Showing papers on "Lie group published in 2004"

Lectures on the Orbit Method

Abstract: Geometry of coadjoint orbits Representations and orbits of the Heisenberg group The orbit method for nilpotent Lie groups Solvable Lie groups Compact Lie groups Miscellaneous Abstract nonsense Smooth manifolds Lie groups and homogeneous manifolds Elements of functional analysis Representation theory References Index.

The geometry of physics : an introduction

Abstract: Preface Part I. Manifolds, Tensors and Exterior Forms: 1. Manifolds and vector fields 2. Tensors and exterior forms 3. Integration of differential forms 4. The Lie derivative 5. The Poincare Lemma and potentials 6. Holonomic and non-holonomic constraints Part II. Geometry and Topology: 7. R3 and Minkowski space 8. The geometry of surfaces in R3 9. Covariant differentiation and curvature 10. Geodesics 11. Relativity, tensors, and curvature 12. Curvature and topology: Synge's theorem 13. Betti numbers and De Rham's theorem 14. Harmonic forms Part III. Lie Groups, Bundles and Chern Forms: 15. Lie groups 16. Vector bundles in geometry and physics 17. Fiber bundles, Gauss-Bonnet, and topological quantization 18. Connections and associated bundles 19. The Dirac equation 20. Yang-Mills fields 21. Betti numbers and covering spaces 22. Chern forms and homotopy groups Appendix: forms in continuum mechanics.

Lie theory for nilpotent L-infinity algebras

Abstract: The Deligne groupoid is a functor from nilpotent differential graded Lie algebras concentrated in positive degrees to groupoids; in the special case of Lie algebras over a field of characteristic zero, it gives the associated simply connected Lie group. We generalize the Deligne groupoid to a functor gamma from L-infinity algebras concentrated in degree >-n to n-groupoids. (We actually construct the nerve of the n-groupoid, which is an enriched Kan complex.) The construction of gamma is quite explicit (it is based on Dupont's proof of the de Rham theorem) and yields higher dimensional analogues of holonomy and of the Campbell-Hausdorff formula. In the case of abelian L-infinity algebras (i.e. chain complexes), the functor gamma is the Dold-Kan simplicial set.

Regularizing Flows for Constrained Matrix-Valued Images

Abstract: Nonlinear diffusion equations are now widely used to restore and enhance images. They allow to eliminate noise and artifacts while preserving large global features, such as object contours. In this context, we propose a differential-geometric framework to define PDEs acting on some manifold constrained datasets. We consider the case of images taking value into matrix manifolds defined by orthogonal and spectral constraints. We directly incorporate the geometry and natural metric of the underlying configuration space (viewed as a Lie group or a homogeneous space) in the design of the corresponding flows. Our numerical implementation relies on structure-preserving integrators that respect intrinsically the constraints geometry. The efficiency and versatility of this approach are illustrated through the anisotropic smoothing of diffusion tensor volumes in medical imaging.

Twisted K-theory

Abstract: Twisted complex K-theory can be defined for a space X equipped with a bundle of complex projective spaces, or, equivalently, with a bundle of C ∗ -algebras. Up to equivalence, the twisting corresponds to an element of H 3 (X; Z). We give a systematic account of the definition and basic properties of the twisted theory, emphasizing some points where it behaves differently from ordinary K-theory. (We omit, however, its relations to classical coho- mology, which we shall treat in a sequel.) We develop an equivariant version of the theory for the action of a compact Lie group, proving that then the twistings are classified by the equivariant cohomology group H 3 G (X; Z). We also consider some basic examples of twisted K-theory classes, related to those appearing in the recent work of Freed-Hopkins-Teleman.

Lie Algebras and Lie Groups

Abstract: In this crucial lecture we introduce the definition of the Lie algebra associated to a Lie group and its relation to that group. All three sections are logically necessary for what follows; §8.1 is essential. We use here a little more manifold theory: specifically, the differential of a map of manifolds is used in a fundamental way in §8.1, the notion of the tangent vector to an arc in a manifold is used in §8.2 and §8.3, and the notion of a vector field is introduced in an auxiliary capacity in §8.3. The Campbell-Hausdorff formula is introduced only to establish the First and Second Principles of §8.1 below; if you are willing to take those on faith the formula (and exercises dealing with it) can be skimmed. Exercises 8.27–8.29 give alternative descriptions of the Lie algebra associated to a Lie group, but can be skipped for now.

Type synthesis of 3R2T 5-DOF parallel mechanisms using the Lie group of displacements

Abstract: The use of lower mobility parallel manipulators with less than six degrees of freedom (DOFs) has drawn a lot of interest in the area of parallel robots In this paper, the type synthesis of 3R2T 5-DOF parallel mechanisms (PMs) is performed systematically using the Lie group of displacements, where R denotes a rotational DOF, and T denotes a translational DOF First, some necessary theoretical fundamentals about the displacement group are recalled Then, a general approach is proposed for the type synthesis of 3R2T 5-DOF PMs The limb kinematic chains, which produce the desired displacement manifolds, are synthesized and enumerated Structural conditions, which guarantee that the intersection of the displacement manifolds generated by the limb is the desired 5-DOF manifold, are presented An exhaustive enumeration of 3R2T 5-DOF symmetrical PMs is obtained Finally, an input selection method is proposed

Integration of twisted Dirac brackets

Abstract: Given a Lie groupoid G over a manifold M, we show that multiplicative 2-forms on G relatively closed with respect to a closed 3-form phi on M correspond to maps from the Lie algebroid of G into T* M satisfying an algebraic condition and a differential condition with respect to the phi-twisted Courant bracket. This correspondence describes, as a special case, the global objects associated to phi-twisted Dirac structures. As applications, we relate our results to equivariant cohomology and foliation theory, and we give a new description of quasi-Hamiltonian spaces and group-valued momentum maps.

Simple Lie Algebras over Fields of Positive Characteristic : I. Structure Theory

Abstract: The problem of classifying the finite-dimensional simple Lie algebras over fields of characteristic p > 0 is a longstanding one. Work on this question during the last 35 years has been directed by the Kostrikin–Shafarevich Conjecture of 1966, which states that over an algebraically closed field of characteristic p > 5 a finite-dimensional restricted simple Lie algebra is classical or of Cartan type. This conjecture was proved for p > 7 by Block and Wilson in 1988. The generalization of the Kostrikin–Shafarevich Conjecture for the general case of not necessarily restricted Lie algebras and p > 7 was announced in 1991 by Strade and Wilson and eventually proved by Strade in 1998. The final Block–Wilson–Strade–Premet Classification Theorem is a landmark result of modern mathematics and can be formulated as follows: Every finite-dimensional simple Lie algebra over an algebraically closed field of characteristic p > 3 is of classical, Cartan, or Melikian type. In the two-volume book, the author is assembling the proof of the Classification Theorem with explanations and references. The goal is a state-of-the-art account on the structure and classification theory of Lie algebras over fields of positive characteristic leading to the forefront of current research in this field. This first volume is devoted to preparing the ground for the classification work to be performed in the second volume. The concise presentation of the general theory underlying the subject matter and the presentation of classification results on a subclass of the simple Lie algebras for all odd primes will make this volume an invaluable source and reference for all research mathematicians and advanced graduate students in algebra.

Geometric Optics on Phase Space

Abstract: Preliminary ToC: Optical Phase Space, Hamiltonian Systems and Lie Algebras.- Lie Groups of Optical Transformation.- The Paraxial Regime.- Hamiltonian Aberrations.

Levy Processes in Lie Groups

Abstract: It is well known that the distribution of a classical Levy process in a Euclidean space \(\mathbb {R}^d\) is determined by a triple of a drift vector, a covariance matrix, and a Levy measure, which are called the characteristics of the Levy process. The triple appears in the Levy-Khinchin formula, which is the Fourier transform of the distribution, or in the pathwise Levy-Ito representation. In the latter representation, the three elements of the triple correspond respectively to a nonrandom drift, a diffusion part, and a pure jump part of the process. A Levy process in a Lie group G cannot be decomposed into three parts as in \(\mathbb {R}^d\), due to the non-commutative nature of G, but the triple representation holds in an infinitesimal sense, in the form of Hunt’s generator formula, to be discussed in §2.1. The relation between Levy measures and jumps of Levy processes is considered in §2.2.

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.

Equivariant normal form for nondegenerate singular orbits of integrable hamiltonian systems

Abstract: We consider an integrable Hamiltonian system with n degrees of freedom whose first integrals are invariant under the symplectic action of a compact Lie group G. We prove that the singular Lagrangian foliation associated to this Hamiltonian system is symplectically equivalent, in a G-equivariant way, to the linearized foliation in a neighborhood of a compact singular nondegenerate orbit. We also show that the nondegeneracy condition is not equivalent to the nonresonance condition for smooth systems.

Freezing Solutions of Equivariant Evolution Equations

Abstract: In this paper we develop numerical methods for integrating general evolution equations u t = F(u), $u(0)=u 0, where F is defined on a dense subspace of some Banach space (generally infinite-dimensional) and is equivariant with respect to the action of a finite-dimensional (not necessarily compact) Lie group. Such equations typically arise from autonomous PDEs on unbounded domains that are invariant under the action of the Euclidean group or one of its subgroups. In our approach we write the solution u(t) as a composition of the action of a time-dependent group element with a "frozen solution" in the given Banach space. We keep the frozen solution as constant as possible by introducing a set of algebraic constraints (phase conditions), the number of which is given by the dimension of the Lie group. The resulting PDAE (partial differential algebraic equation) is then solved by combining classical numerical methods, such as restriction to a bounded domain with asymptotic boundary conditions, half-explicit Eu...

Nonholonomic LR Systems as Generalized Chaplygin Systems with an Invariant Measure and Flows on Homogeneous Spaces

Abstract: We consider a class of dynamical systems on a compact Lie group G with a left-invariant metric and right-invariant nonholonomic constraints (so-called LR systems) and show that, under a generic condition on the constraints, such systems can be regarded as generalized Chaplygin systems on the principle bundle G \to Q = G/H, H being a Lie subgroup. In contrast to generic Chaplygin systems, the reductions of our LR systems onto the homogeneous space Q always possess an invariant measure. We study the case G = SO(n), when LR systems are ultidimensional generalizations of the Veselova problem of a nonholonomic rigid body motion which admit a reduction to systems with an invariant measure on the (co)tangent bundle of Stiefel varieties V(k, n) as the corresponding homogeneous spaces. For k = 1 and a special choice of the left-invariant metric on SO(n), we prove that after a time substitution the reduced system becomes an integrable Hamiltonian system describing a geodesic flow on the unit sphere Sn-1. This provides a first example of a nonholonomic system with more than two degrees of freedom for which the celebrated Chaplygin reducibility theorem is applicable for any dimension. In this case we also explicitly reconstruct the motion on the group SO(n).

A globally convergent numerical algorithm for computing the centre of mass on compact Lie groups

Abstract: Motivated by applications in fuzzy control, robotics and vision, this paper considers the problem of computing the centre of mass (precisely, the Karcher mean) of a set of points defined on a compact Lie group, such as the special orthogonal group consisting of all orthogonal matrices with unit determinant. An iterative algorithm, whose derivation is based on the geometry of the problem, is proposed. It is proved to be globally convergent. Interestingly, the proof starts by showing the algorithm is actually a Riemannian gradient descent algorithm with fixed step size.

On the geometry of Riemannian manifolds with a Lie structure at infinity

Abstract: We study a generalization of the geodesic spray and give conditions for noncomapct manifolds with a Lie structure at infinity to have positive injectivity radius. We also prove that the geometric operators are generated by the given Lie algebra of vector fields. This is the first one in a series of papers devoted to the study of the analysis of geometric differential operators on manifolds with Lie structure at infinity.

Higher gauge theory I: 2-Bundles

Abstract: I categorify the definition of fibre bundle, replacing smooth manifolds with differentiable categories, Lie groups with coherent Lie 2-groups, and bundles with a suitable notion of 2-bundle. To link this with previous work, I show that certain 2-categories of principal 2-bundles are equivalent to certain 2-categories of (nonabelian) gerbes. This relationship can be (and has been) extended to connections on 2-bundles and gerbes. The main theorem, from a perspective internal to this paper, is that the 2-category of 2-bundles over a given 2-space under a given 2-group is (up to equivalence) independent of the fibre and can be expressed in terms of cohomological data (called 2-transitions). From the perspective of linking to previous work on gerbes, the main theorem is that when the 2-space is the 2-space corresponding to a given space and the 2-group is the automorphism 2-group of a given group, then this 2-category is equivalent to the 2-category of gerbes over that space under that group (being described by the same cohomological data).

Invariant Finsler metrics on homogeneous manifolds

Abstract: In this paper, we study invariant Finsler metrics on homogeneous manifolds. We first give an algebraic description of these metrics and obtain a necessary and sufficient condition for a homogeneous manifold to have invariant Finsler metrics. As a special case, we study bi-invariant Finsler metrics on Lie groups and obtain a necessary and sufficient condition for a Lie group to have bi-invariant Finsler metrics. Finally, we provide some conditions for a homogeneous manifold to admit invariant non-Riemannian Finsler metrics and present some interesting examples.

Probabilities and Statistics on Riemannian Manifolds : A Geometric approach

Abstract: Measurements of geometric primitives are often noisy in real applications and we need to use statistics either to reduce the uncertainty (estimation), to compare measurements, or to test hypotheses. Unfortunately, geometric primitives often belong to manifolds that are not vector spaces. In previous works , we used invariance requirements to develop some basic probability tools on transformation groups and homogeneous manifolds that avoids paradoxes. In this paper, we consider the Riemannian metric as the basic structure for the manifold. Based on this metric, we develop the notions of mean value and covariance matrix of a random element, normal law, Mahalanobis distance and ^2 test. We provide a simple (but highly non trivial) characterization of Karcher means and an original gradient descent algorithm to efficiently compute them. The notion of Normal law we propose is based on the the minimization of the information knowing the mean and covariance of the distribution. The resulting family of pdfs spans the whole range from uniform (on compact manifolds) to the point mass distribution. Moreover, we were able to provide tractable approximations (with their limits) for small variances which show that we can effectively implement and work with these definitions. To come back to more practical cases, we then reconsider the case of connected Lie groups and homogeneous manifolds. In our Riemannian context, we investigate the use of invariance principles to choose the metric: we show that it can provide the stability of our statistical definitions w.r.t. geometric operations (composition, inversion and action of transformations). However, an invariant metric does not always exists for homogeneous manifolds, nor does a left and right invariant metric for non-compact Lie groups. In this case, we cannot guaranty the full consistency of geometric and statistical operations. Thus, future work will have to concentrate on constraints weaker than invariance.

A singularity removal theorem for Yang-Mills fields in higher dimensions

Abstract: The purpose of this paper is to investigate the small-energy behavior of weakly Yang-Mills fields in Rn for n > 4, and in particular to extend the singularity removal theorem of Uhlenbeck [7] to higher dimensions. Fix n > 4, and let ft be some bounded domain in Rn; typically we shall restrict our attention to the cubes ft = [-1, 2]n or ft = [0, l]n. Let G be a fixed finite-dimensional compact Lie group; it will be convenient to consider G as embedded in some large unitary group U(N). Let g be the Lie algebra of G. We define a connection on ft to be a section A of T*ft ? q (i.e., a g-valued 1-form) which is locally L2. For any connection A let (1) F(A):=dA + AAA

Anosov automorphisms on compact nilmanifolds associated with graphs

Abstract: We associate with each graph (S, E) a 2-step simply connected nilpotent Lie group N and a lattice Γ in N. We determine the group of Lie automorphisms of N and apply the result to describe a necessary and sufficient condition, in terms of the graph, for the compact nilmanifold N/F to admit an Anosov automorphism. Using the criterion we obtain new examples of compact nilmanifolds admitting Anosov automorphisms, and conclude that for every n > 17 there exist a n-dimensional 2-step simply connected nilpotent Lie group N which is indecomposable (not a direct product of lower dimensional nilpotent Lie groups), and a lattice Γ in N such that N/F admits an Anosov automorphism; we give also a lower bound on the number of mutually nonisomorphic Lie groups N of a given dimension, satisfying the condition. Necessary and sufficient conditions are also described for a compact nilmanifold as above to admit ergodic automorphisms.

Continuous symmetries of Lagrangians and exact solutions of discrete equations

Abstract: One of the difficulties encountered when studying physical theories in discrete space–time is that of describing the underlying continuous symmetries (like Lorentz, or Galilei invariance). One of the ways of addressing this difficulty is to consider point transformations acting simultaneously on difference equations and lattices. In a previous article we have classified ordinary difference schemes invariant under Lie groups of point transformations. The present article is devoted to an invariant Lagrangian formalism for scalar single-variable difference schemes. The formalism is used to obtain first integrals and explicit exact solutions of the schemes. Equations invariant under two- and three-dimensional groups of Lagrangian symmetries are considered

Symmetry classification of KdV-type nonlinear evolution equations

Abstract: Group classification of a class of third-order nonlinear evolution equations generalizing KdV and mKdV equations is performed. It is shown that there are two equations admitting simple Lie algebras of dimension three. Next, we prove that there exist only four equations invariant with respect to Lie algebras having nontrivial Levi factors of dimension four and six. Our analysis shows that there are no equations invariant under algebras which are semi-direct sums of Levi factor and radical. Making use of these results we prove that there are three, nine, thirty-eight, fifty-two inequivalent KdV-type nonlinear evolution equations admitting one-, two-, three-, and four-dimensional solvable Lie algebras, respectively. Finally, we perform a complete group classification of the most general linear third-order evolution equation.

Twisted K-Theory of Lie Groups

Abstract: I determine the twisted K-theory of all compact simply connected simple Lie groups. The computation reduces via the Freed-Hopkins-Teleman theorem [1] to the CFT prescription, and thus explains why it gives the correct result. Finally I analyze the exceptions noted by Bouwknegt et al. [2].