter maps: dynamical point of view

Abstract: A review of some recent results on the dynamical theory of the Yang- Baxter maps (also known as set-theoretical solutions to the quantum Yang-Baxter equation) is given. The central question is the integrability of the transfer dynamics. The relations with matrix factorisations, matrix KdV solitons, Poisson Lie groups, geometric crystals and tropical combinatorics are discussed and demonstrated on several concrete examples.

Sampling theorems on locally compact groups from oscillation estimates

Abstract: We present a general approach to derive sampling theorems on locally compact groups from oscillation estimates. We focus on the L 2-stability of the sampling operator by using notions from frame theory. This approach yields particularly simple and transparent reconstruction procedures. We then apply these methods to the discretization of discrete series representations and to Paley–Wiener spaces on stratified Lie groups.

Unitary representations of super Lie groups and applications to the classification and multiplet structure of super particles

Abstract: It is well known that the category of super Lie groups (SLG) is equivalent to the category of super Harish-Chandra pairs (SHCP). Using this equivalence, we define the category of unitary representations (UR's) of a super Lie group. We give an extension of the classical inducing construction and Mackey imprimitivity theorem to this setting. We use our results to classify the irreducible unitary representations of semidirect products of super translation groups by classical Lie groups, in particular of the super Poincare groups in arbitrary dimension and signature. Finally we compare our results with those in the physical literature on the structure and classification of super multiplets.

Sobolev spaces on lie manifolds and regularity for polyhedral domains

Abstract: We study some basic analytic questions related to dif- ferential operators on Lie manifolds, which are manifolds whose large scale geometry can be described by a a Lie algebra of vector fields on a compactification. We extend to Lie manifolds several classical results on Sobolev spaces, elliptic regularity, and mapping properties of pseudodifferential operators. A tubular neighborhood theorem for Lie submanifolds allows us also to extend to regular open subsets of Lie manifolds the classical results on traces of functions in suitable Sobolev spaces. Our main application is a regularity result on poly- hedral domains P ⊂ R 3 using the weighted Sobolev spaces K m (P). In particular, we show that there is no loss of K m -regularity for solutions of strongly elliptic systems with smooth coefficients. For the proof, we identify K m (P) with the Sobolev spaces on P associated to the metric r −2 P gE, where gE is the Euclidean metric and rP(x) is a smoothing of the Euclidean distance from x to the set of singular points of P. A suitable compactification of the interior of P then becomes a regular open subset of a Lie manifold. We also obtain the well-posedness of a non-standard boundary value problem on a smooth, bounded do- main with boundary O ⊂ R n using weighted Sobolev spaces, where the weight is the distance to the boundary.

Geometry of, and via, symmetries

Abstract: It is well known that Lie groups and homogeneous spaces provide a rich source of interesting examples for a variety of geometric aspects. Likewise it is often the case that topological and geometric restrictions yield the existence of isometries in a more or less direct way. The most obvious example of this is the group of deck transformations of the universal cover of a nonsimply connected manifold. More subtle situations arise in the contexts of rigidity problems and of collapsing with bounded curvature. Our main purpose here is to present the view point that the geometry of isometry groups provide a natural and useful link between theory and examples in Riemannian geometry. This fairly unexplored territory is fascinating and interesting in its own right. At the same time it enters naturally when such groups arise in settings as above. More importantly, perhaps, this study also provides a systematic search for geometrically interesting examples, where the group of isometries is short of acting transitively in contrast to the case of homogeneous spaces mentioned above. Although the general philosophy presented here applies to many different situations, we will illustrate our point of view primarily within the context of manifolds with nonnegative or positive curvature. We have divided our presentation into five sections. The first section is concerned with basic equivariant Riemannian geometry of smooth compact transformation groups, including a treatment of Alexandrov geometry of orbit spaces. Section two is the heart of the subject. It deals with the geometry and topology in the presence of symmetries. It is here we explain our guiding principle which provides a systematic search for new constructions and examples of manifolds of positive or nonnegative curvature. In the third section we exhibit all the known constructions and examples of such manifolds. The topic of section four is geometry via symmetries. We display three different types of problems in which symmetries are not immediately present from the outset, but where their emergence is crucial to their solutions. In the last section we discuss a number of open problems and conjectures related either directly, potentially or at least in spirit to the subject presented here. Our exposition assumes basic knowledge of Riemannian geometry, and a rudimentary familiarity with Lie groups. Although we use Alexandrov geometry of spaces with a lower curvature bound our treatment does not require prior knowledge of this subject. Our intentions have been that anyone with these prerequisites will be able to get an impression of the subject, and guided by the references provided here will be able to go as far as their desires will take them.

Explicit Magnus expansions for nonlinear equations

Abstract: In this paper we develop and analyse new explicit Magnus expansions for the nonlinear equation Y' = A(t, Y)Y defined in a matrix Lie group. In particular, integration methods up to order four are presented in terms of integrals which can be either evaluated exactly or replaced by conveniently adapted quadrature rules. The structure of the algorithm allows us to change the step size and even the order during the integration process, thus improving its efficiency. Several examples are considered, including isospectral flows and highly oscillatory nonlinear differential equations.

Structured Eigenvalue Condition Numbers

Abstract: This paper investigates the effect of structure-preserving perturbations on the eigenvalues of linearly and nonlinearly structured eigenvalue problems. Particular attention is paid to structures that form Jordan algebras, Lie algebras, and automorphism groups of a scalar product. Bounds and computable expressions for structured eigenvalue condition numbers are derived for these classes of matrices, which include complex symmetric, pseudo-symmetric, persymmetric, skew-symmetric, Hamiltonian, symplectic, and orthogonal matrices. In particular we show that under reasonable assumptions on the scalar product, the structured and unstructured eigenvalue condition numbers are equal for structures in Jordan algebras. For Lie algebras, the effect on the condition number of incorporating structure varies greatly with the structure. We identify Lie algebras for which structure does not affect the eigenvalue condition number.

Contractions of Low-Dimensional Lie Algebras

Abstract: Theoretical background of continuous contractions of finite-dimensional Lie algebras is rigorously formulated and developed. In particular, known necessary criteria of contractions are collected and new criteria are proposed. A number of requisite invariant and semi-invariant quantities are calculated for wide classes of Lie algebras including all low-dimensional Lie algebras. An algorithm that allows one to handle one-parametric contractions is presented and applied to low-dimensional Lie algebras. As a result, all one-parametric continuous contractions for the both complex and real Lie algebras of dimensions not greater than four are constructed with intensive usage of necessary criteria of contractions and with studying correspondence between real and complex cases. Levels and co-levels of low-dimensional Lie algebras are discussed in detail. Properties of multi-parametric and repeated contractions are also investigated.

Semiclassical differential structures

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.

Testing the Master Constraint Programme for Loop Quantum Gravity III. SL(2,R) Models

Abstract: This is the third paper in our series of five in which we test the master constraint programme for solving the Hamiltonian constraint in loop quantum gravity. In this work, we analyse models which, despite the fact that the phase space is finite dimensional, are much more complicated than in the second paper. These are systems with an gauge symmetry and the complications arise because non-compact semisimple Lie groups are not amenable (have no finite translation invariant measure). This leads to severe obstacles in the refined algebraic quantization programme (group averaging) and we see a trace of that in the fact that the spectrum of the master constraint does not contain the point zero. However, the minimum of the spectrum is of order 2 which can be interpreted as a normal ordering constant arising from first class constraints (while second class systems lead to normal ordering constants). The physical Hilbert space can then be obtained after subtracting this normal ordering correction.

A Canonical Compatible Metric for Geometric Structures on Nilmanifolds

Abstract: Let (N, γ) be a nilpotent Lie group endowed with an invariant geometric structure (cf. symplectic, complex, hypercomplex or any of their ‘almost’ versions). We define a left invariant Riemannian metric on N compatible with γ to be minimal, if it minimizes the norm of the invariant part of the Ricci tensor among all compatible metrics with the same scalar curvature. We prove that minimal metrics (if any) are unique up to isometry and scaling, they develop soliton solutions for the ‘invariant Ricci’ flow and are characterized as the critical points of a natural variational problem. The uniqueness allows us to distinguish two geometric structures with Riemannian data, giving rise to a great deal of invariants.

Cartan decomposition of the moment map

Abstract: We investigate a class of actions of real Lie groups on complex spaces. Using moment map techniques we establish the existence of a quotient and a version of Luna’s slice theorem as well as a version of the Hilbert–Mumford criterion. A global slice theorem is proved for proper actions. We give new proofs of results of Mostow on decompositions of groups and homogeneous spaces.

Smoothness Analysis of Subdivision Schemes by Proximity

Abstract: Linear curve subdivision schemes may be perturbed in various ways, for example, by modifying them such as to work in a manifold, surface, or group. The analysis of such perturbed and often nonlinear schemes "T" is based on their proximity to the linear schemes "S" which they are derived from. This paper considers two aspects of this problem: One is to find proximity inequalities which together with Ck smoothness of S imply Ck smoothness of T. The other is to verify these proximity inequalities for several ways to construct the nonlinear scheme T analogous to the linear scheme S. The first question is treated for general k, whereas the second one is treated only in the case k = 2. The main result of the paper is that convergent geodesic/projection/Lie group analogues of a certain class of factorizable linear schemes have C2 limit curves.

Equivalences of Classifying Spaces Completed at the Prime Two

Abstract: Introduction Higher limits over orbit categories Reduction to simple groups A relative version of $\Lambda$-functors Subgroups which contribute to higher limits Alternating groups Groups of Lie type in characteristic two Classical groups of Lie type in odd characteristic Exceptional groups of Lie type in odd characteristic Sproadic groups Computations of $\textrm{lim}^1(\mathcal{Z}_G)$ Bibliography.

Integrable and superintegrable systems with spin

Abstract: A system of two particles with spin s=0 and s=12, respectively, moving in a plane is considered. It is shown that such a system with a nontrivial spin-orbit interaction can allow an eight dimensional Lie algebra of first-order integrals of motion. The Pauli equation is solved in this superintegrable case and reduced to a system of ordinary differential equations when only one first-order integral exists.

Noether Symmetries Versus Killing Vectors and Isometries of Spacetimes

Abstract: Symmetries of spacetime manifolds which are given by Killing vectors are compared with the symmetries of the Lagrangians of the respective spacetimes. We find the point generators of the one parameter Lie groups of transformations that leave invariant the action integral corresponding to the Lagrangian (Noether symmetries). In the examples considered, it is shown that the Noether symmetries obtained by considering the Larangians provide additional symmetries which are not provided by the Killing vectors. It is conjectured that these symmetries would always provide a larger Lie algebra of which the KV symmetres will form a subalgebra.