Showing papers on "Lie group published in 2003"
01 Jan 2003
TL;DR: The symmetry group of a system is often a Lie group, a smooth manifold endowed with a group structure in such a way that operations of inversion and group multiplication are smooth as discussed by the authors.
Abstract: An important concept in physics is that of symmetry, whether it be rotational symmetry for many physical systems or Lorentz symmetry in relativistic systems. In many cases, the group of symmetries of a system is a continuous group, that is, a group that is parameterized by one or more real parameters. More precisely, the symmetry group is often a Lie group, that is, a smooth manifold endowed with a group structure in such a way that operations of inversion and group multiplication are smooth. The tangent space at the identity in a Lie group has a natural “bracket” operation that makes the tangent space into a Lie algebra. The Lie algebra of a Lie group encodes many of the properties of the Lie group, and yet the Lie algebra is easier to work with because it is a linear space.
520 citations
Book•
16 Dec 2003
TL;DR: In this paper, the Symplectic Slice Theorem and Singular reduction and the stratification theorem are used to define a regular symplectic reduction theory for Lie Group actions.
Abstract: Introduction.- Manifolds and smooth structures.- Lie group actions.- Pseudogroups and groupoids.- The standard momentum map.- Generalizations of the momentum map.- Regular symplectic reduction theory.- The Symplectic Slice Theorem.- Singular reduction and the stratification theorem.- Optimal reduction.- Poisson reduction.- Dual Pairs.- Bibliography.- Index.
450 citations
TL;DR: In this article, a class of examples that can be shown to satisfy the requirements of mirror symmetry were presented, and whose Hodge numbers are also equal. And these examples are of great interest in their own right: they are spaces of flat SLr-connections on a smooth curve.
Abstract: Among the major mathematical approaches to mirror symmetry are those of Batyrev-Borisov and Strominger-Yau-Zaslow (SYZ). The first is explicit and amenable to computation but is not clearly related to the physical motivation; the second is the opposite. Furthermore, it is far from obvious that mirror partners in one sense will also be mirror partners in the other. This paper concerns a class of examples that can be shown to satisfy the requirements of SYZ, but whose Hodge numbers are also equal. This provides significant evidence in support of SYZ. Moreover, the examples are of great interest in their own right: they are spaces of flat SLr-connections on a smooth curve. The mirror is the corresponding space for the Langlands dual group PGLr. These examples therefore throw a bridge from mirror symmetry to the duality theory of Lie groups and, more broadly, to the geometric Langlands program.
291 citations
TL;DR: In this paper, it was shown that the Kortewegde Vries equation, the Camassa-Holm equation, and the Hunter-Saxton equation have the same symmetry group and similar bihamiltonian structures.
266 citations
TL;DR: A general control design approach for the stabilization of controllable driftless nonlinear systems on finite dimensional Lie groups is presented, based on the concept of bounded transverse functions, the existence of which is equivalent to the system's controllability.
Abstract: A general control design approach for the stabilization of controllable driftless nonlinear systems on finite dimensional Lie groups is presented. The approach is based on the concept of bounded transverse functions, the existence of which is equivalent to the system's controllability. Its outcome is the practical stabilization of any trajectory, i.e., not necessarily a solution of the control system, in the state-space. The possibility of applying the approach to an arbitrary controllable smooth driftless system follows in turn from the fact that any controllable homogeneous approximation of this system can be lifted (via a dynamic extension) to a system on a Lie group. Illustrative examples are given.
239 citations
18 Jun 2003
TL;DR: This paper shows that medial descriptions are in fact elements of a Lie group, and develops methodology based on Lie groups for the statistical analysis of medially-defined anatomical objects.
Abstract: Principal component analysis has proven to be useful for understanding geometric variability in populations of parameterized objects. The statistical framework is well understood when the parameters of the objects are elements of a Euclidean vector space. This is certainly the case when the objects are described via landmarks or as a dense collection of boundary points. We have been developing representations of geometry based on the medial axis description or m-rep. Although this description has proven to be effective, the medial parameters are not naturally elements of a Euclidean space. In this paper we show that medial descriptions are in fact elements of a Lie group. We develop methodology based on Lie groups for the statistical analysis of medially-defined anatomical objects.
196 citations
TL;DR: In this article, the authors provide a first taste of the theory of Lie groups as an appetiser for a more substantial further course in linear algebra and differential geometry, and provide an approach that is accessible to readers equipped with a standard undergraduate toolkit of algebra and analysis.
Abstract: Aimed at advanced undergraduate and beginning graduate students, this book provides a first taste of the theory of Lie groups as an appetiser for a more substantial further course. Lie theoretic ideas lie at the heart of much of standard undergraduate linear algebra and exposure to them can inform or motivate the study of the latter.The main focus is on matrix groups, i.e., closed subgroups of real and complex general linear groups. The first part studies examples and describes the classical families of simply connected compact groups. The second part introduces the idea of a lie group and studies the associated notion of a homogeneous space using orbits of smooth tions. Throughout, the emphasis is on providing an approach that is accessible to readers equipped with a standard undergraduate toolkit of algebra and analysis. Although the formal prerequisites are kept as low level as possible, the subject matter is sophisticated and contains many of the key themes of the fully developed theory, preparing students for a more standard and abstract course in Lie theory and differential geometry.
185 citations
Book•
20 Oct 2003TL;DR: Lie groups Maximal tori and the classification theorem The geometry of a compact Lie group Homogeneous spaces as mentioned in this paper and the geometry of reductive homogeneous spaces Symmetric spaces Generalized flag manifolds Advanced topics Bibliography Index
Abstract: Lie groups Maximal tori and the classification theorem The geometry of a compact Lie group Homogeneous spaces The geometry of a reductive homogeneous space Symmetric spaces Generalized flag manifolds Advanced topics Bibliography Index.
181 citations
Posted Content•
TL;DR: In this article, it was shown that multiplicative 2-forms on a Lie groupoid over a manifold are relatively closed with respect to a closed 3-form on the manifold and correspond to maps from the Lie algebroid of the groupoid into the cotangent bundle of the manifold.
Abstract: The correspondence between Poisson structures and symplectic groupoids, analogous to the one of Lie algebras and Lie groups, plays an important role in Poisson geometry; it offers, in particular, a unifying framework for the study of hamiltonian and Poisson actions. In this paper, we extend this correspondence to the context of Dirac structures twisted by a closed 3-form.
More generally, 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 the cotangent bundle $T^*M$ of $M$, 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 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.
118 citations
TL;DR: Cauchon et al. as discussed by the authors constructed the derivative-elimination algorithm, which consists of a sequence of changes of variables inside the division ring F=Fract(R), starting with the indeterminates (X1, X2, XN) and terminating with new variables (T1, T1, N) to generate some quantum-affine space R such that F= Fract(R ).
116 citations
TL;DR: In this paper, a variational principle for the Euler-Poincare equations on a Lie algebra g of a Lie group G obtained by reducing Hamilton's principle on G by the action of G by, say, left multiplication is given.
Abstract: As is well-known, there is a variational principle for the
Euler–Poincare equations on a Lie algebra g of a Lie group G obtained by reducing Hamilton’s principle on G by the action of G by, say, left multiplication. The purpose of this paper is to give a variational principle
for the Lie–Poisson equations on g*, the dual of g, and also to
generalize this construction.
The more general situation is that in which the original configuration
space is not a Lie group, but rather a configuration manifold Q
on which a Lie group G acts freely and properly, so that Q → Q/G
becomes a principal bundle. Starting with a Lagrangian system on TQ
invariant under the tangent lifted action of G, the reduced equations on
(TQ)/G, appropriately identified, are the Lagrange–Poincare equations.
Similarly, if we start with a Hamiltonian system on T*Q, invariant under
the cotangent lifted action of G, the resulting reduced equations on
(T*Q)/G are called the Hamilton–Poincare equations.
Amongst our new results, we derive a variational structure for the
Hamilton–Poincare equations, give a formula for the Poisson structure
on these reduced spaces that simplifies previous formulas of Montgomery,
and give a new representation for the symplectic structure on the associated
symplectic leaves. We illustrate the formalism with a simple, but
interesting example, that of a rigid body with internal rotors.
TL;DR: In this article, a Cartan differential calculus for triangular Jacobi algebras is presented, and the notion of a Courant-Jacobi algebroid and Dirac Jacobi structure is defined.
Abstract: Jacobi algebroids (i.e. 'Jacobi versions' of Lie algebroids) are studied in the context of graded Jacobi brackets on graded commutative algebras. This unifies various concepts of graded Lie structures in geometry and physics. A method of describing such structures by classical Lie algebroids via certain gauging (in the spirit of E Witten's gauging of the exterior derivative) is developed. One constructs a corresponding Cartan differential calculus (graded commutative one) in a natural manner. This, in turn, gives canonical generating operators for triangular Jacobi algebroids. One gets, in particular, the Lichnerowicz–Jacobi homology operators associated with classical Jacobi structures. Courant–Jacobi brackets are obtained in a similar way and used to define an abstract notion of a Courant–Jacobi algebroid and Dirac–Jacobi structure.
TL;DR: A new format of Lie group methods is proposed which does not involve commutators and which uses a much lower number of exponentials than those proposed by Crouch and Grossman, and which may be particularly useful when applied to problems on homogeneous manifolds with large isotropy groups, or when used for stiff problems.
TL;DR: In this article, the Lagrangians explicitly to first order in the deformation parameter are constructed for arbitrary compact Lie groups using the concept of enveloping algebra-valued gauge transformations and Seiberg-Witten formalism.
Abstract: This study of gauge field theories on kappa-deformed Minkowski spacetime extends previous work on field theories on this example of a noncommutative spacetime. We construct deformed gauge theories for arbitrary compact Lie groups using the concept of enveloping algebra-valued gauge transformations and the Seiberg-Witten formalism. Derivative-valued gauge fields lead to field strength tensors as the sum of curvature- and torsion-like terms. We construct the Lagrangians explicitly to first order in the deformation parameter. This is the first example of a gauge theory that possesses a deformed Lorentz covariance.
09 Dec 2003
TL;DR: In this article, the authors considered a Lie group formulation for the problem of control of formations using the planar Frenet-Serret equations of motion, which capture the evolution of both vehicle position and orientation for unit-speed motion subject to curvature (steering) control.
Abstract: We consider a Lie group formulation for the problem of control of formations. Vehicle trajectories are described using the planar Frenet-Serret equations of motion, which capture the evolution of both vehicle position and orientation for unit-speed motion subject to curvature (steering) control. The Lie group structure can be exploited to determine the set of all possible (relative) equilibria for arbitrary G-invariant curvature controls, where G=SE(2) is a symmetry group for the control law. The main result is a convergence result for n vehicles (for finite n), using a Lyapunov function which for n=2, has been previously shown to yield global convergence. A continuum formulation of the basic equations is also presented.
TL;DR: In this article, the authors present a general method for analysing and numerically solving partial differential equations with self-similar solutions, which employs ideas from symmetry reduction in geometric mechanics and involves separating the dynamics on the shape space from those on the group space.
Abstract: We present a general method for analysing and numerically solving partial differential equations with self-similar solutions. The method employs ideas from symmetry reduction in geometric mechanics, and involves separating the dynamics on the shape space (which determines the overall shape of the solution) from those on the group space (which determines the size and scale of the solution). The method is computationally tractable as well, allowing one to compute self-similar solutions by evolving a dynamical system to a steady state, in a scaled reference frame where the self-similarity has been factored out. More generally, bifurcation techniques can be used to find self-similar solutions, and determine their behaviour as parameters in the equations are varied. The method is given for an arbitrary Lie group, providing equations for the dynamics on the reduced space, for reconstructing the full dynamics and for determining the resulting scaling laws for self-similar solutions. We illustrate the technique with a numerical example, computing self-similar solutions of the Burgers equation.
TL;DR: In this paper, the authors discuss the relationship between commutative and non-com- mutative integrability of Hamiltonian systems and construct new examples of integrable geodesic flows on Riemannian manifolds.
Abstract: The purpose of this paper is to discuss the relationship between commutative and noncom- mutative integrability of Hamiltonian systems and to construct new examples of integrable geodesic flows on Riemannian manifolds. In particular, we prove that the geodesic flow of the bi-invariant metric on any bi-quotient of a compact Lie group is integrable in the noncommutative sense by means of polynomial integrals, and therefore, in the classical commutative sense by means of C ∞ -smooth integrals. Mathematics Subject Classifications (2000): 37J35, 37J15, 70H06, 70H33, 53D20, 53D25.
TL;DR: In this paper, the authors give an explicit formula for the cohomology ring of the symplectic quotient M/G in terms of the cohology ring and fixed point data, and show that the reduced space is torsion-free.
Abstract: Let (M,ω) be a symplectic manifold, equipped with a Hamiltonian action of a compact Lie group G. We give an explicit formula for the cohomology ring of the symplectic quotient M//G in terms of the cohomology ring of M and fixed point data. Under certain conditions, our formula also holds for the integral cohomology ring, and can be used to show that the cohomology of the reduced space is torsion-free.
TL;DR: In this article, the authors show how the Lie group methods look when applied to the rigid body equations in particular and indicate how the methods work in general, and provide numerical experiments which give an idea of the performance of Lie group method compared to other known integration schemes.
TL;DR: In this paper, the wave map equation between two Riemannian manifold is derived from the same geometric considerations as the harmonic map equation, and a new multiplication theorem for mixed Lebesgue-Besov spaces is shown.
Abstract: We construct a gauge theoretic change of variables for the wave map from R × R into a compact group or Riemannian symmetric space, prove a new multiplication theorem for mixed Lebesgue-Besov spaces, and show the global well-posedness of a modified wave map equation n ≥ 4 for small critical initial data. We obtain global existence and uniqueness for the Cauchy problem of wave maps into compact Lie groups and symmetric spaces with small critical initial data and n ≥ 4. 0. Introduction The wave map equation between two Riemannian manifoldsthe wave equation version of the evolution equations which are derived from the same geometric considerations as the harmonic map equation between two Riemannian manifoldshas been studied by a number of mathematicians in the last decade. The work of Klainerman and Machedon and Klainerman and Selberg [5] [6] [8] studying the Cauchy problem for regular data is probably the best known. The more recent work of Tataru [15], [16] and Tao [13] [14] relies and further develops deep ideas from harmonic analysis in Tao’s case in conjunction with gauge theoretic geometric methods and thus seems very promising. Keel and Tao studied the one (spatial) dimensional case in [4]. In [13], Tao established the global regularity for wave maps from R × R into the sphere S when n ≥ 5. Similar results to those of Tao were obtained by Klainerman and Rodniansky [7] for target manifolds that admit a bounded parallelizable structure. In this paper we are interested in revisiting this work. We study the Cauchy problem for wave maps from R × R into a (compact) Lie group (or Riemannian symmetric 1991 Mathematics Subject Classification. Primary 35J10, Secondary 45B15, 42B35.
Book•
12 Sep 2003
TL;DR: In this paper, De Giorgi et al. introduce general formalism and general formality, and show that Lie groups and Lie algebras can be represented by Lie groups.
Abstract: I Introduction.- II General Formalism.- II.1 Lie groups and Lie algebras.- II.2 Subelliptic operators.- II.3 Subelliptic kernels.- II.4 Growth properties.- II.5 Real operators.- II.6 Local bounds on kernels.- II.7 Compact groups.- II.8 Transference method.- II.9 Nilpotent groups.- II.10 De Giorgi estimates.- II.11 Almost periodic functions.- II.12 Interpolation.- Notes and Remarks.- III Structure Theory.- III.1 Complementary subspaces.- III.2 The nilshadow algebraic structure.- III.3 Uniqueness of the nilshadow.- III.4 Near-nilpotent ideals.- III.5 Stratified nilshadow.- III.6 Twisted products.- III.7 The nilshadow analytic structure.- Notes and Remarks.- IV Homogenization and Kernel Bounds.- IV.1 Subelliptic operators.- IV.2 Scaling.- IV.3 Homogenization correctors.- IV.4 Homogenized operators.- IV.5 Homogenization convergence.- IV.6 Kernel bounds stratified nilshadow.- IV.7 Kernel bounds general case.- Notes and Remarks.- V Global Derivatives.- V.1 L2-bounds.- V.1.1 Compact derivatives.- V.1.2 Nilpotent derivatives.- V.2 Gaussian bounds.- V.3 Anomalous behaviour.- Notes and Remarks.- VI Asymptotics.- VI. 1 Asymptotics of semigroups.- VI.2 Asymptotics of derivatives.- Notes and Remarks.- Appendices.- A.1 De Giorgi estimates.- A.2 Morrey and Campanato spaces.- A.3 Proof of Theorem II.10.5.- A.4 Rellich lemma.- Notes and Remarks.- References.- Index of Notation.
Proceedings Article•
13 Oct 2003TL;DR: This paper considers the images taken from pairs of parabolic catadioptric cameras separated by discrete motions, and presents a method to estimate a matrix in this space, so as to obtain an estimate of the motion.
Abstract: In this paper we consider the images taken from pairs ofparabolic catadioptric cameras separated by discrete motions.Despite the nonlinearity of the projection model, theepipolar geometry arising from such a system, like the perspectivecase, can be encoded in a bilinear form, the catadioptricfundamental matrix. We show that all such matriceshave equal Lorentzian singular values, and they definea nine-dimensional manifold in the space of 4 × 4 matrices.Furthermore, this manifold can be identified with a quotientof two Lie groups. We present a method to estimate a matrixin this space, so as to obtain an estimate of the motion.We show that the estimation procedures are robust to modestdeviations from the ideal assumptions.
TL;DR: In this article, a unified algebraic construction of the classical Smorodinsky-Winternitz systems on the ND sphere, Euclidean and hyperbolic spaces through the Lie groups SO(N + 1), ISO(N) and SO (N, 1) is presented.
Abstract: A unified algebraic construction of the classical Smorodinsky–Winternitz systems on the ND sphere, Euclidean and hyperbolic spaces through the Lie groups SO(N + 1), ISO(N) and SO(N, 1) is presented. Firstly, general expressions for the Hamiltonian and its integrals of motion are given in a linear ambient space N+1, and secondly they are expressed in terms of two geodesic coordinate systems on the ND spaces themselves, with an explicit dependence on the curvature as a parameter. On the sphere, the potential is interpreted as a superposition of N + 1 oscillators. Furthermore, each Lie algebra generator provides an integral of motion and a set of 2N − 1 functionally independent ones are explicitly given. In this way the maximal superintegrability of the ND Euclidean Smorodinsky–Winternitz system is shown for any value of the curvature.
TL;DR: For T an abelian compact Lie group, the authors gave a description of T-equivariant K-theory with complex coefficients in terms of equivariant cohomology.
Abstract: For T an abelian compact Lie group, we give a description of T-equivariant K-theory with complex coefficients in terms of equivariant cohomology. In the appendix we give applications of this by extending results of Chang-Skjelbred and Goresky-Kottwitz-MacPherson from equivariant cohomology to equivariant K-theory.
TL;DR: The structure theory of finitely generated torsion -modules up to pseudo-isomorphism was studied in this article, which is largely parallel to the classical theory when is abelian (except for basic differences which occur for those torsions which do not possess a nonzero global annihilator).
Abstract: Let be a compact -valued -adic Lie group, and let be its Iwasawa algebra. The present paper establishes results about the structure theory of finitely generated torsion -modules, up to pseudo-isomorphism, which are largely parallel to the classical theory when is abelian (except for basic differences which occur for those torsion modules which do not possess a non-zero global annihilator). We illustrate our general theory by concrete examples of such modules arising from the Iwasawa theory of elliptic curves without complex multiplication over the field generated by all of their -power torsion points.AMS 2000 Mathematics subject classification: Primary 11G05; 11R23; 16D70; 16E65; 16W70
01 Jan 2003
TL;DR: In this article, the authors considered the images taken from pairs of parabolic catadioptric cameras separated by discrete motions and showed that the epipolar geometry arising from such a system, like the perspective case, can be encoded in a bilinear form, the catadiophoric fundamental matrix, and they showed that all such matrices have equal Lorentzian singular values.
Abstract: In this paper we consider the images taken from pairs of parabolic catadioptric cameras separated by discrete motions. Despite the nonlinearity of the projection model, the epipolar geometry arising from such a system, like the perspective case, can be encoded in a bilinear form, the catadioptric fundamental matrix. We show that all such matrices have equal Lorentzian singular values, and they define a nine-dimensional manifold in the space of 4 /spl times/ 4 matrices. Furthermore, this manifold can be identified with a quotient of two Lie groups. We present a method to estimate a matrix in this space, so as to obtain an estimate of the motion. We show that the estimation procedures are robust to modest deviations from the ideal assumptions.
TL;DR: In this paper, a Lie-group formulation for the kinematics and dynamics of constrained mechanical systems (CMSs) is presented, which is based on the screw system of the MSs.
Abstract: A Lie-group formulation for the kinematics and dynamics ofholonomic constrained mechanical systems (CMS) is presented. The kinematics ofrigid multibody systems (MBS) is described in terms of the screw system of theMBS. Using Lie-algebraic properties of screw algebra, isomorphicto se(3), allows a purely algebraic derivation of the Lagrangian motion equations. As such the Lie-group SE(3) ⊗... ⊗ SE(3) (n copies) is theambient space of a MBS consisting of n rigid bodies. Any parameterizationof the ambient space corresponds to a chart on the MBS configuration space ℝn. The key to combine differential geometric and Lie-algebraic approaches is the existence of kinematic basic functions whichare push forward maps from the tangent bundle Tℝn to the Lie-algebra of the ambient space.
20 Jul 2003
TL;DR: Analogous to principal component analysis of covariance in Euclidean spaces, principal geodesic analysis on Lie groups is defined for the study of anatomical variability in medially-defined objects and results of applying this framework on a population of hippocampi in a schizophrenia study are presented.
Abstract: The Gaussian distribution is the basis for many methods used in the statistical analysis of shape. One such method is principal component analysis, which has proven to be a powerful technique for describing the geometric variability of a population of objects. The Gaussian framework is well understood when the data being studied are elements of a Euclidean vector space. This is the case for geometric objects that are described by landmarks or dense collections of boundary points. We have been using medial representations, or m-reps, for modelling the geometry of anatomical objects. The medial parameters are not elements of a Euclidean space, and thus standard PCA is not applicable. In our previous work we have shown that the m-rep model parameters are instead elements of a Lie group. In this paper we develop the notion of a Gaussian distribution on this Lie group. We then derive the maximum likelihood estimates of the mean and the covariance of this distribution. Analogous to principal component analysis of covariance in Euclidean spaces, we define principal geodesic analysis on Lie groups for the study of anatomical variability in medially-defined objects. Results of applying this framework on a population of hippocampi in a schizophrenia study are presented.
TL;DR: The authors obtained similar results for equivariant observations on compact group extensions of hyperbolic basic sets using the Ruelle transfer operator, and showed that such observations satisfy strong statistical properties such as exponential decay of correlations, central limit theorems and invariance principles.
Abstract: Holder continuous observations on hyperbolic basic sets satisfy strong statistical properties such as exponential decay of correlations, central limit theorems and invariance principles (approximation by Brownian motion). Using an equivariant version of the Ruelle transfer operator studied by Parry and Pollicott, we obtain similar results for equivariant observations on compact group extensions of hyperbolic basic sets.
TL;DR: In this article, a characterization of block-diagonalizable pseudo-unitary operators with finite-dimensional diagonal blocks is presented, and a quantum system with a 2-times 2-dimensional dynamical group is discussed.
Abstract: We consider pseudo-unitary quantum systems and discuss various properties of pseudo-unitary operators. In particular we prove a characterization theorem for block-diagonalizable pseudo-unitary operators with finite-dimensional diagonal blocks. Furthermore, we show that every pseudo-unitary matrix is the exponential of $i=\sqrt{-1}$ times a pseudo-Hermitian matrix, and determine the structure of the Lie groups consisting of pseudo-unitary matrices. In particular, we present a thorough treatment of $2\times 2$ pseudo-unitary matrices and discuss an example of a quantum system with a $2\times 2$ pseudo-unitary dynamical group. As other applications of our general results we give a proof of the spectral theorem for symplectic transformations of classical mechanics, demonstrate the coincidence of the symplectic group $Sp(2n)$ with the real subgroup of a matrix group that is isomorphic to the pseudo-unitary group U(n,n), and elaborate on an approach to second quantization that makes use of the underlying pseudo-unitary dynamical groups.