Showing papers on "Lie group published in 2000"

Lie-group methods

Abstract: Many differential equations of practical interest evolve on Lie groups or on manifolds acted upon by Lie groups. The retention of Lie-group structure under discretization is often vital in the recovery of qualitatively correct geometry and dynamics and in the minimization of numerical error. Having introduced requisite elements of differential geometry, this paper surveys the novel theory of numerical integrators that respect Lie-group structure, highlighting theory, algorithmic issues and a number of applications.

The Poisson formula for groups with hyperbolic properties

Abstract: The Poisson boundary of a group G with a probability measure „ is the space of ergodic components of the time shift in the path space of the associated random walk. Via a generalization of the classical Poisson formula it gives an integral representation of bounded „-harmonic functions on G. In this paper we develop a new method of identifying the Poisson boundary based on entropy estimates for conditional random walks. It leads to simple purely geometric criteria of boundary maximality which bear hyperbolic nature and allow us to identify the Poisson boundary with natural topological boundaries for several classes of groups: word hyperbolic groups and discontinuous groups of isometries of Gromov hyperbolic spaces, groups with inflnitely many ends, cocompact lattices in Cartan-Hadamard manifolds, discrete subgroups of semisimple Lie groups.

Symmetry Methods for Differential Equations: A Beginner's Guide

Abstract: 1. Introduction to symmetries 1.1. Symmetries of planar objects 1.2. Symmetries of the simplest ODE 1.3. The symmetry condition for first-order ODEs 1.4. Lie symmetries solve first-order ODEs 2. Lie symmetries of first order ODEs 2.1. The action of Lie symmetries on the plane 2.2. Canonical coordinates 2.3. How to solve ODEs with Lie symmetries 2.4. The linearized symmetry condition 2.5. Symmetries and standard methods 2.6. The infinitesimal generator 3. How to find Lie point symmetries of ODEs 3.1 The symmetry condition. 3.2. The determining equations for Lie point symmetries 3.3. Linear ODEs 3.4. Justification of the symmetry condition 4. How to use a one-parameter Lie group 4.1. Reduction of order using canonical coordinates 4.2. Variational symmetries 4.3. Invariant solutions 5. Lie symmetries with several parameters 5.1. Differential invariants and reduction of order 5.2. The Lie algebra of point symmetry generators 5.3. Stepwise integration of ODEs 6. Solution of ODEs with multi-parameter Lie groups 6.1 The basic method: exploiting solvability 6.2. New symmetries obtained during reduction 6.3. Integration of third-order ODEs with sl(2) 7. Techniques based on first integrals 7.1. First integrals derived from symmetries 7.2. Contact symmetries and dynamical symmetries 7.3. Integrating factors 7.4. Systems of ODEs 8. How to obtain Lie point symmetries of PDEs 8.1. Scalar PDEs with two dependent variables 8.2. The linearized symmetry condition for general PDEs 8.3. Finding symmetries by computer algebra 9. Methods for obtaining exact solutions of PDEs 9.1. Group-invariant solutions 9.2. New solutions from known ones 9.3. Nonclassical symmetries 10. Classification of invariant solutions 10.1. Equivalence of invariant solutions 10.2. How to classify symmetry generators 10.3. Optimal systems of invariant solutions 11. Discrete symmetries 11.1. Some uses of discrete symmetries 11.2. How to obtain discrete symmetries from Lie symmetries 11.3. Classification of discrete symmetries 11.4. Examples.

Controllability and motion algorithms for underactuated Lagrangian systems on Lie groups

Abstract: We provide controllability tests and motion control algorithms for underactuated mechanical control systems on Lie groups with Lagrangian equal to kinetic energy. Examples include satellite and underwater vehicle control systems with the number of control inputs less than the dimension of the configuration space. Local controllability properties of these systems are characterized, and two algebraic tests are derived in terms of the symmetric product and the Lie bracket of the input vector fields. Perturbation theory is applied to compute approximate solutions for the system under small-amplitude forcing; in-phase signals play a crucial role in achieving motion along symmetric product directions. Motion control algorithms are then designed to solve problems of point-to-point reconfiguration, static interpolation and exponential stabilization. We illustrate the theoretical results and the algorithms with applications to models of planar rigid bodies, satellites and underwater vehicles.

Renormalization in quantum field theory and the Riemann-Hilbert problem II: the $\beta$-function, diffeomorphisms and the renormalization group

Abstract: We showed in part I (hep-th/9912092) that the Hopf algebra ${\cal H}$ of Feynman graphs in a given QFT is the algebra of coordinates on a complex infinite dimensional Lie group $G$ and that the renormalized theory is obtained from the unrenormalized one by evaluating at $\ve=0$ the holomorphic part $\gamma_+(\ve)$ of the Riemann-Hilbert decomposition $\gamma_-(\ve)^{-1}\gamma_+(\ve)$ of the loop $\gamma(\ve)\in G$ provided by dimensional regularization. We show in this paper that the group $G$ acts naturally on the complex space $X$ of dimensionless coupling constants of the theory. More precisely, the formula $g_0=gZ_1Z_3^{-3/2}$ for the effective coupling constant, when viewed as a formal power series, does define a Hopf algebra homomorphism between the Hopf algebra of coordinates on the group of formal diffeomorphisms to the Hopf algebra ${\cal H}$. This allows first of all to read off directly, without using the group $G$, the bare coupling constant and the renormalized one from the Riemann-Hilbert decomposition of the unrenormalized effective coupling constant viewed as a loop of formal diffeomorphisms. This shows that renormalization is intimately related with the theory of non-linear complex bundles on the Riemann sphere of the dimensional regularization parameter $\ve$. It also allows to lift both the renormalization group and the $\beta$-function as the asymptotic scaling in the group $G$. This exploits the full power of the Riemann-Hilbert decomposition together with the invariance of $\gamma_-(\ve)$ under a change of unit of mass. This not only gives a conceptual proof of the existence of the renormalization group but also delivers a scattering formula in the group $G$ for the full higher pole structure of minimal subtracted counterterms in terms of the residue.

Ergodic Theory and Topological Dynamics of Group Actions on Homogeneous Spaces

Abstract: The study of geodesic flows on homogenous spaces is an area of research that has yielded some fascinating developments This book, first published in 2000, focuses on many of these, and one of its highlights is an elementary and complete proof (due to Margulis and Dani) of Oppenheim's conjecture Also included here: an exposition of Ratner's work on Raghunathan's conjectures; a complete proof of the Howe-Moore vanishing theorem for general semisimple Lie groups; a new treatment of Mautner's result on the geodesic flow of a Riemannian symmetric space; Mozes' result about mixing of all orders and the asymptotic distribution of lattice points in the hyperbolic plane; Ledrappier's example of a mixing action which is not a mixing of all orders The treatment is as self-contained and elementary as possible It should appeal to graduate students and researchers interested in dynamical systems, harmonic analysis, differential geometry, Lie theory and number theory

Holomorphy and Convexity in Lie Theory

Abstract: representations theory convex geometry and representations of vector spaces convex geometry of lie algebras highest weight representaitons of lie algebras, lie groups ad semigroups complex geometry and representation theory.

The word and Riemannian metrics on lattices of semisimple groups

Abstract: Let G be a semisimple Lie group of rank ⩾2 and Γ an irreducible lattice. Γ has two natural metrics: a metric inherited from a Riemannian metric on the ambient Lie group and a word metric defined with respect to some finite set of generators. Confirming a conjecture of D. Kazhdan (cf. Gromov [Gr2]) we show that these metrics are Lipschitz equivalent. It is shown that a cyclic subgroup of Γ is virtually unipotent if and only if it has exponential growth with respect to the generators of Γ.

On cobordism of manifolds with corners

Abstract: This work sets up a cobordism theory for manifolds with corners and gives an identication with the homotopy of a certain limit of Thom spectra. It thereby creates a geometrical interpretation of Adams-Novikov resolutions and lays the foundation for investigating the chromatic status of the elements so realized. As an application Lie groups together with their left invariant framings are calculated by regarding them as corners of manifolds with interesting Chern numbers. The work also shows how elliptic cohomology can provide useful invariants for manifolds of codimension 2.

Coadjoint orbits, moment polytopes, and the Hilbert-Mumford criterion

Abstract: Consider a compact Lie group and a closed subgroup. Generalizing a result of Klyachko, we give a necessary and sufficient criterion for a coadjoint orbit of the subgroup to be contained in the projection of a given coadjoint orbit of the ambient group. The criterion is couched in terms of the ``relative'' Schubert calculus of the flag varieties of the two groups.

Lie's structural approach to PDE systems

Abstract: Preface 1. Introduction and summary 2. PDE systems, pfaffian systems and vector field systems 3. Cartan's local existence theorem 4. Involutivity and the prolongation theorem 5. Drach's classification, second order PDEs in one dependent variable and Monge characteristics 6. Integration of vector field systems n satisfying dim n' = dim n + 1 7. Higher order contact transformations 8. Local Lie groups 9. Structural classification of 3-dimensional Lie algebras over the complex numbers 10. Lie equations and Lie vector field systems 11. Second order PDEs in one dependent and two independent variables 12. Hyperbolic PDEs with Monge systems admitting 2 or 3 first integrals 13. Classification of hyperbolic Goursat equations 14. Cartan's theory of Lie pseudogroups 15. The equivalence problem 16. Parabolic PDEs for which the Monge system admits at least two first integrals 17. The equivalence problem for general 3-dimensional pfaffian systems in five variables 18. Involutive second order PDE systems in one dependent and three independent variables, solved by the method of Monge Bibliography Index.

Lie group classification of second-order ordinary difference equations

Abstract: A group classification of invariant difference models, i.e., difference equations and meshes, is presented. In the continuous limit the results go over into Lie’s classification of second-order ordinary differential equations. The discrete model is a three point one and we show that it can be invariant under Lie groups of dimension 0⩽n⩽6.

Banach space representations and Iwasawa theory

Abstract: The lack of a $p$-adic Haar measure causes many methods of traditional representation theory to break down when applied to continuous representations of a compact $p$-adic Lie group $G$ in Banach spaces over a given $p$-adic field $K$. For example, Diarra showed that the abelian group $G=\dZ$ has an enormous wealth of infinite dimensional, topologically irreducible Banach space representations. We therefore address the problem of finding an additional ''finiteness'' condition on such representations that will lead to a reasonable theory. We introduce such a condition that we call ''admissibility''. We show that the category of all admissible $G$-representations is reasonable -- in fact, it is abelian and of a purely algebraic nature -- by showing that it is anti-equivalent to the category of all finitely generated modules over a certain kind of completed group ring $K[[G]]$. As an application of our methods we determine the topological irreducibility as well as the intertwining maps for representations of $GL_2(\dZ)$ obtained by induction of a continuous character from the subgroup of lower triangular matrices.

Manin Pairs and Moment Maps

Abstract: A Lie group $G$ in a group pair ($D, G$), integrating the Lie algebra $\mathfrak{g}$ in a Manin pair ($\mathfrak{d,g}$), has a quasi-Poisson structure. We define the quasi-Poisson actions of such Lie groups $G$, and show that they generalize the Poisson actions of Poisson Lie groups. We define and study the moment maps for those quasi-Poisson actions which are hamiltonian. These moment maps take values in the homogeneous space $D/G$. We prove an analogue of the hamiltonian reduction theorem for quasi-Poisson group actions, and we study the symplectic leaves of the orbit spaces of hamiltonian quasi-Poisson spaces.

Lie groups, Calabi-Yau threefolds, and F-theory

Abstract: The F-theory vacuum constructed from an elliptic Calabi-Yau threefold with section yields an effective six-dimensional theory. The Lie algebra of the gauge sector of this theory and its representation on the space of massless hypermultiplets are shown to be determined by the intersection theory of the homology of the Calabi-Yau threefold. (Similar statements hold for M-theory and the type IIA string compactified on the threefold, where there is also a dependence on the expectation values of the Ramond-Ramond fields.) We describe general rules for computing the hypermultiplet spectrum of any F-theory vacuum, including vacua with non-simply-laced gauge groups. The case of monodromy acting on a curve of A_even singularities is shown to be particularly interesting and leads to some unexpected rules for how 2-branes are allowed to wrap certain 2-cycles. We also review the peculiar numerical predictions for the geometry of elliptic Calabi-Yau threefolds with section which arise from anomaly cancellation in six dimensions.

Similarity and conditional similarity reductions of a (2+1)-dimensional KdV equation via a direct method

Abstract: To get the similarity solutions of a nonlinear physical equation, one may use the classical Lie group approach, nonclassical Lie group approach and the Clarkson and Kruskal (CK) direct method. In this paper the direct method is modified to get the similarity and conditional similarity reductions of a (2+1) dimensional KdV-type equation. Ten types of usual similarity reductions [including the (1+1)-dimensional shallow water wave equation and the variable KdV equation] and six types of conditional similarity reductions of the (2+1)-dimensional KdV equation are obtained. Some special solutions of the conditional similarity reduction equations are found to show the nontriviality of the conditional similarity reduction approach. The conditional similarity solutions cannot be obtained by using the nonclassical Lie group approach in its present form. How to modify the nonclassical Lie group approach to obtain the conditional similarity solutions is still open.

The initial value problem for cohomogeneity one Einstein metrics

Abstract: The PDE Ric(g) = λ · g for a Riemannian Einstein metric g on a smooth manifold M becomes an ODE if we require g to be invariant under a Lie group G acting properly on M with principal orbits of codimension one. A singular orbit of the G-action gives a singularity of this ODE. Generically, an equation with such type of singularity has no smooth solution at the singularity. However, in our case, the very geometric nature of the equation makes it solvable. More precisely, we obtain a smooth G-invariant Einstein metric (with any Einstein constant λ) in a tubular neighbourhood around a singular orbit Q ⊂ M for any prescribed G-invariant metric gQ and second fundamental form LQ on Q, provided that the following technical condition is satisfied (which is very often the case): the representations of the principal isotropy group on the tangent and the normal space of the singular orbit Q have no common sub-representations. This Einstein metric is not uniquely determined by the initial data gQ and LQ; in fact, one may prescribe initial derivatives of higher degree, and examples show that this degree can be arbitrarily high. The proof involves a blend of ODE techniques and representation theory of the principal and singular isotropy groups.

On the Grushin operator and hyperbolic symmetry

Abstract: Complexity of geometric symmetry for differential operators with mixed homogeniety is examined here. Sharp Sobolev estimates are calculated for the Grushin operator in low dimensions using hyperbolic symmetry and conformal geometry. Considerable interest exists in understanding differential operators with mixed homogeneity. A simple example is the Grushin operator on 1R2 02 4t2 02 G= t2 + 9Ox2 The purpose of this note is to demonstrate the complexity of geometric symmetry that may exist for operators defined on Lie groups. Here the existence of an underlying SL(2, R) symmetry for AG is used to compute the sharp constant for the associated L2 Sobolev inequality. Theorem 1. For f G C1(1R2) (1) [lf 1L6(R2) < ? 2/j [( At ) + 4t2( )] dxdt This inequality is sharp, and an extremal is given by [(1 + It12)2 + IX12] -1/4 This result follows from the analysis of a Sobolev inequality on SL(2, R)/SO(2). But the hyperbolic embedding estimate requires some interpretation to take into account cancellation effects. It will be essential to include contibutions to the hyperbolic Dirichlet form from non-L2 functions. Let z = x + iy denote a point in the upper half-plane Rl+ j _2 H M SL(2, R)/SO(2). Here the invariant distance is given by the Poincare metric

Regularity near the characteristic set in the non-linear Dirichlet problem and conformal geometry of sub-Laplacians on Carnot groups

Abstract: This paper constitutes the first part of a project devoted to the study of a class of nonlinear sub-elliptic problems which arise in function theory on CR manifolds. The infinitesimal groups naturally associated with these problems are non-commutative Lie groups whose Lie algebra admits a stratification. The fundamental role of such groups in analysis was envisaged by E. M. Stein [72] in his address at the Nice International Congress of Mathematicians in 1970, see also the recent monograph [73]. There has been since a tremendous development in the analysis of the so-called stratified nilpotent Lie groups, nowadays also known as Carnot groups, and in the study of the sub-elliptic partial differential equations, both linear and non-linear, which arise in this connection. Despite all the progress, our understanding of a large number of basic questions is not to present day as substantial as one may desire. Such situation is due primarily to the complexity of the underlying sub-Riemannian geometry, on the one hand, and to the considerable obstacles which are imposed by non-commutativity and by the presence of characteristic points on the other. To introduce the problems studied in this paper we recall that a Carnot group G is a simply connected nilpotent Lie group such that its Lie algebra g admits a stratification g = r ⊕ j=1 Vj , with [V1, Vj ] = Vj+1 for 1 ≤ j < r , [V1, Vr ] = {0}.

Approximating the exponential from a Lie algebra to a Lie group

Abstract: Consider a differential equation y' = A(t,y)y, y(0) = y0 with y 0 ∈ G and A: R + × G → g, where g is a Lie algebra of the matricial Lie group G. Every B ∈ g can be mapped to G by the matrix exponential map exp (tB) with t ∈ R. Most numerical methods for solving ordinary differential equations (ODEs) on Lie groups are based on the idea of representing the approximation y n of the exact solution y(t n ), t n ∈ R + , by means of exact exponentials of suitable elements of the Lie algebra, applied to the initial value y 0 . This ensures that .y n ∈ G. When the exponential is difficult to compute exactly, as is the case when the dimension is large, an approximation of exp (tB) plays an important role in the numerical solution of ODEs on Lie groups. In some cases rational or polynomial approximants are unsuitable and we consider alternative techniques, whereby exp (tB) is approximated by a product of simpler exponentials. In this paper we present some ideas based on the use of the Strang splitting for the approximation of matrix exponentials. Several cases of g and G are considered, in tandem with general theory. Order conditions are discussed, and a number of numerical experiments conclude the paper.

Lectures on Lie Groups

Abstract: Linear groups and linear representations Lie groups and Lie algebras orbital geometry of the adjoint action coxeter groups, Weyl reduction and Weyl formulas structural theory of compact Lie algebras and compact connected Lie groups.

BRST Cohomology and Phase Space Reduction in Deformation Quantization

Abstract: In this article we consider quantum phase space reduction when zero is a regular value of the momentum map. By analogy with the classical case we define the BRST cohomology in the framework of deformation quantization. We compute the quantum BRST cohomology in terms of a “quantum” Chevalley–Eilenberg cohomology of the Lie algebra on the constraint surface. To prove this result, we construct an explicit chain homotopy, both in the classical and quantum case, which is constructed out of a prolongation of functions on the constraint surface. We have observed the phenomenon that the quantum BRST cohomology cannot always be used for quantum reduction, because generally its zero part is no longer a deformation of the space of all smooth functions on the reduced phase space. But in case the group action is “sufficiently nice”, e.g. proper (which is the case for all compact Lie group actions), it is shown for a strongly invariant star product that the BRST procedure always induces a star product on the reduced phase space in a rather explicit and natural way. Simple examples and counterexamples are discussed.

Application of Lie Algebras to Visual Servoing

Abstract: A novel approach to visual servoing is presented, which takes advantage of the structure of the Lie algebra of affine transformations The aim of this project is to use feedback from a visual sensor to guide a robot arm to a target position The target position is learned using the principle of ‘teaching by showing’ in which the supervisor places the robot in the correct target position and the system captures the necessary information to be able to return to that position The sensor is placed in the end effector of the robot, the ‘camera-in-hand’ approach, and thus provides direct feedback of the robot motion relative to the target scene via observed transformations of the scene These scene transformations are obtained by measuring the affine deformations of a target planar contour (under the weak perspective assumption), captured by use of an active contour, or snake Deformations of the snake are constrained using the Lie groups of affine and projective transformations Properties of the Lie algebra of affine transformations are exploited to provide a novel method for integrating observed deformations of the target contour These can be compensated with appropriate robot motion using a non-linear control structure The local differential representation of contour deformations is extended to allow accurate integration of an extended series of small perturbations This differs from existing approaches by virtue of the properties of the Lie algebra representation which implicitly embeds knowledge of the three-dimensional world within a two-dimensional image-based system These techniques have been implemented using a video camera to control a 5 DoF robot arm Experiments with this implementation are presented, together with a discussion of the results

Geometric quantization and the generalized Segal-Bargmann transform for Lie groups of compact type

Abstract: Let K be a connected Lie group of compact type and let T*(K) be its cotangent bundle. This paper considers geometric quantization of T*(K), first using the vertical polarization and then using a natural Kahler polarization obtained by identifying T*(K) with the complexified group K_C. The first main result is that the Hilbert space obtained by using the Kahler polarization is naturally identifiable with the generalized Segal-Bargmann space introduced by the author from a different point of view, namely that of heat kernels. The second main result is that the pairing map of geometric quantization coincides with the generalized Segal-Bargmann transform introduced by the author. This means that in this case the pairing map is a constant multiple of a unitary map. For both results it is essential that the half-form correction be included when using the Kahler polarization. Together with results of the author with B. Driver, these results may be seen as an instance of "quantization commuting with reduction."

Symmetries and first integrals of ordinary difference equations

Abstract: This paper describes a new symmetry-based approach to solving a given ordinary difference equation. By studying the local structure of the set of solutions, we derive a systematic method for determining one-parameter Lie groups of symmetries in closed form. Such groups can be used to achieve successive reductions of order. If there are enough symmetries, the difference equation can be completely solved. Several examples are used to illustrate the technique for transitive and intransitive symmetry groups. It is also shown that every linear second-order ordinary difference equation has a Lie algebra of symmetry generators that is isomorphic to sl(3). The paper concludes with a systematic method for constructing first integrals directly, which can be used even if no symmetries are known.

Isoparametric submanifolds and a Chevalley-type restriction theorem

Abstract: We define and study isoparametric submanifolds of general ambient spaces and of arbitrary codimension. In particular we study their behaviour with respect to Rie- mannian submersions and their lift into a Hilbert space. These results are used to prove a Chevalley type restriction theorem which relates by restriction eigenfunctions of the Lapla- cian on a compact Riemannian manifold which contains an isoparametric submanifold with flat sections to eigenfunctions of the Laplacian of a section. A simple example of such an isoparametric foliation is given by the conjugacy classes of a compact Lie group and in that case the restriction theorem is a (well known) fundamental result in representation theory. As an application of the restriction theorem we show that isoparametric submanifolds with flat sections in compact symmetric spaces are level sets of eigenfunctions of the Laplacian and are hence related to representation theory. In addition we also get the following results. Isoparametric submanifolds in Hilbert space have globally flat normal bundle, and a general result about Riemannian submersions which says that focal distances do not change if a submanifold of the base is lifted to the total space.

Group representations and the Euler characteristic of elliptically fibered Calabi-Yau threefolds

Abstract: To every elliptic Calabi-Yau threefold with a section $X$ there can be associated a Lie group $G$ and a representation $\rho$ of that group. The group is determined from the Weierstrass model, which has singularities that are generically rational double points; these double points lead to local factors of $G$ which are either the corresponding A-D-E groups or some associated non-simply laced groups. The representation $\rho$ is a sum of representations coming from the local factors of $G$, and of other representations which can be associated to the points at which the singularities are worse than generic. This construction first arose in physics, and the requirement of anomaly cancellation in the associated physical theory makes some surprising predictions about the connection between $X$ and $\rho$. In particular, an explicit formula (in terms of $\rho$) for the Euler characteristic of $X$ is predicted. We give a purely mathematical proof of that formula in this paper, introducing along the way a new invariant of elliptic Calabi-Yau threefolds. We also verify the other geometric predictions which are consequences of anomaly cancellation, under some (mild) hypotheses about the types of singularities which occur. As a byproduct we also discover a novel relation between the Coxeter number and the rank in the case of the simply laced groups in the ``exceptional series'' studied by Deligne.