Showing papers on "Lie group published in 1995"

Equivalence, invariants, and symmetry

Abstract: 1 Geometric foundations 2 Lie groups 3 Representation theory 4 Jets and contact transformations 5 Differential invariants 6 Symmetries of differential equations 7 Symmetries of variational problems 8 Equivalence of coframes 9 Formulation of equivalence problems 10 Cartan's equivalence method 11 Involution 12 Prolongation of equivalence problems 13 Differential systems 14 Frobenius' theorem 15 The Cartan-Kahler existence theorem

Controllability of molecular systems

Abstract: In this paper we analyze the controllability of quantum systems arising in molecular dynamics. We model these systems as systems with finite numbers of levels, and examine their controllability. To do this we pass to their unitary generators and use results on the controllability of invariant systems on Lie groups. Examples of molecular systems, modeled as finite-dimensional control systems, are provided. A simple algorithm to detect the controllability of a molecular system is provided. Finally, we apply this algorithm to a five-level system.

A Lie group formulation of robot dynamics

Abstract: In this article we present a unified geometric treatment of robot dynamics. Using standard ideas from Lie groups and Rieman nian geometry, we formulate the equations of motion for an open chain manipulator both recursively and in closed form. The recursive formulation leads to an O(n) algorithm that ex presses the dynamics entirely in terms of coordinate-free Lie algebraic operations. The Lagrangian formulation also ex presses the dynamics in terms of these Lie algebraic operations and leads to a particularly simple set of closed-form equations, in which the kinematic and inertial parameters appear explic itly and independently of each other. The geometric approach permits a high-level, coordinate-free view of robot dynamics that shows explicitly some of the connections with the larger body of work in mathematics and physics. At the same time the resulting equations are shown to be computationally ef fective and easily differentiated and factored with respect to any of the robot parameters. This latter fe...

Contact metric manifolds satisfying a nullity condition

Abstract: This paper presents a study of contact metric manifolds for which the characteristic vector field of the contact structure satisfies a nullity type condition, condition (*) below. There are a number of reasons for studying this condition and results concerning it given in the paper: There exist examples in all dimensions; the condition is invariant underD-homothetic deformations; in dimensions>5 the condition determines the curvature completely; and in dimension 3 a complete, classification is given, in particular these include the 3-dimensional unimodular Lie groups with a left invariant metric.

Proportional Derivative (PD) Control on the Euclidean Group

Abstract: In this paper we study the stabilization problem for control systems defined on SE(3) (the special Euclidean group of rigid-body motions) and its subgroups. Assuming one actuator is available for each degree of freedom, we exploit geometric properties of Lie groups (and corresponding Lie algebras) to generalize the classical proportional derivative (PD) control in a coordinate-free way. For the SO(3) case, the compactness of the group gives rise to a natural metric structure and to a natural choice of preferred control direction: an optimal (in the sense of geodesic) solution is given to the attitude control problem. In the SE(3) case, no natural metric is uniquely defined, so that more freedom is left in the control design. Different formulations of PD feedback can be adopted by extending the SO(3) approach to the whole of SE(3) or by breaking the problem into a control problem on SO(3) x R^3. For the simple SE(2) case, simulations are reported to illustrate the behavior of the different choices. We also discuss the trajectory tracking problem and show how to reduce it to a stabilization problem, mimicking the usual approach in R^n. Finally, regarding the case of underactuated control systems, we derive linear and homogeneous approximating vector fields for standard systems on SO(3) and SE(3).

Lie group actions in complex analysis

Abstract: Introduction - Lie-theory - Automorphism groups - Compact homogeneous manifolds - Homogeneous vector bundles - Function theory on homogeneous manifolds - Concluding remarks.

The dynamic interpolation problem: On Riemannian manifolds, Lie groups, and symmetric spaces

Abstract: We consider the dynamic interpolation problem for nonlinear control systems modeled by second-order differential equations whose configuration space is a Riemannian manifoldM. In this problem we are given an ordered set of points inM and would like to generate a trajectory of the system through the application of suitable control functions, so that the resulting trajectory in configuration space interpolates the given set of points. We also impose smoothness constraints on the trajectory and typically ask that the trajectory be also optimal with respect to some physically interesting cost function. Here we are interested in the situation where the trajectory is twice continuously differentiable and the Lagrangian in the optimization problem is given by the norm squared acceleration along the trajectory. The special cases whereM is a connected and compact Lie group or a homogeneous symmetric space are studied in more detail.

Motion control of drift-free, left-invariant systems on Lie groups

Abstract: In this paper we address the constructive controllability problem for drift-free, left-invariant systems on finite-dimensional Lie groups with fewer controls than state dimension. We consider small (/spl epsiv/) amplitude, low-frequency, periodically time-varying controls and derive average solutions for system behavior. We show how the pth-order average formula can be used to construct open-loop controls for point-to-point maneuvering of systems which require up to (p-1) iterations of Lie brackets to satisfy the Lie algebra controllability rank condition. In the cases p=2,3, we give algorithms for constructing these controls as a function of structure constants that define the control authority, i.e., the actuator capability, of the system. The algorithms are based on a geometric interpretation of the average formulas and produce sinusoidal controls that solve the constructive controllability problem with O(/spl epsiv//sup P/) accuracy in general (exactly if the Lie algebra is nilpotent). The methodology is applicable to a variety of control problems and is illustrated for the motion control problem of an autonomous underwater vehicle with as few as three control inputs. >

Conformal Field Theory and Integrable Systems Associated to Elliptic Curves

Abstract: It has become clear over the years that quantum groups (i.e., quasitriangular Hopf algebras, see [D]) and their semiclassical counterpart, Poisson Lie groups, are an essential algebraic structure underlying three related subjects: integrable models of statistical mechanics, conformal field theory, and integrable models of quantum field theory in 1+1 dimensions. Still, some points remain obscure from the point of view of Hopf algebras. In particular, integrable models associated with elliptic curves are still poorly understood. We propose her an elliptic version of quantum groups, based on the relation to conformal field theory, which hopefully will be helpful to complete the picture.

Algebraic theory of molecules

Abstract: An algebraic formulation of quantum mechanics is presented. In this formulation, operators of interest are expanded onto elements of an algebra, G. For bound state problems in nu dimensions the algebra G is taken to be U(nu + 1). Applications to the structure of molecules are presented.

Coherent states and their generalizations - A mathematical overview

Abstract: We present a survey of the theory of coherent states (CS); and some of their generalizations, with emphasis on the mathematical structure, rather than on physical applications. Starting from the standard theory of CS over Lie groups, we develop a general formalism, in which CS are associated to group representations which are square integrable over a homogeneous space. A further step allows us to dispense with the group context altogether, and thus obtain the so-called reproducing triples and continuous frames introduced in some earlier work. We discuss in detail a number of concrete examples, namely semisimple Lie groups, the relativity groups and various types of wavelets. Finally we turn to some physical applications, centering on quantum measurement and the quantization/dequantization problem, that is, the transition from the classical to the quantum level and vice versa.

Bézier Curves on Riemannian Manifolds and Lie Groups with Kinematics Applications

Abstract: In this article we generalize the concept of Bezier curves to curved spaces, and illustrate this generalization with an application in kinematics. We show how De Casteljau's algorithm for constructing Bezier curves can be extended in a natural way to Riemannian manifolds. We then consider a special class of Riemannian manifold, the Lie groups. Because of their group structure Lie groups admit an elegant, efficient recursive algorithm for constructing Bezier curves. Spatial displacements of a rigid body also form a Lie group, and can therefore be interpolated (in the Bezier sense) using this recursive algorithm. We apply this algorithm to the kinematic problem of trajectory generation or motion interpolation for a moving rigid body. The orientation trajectory of motions generated in this way have the important property of being invariant with respect to choices of inertial and body-fixed reference frames

On the space of ergodic invariant measures of unipotent flows

Abstract: Let G be a Lie group and Γ be a discrete subgroup. We show that if {μn} is a convergent sequence of probability measures on G/Γ which are invariant and ergodic under actions of unipotent one-parameter subgroups, then the limit μ of such a sequence is supported on a closed orbit of the subgroup preserving it, and is invariant and ergodic for the action of a unipotent one-parameter subgroup of G.

Nonsmooth critical point theory and quasilinear elliptic equations

Abstract: These lectures are devoted to a generalized critical point theory for nonsmooth functionals and to existence of multiple solutions for quasilinear elliptic equations. If f is a continuous function defined on a metric space, we define the weak slope |df|(u), an extended notion of norm of the Frechet derivative. Generalized notions of critical point and Palais-Smale condition are accordingly introduced. The Deformation Theorem and the Noncritical Interval Theorem are proved in this setting. The case in which f is invariant under the action of a compact Lie group is also considered. Mountain pass theorems for continuous functionals are proved. Estimates of the number of critical points of f by means of the relative category are provided. A partial extension of these techniques to lower semicontinuous functionals is outlined. The second part is mainly concerned with functionals of the Calculus of Variations depending quadratically on the gradient of the function. Such functionals are naturally continuous, but not locally Lipschitz continuous on H 0 1 . When f is even and suitable qualitative conditions are satisfied, we prove the existence of infinitely many solutions for the associated Euler equation. The regularity of such solutions is also studied.

Lie-Butcher theory for Runge-Kutta methods

Abstract: Runge-Kutta methods are formulated via coordinate independent operations on manifolds. It is shown that there is an intimate connection between Lie series and Lie groups on one hand and Butcher's celebrated theory of order conditions on the other. In Butcher's theory the elementary differentials are represented as trees. In the present formulation they appear as commutators between vector fields. This leads to a theory for the order conditions, which can be developed in a completely coordinate free manner. Although this theory is developed in a language that is not widely used in applied mathematics, it is structurally simple. The recursion for the order conditions rests mainly on three lemmas, each with very short proofs. The techniques used in the analysis are prepared for studying RK-like methods on general Lie groups and homogeneous manifolds, but these themes are not studied in detail within the present paper.

Quantum SO(3)-invariants dominate the SU(2)-invariant of Casson and Walker

Abstract: For a compact Lie group G, E. Witten proposed topological invariants of a threemanifold (quantum G-invariants) in 1988 by using the Chern-Simons functional and the Feynman path integral [30]. See also [2]. N. Yu. Reshetikhin and V. G. Turaev gave a mathematical proof of existence of such invariants for G = SU(2) [28]. R. Kirby and P. Melvin found that the quantum SU(2)-invariant associated to q = exp(2π √ − 1/r) with r odd splits into the product of the quantum SO(3)-invariant and [15]. For other approaches to these invariants, see [3, 4, 5, 16, 22, 27].

Spherical functions on affine Lie groups

Abstract: We describe vector valued conjugacy equivariant functions on a group K in two cases -- K is a compact simple Lie group, and K is an affine Lie group. We construct such functions as weighted traces of certain intertwining operators between representations of K. For a compact group $K$, Peter-Weyl theorem implies that all equivariant functions can be written as linear combinations of such traces. Next, we compute the radial parts of the Laplace operators of $K$ acting on conjugacy equivariant functions and obtain a comple- tely integrable quantum system with matrix coefficients, which in a special case coincides with the trigonometric Calogero-Sutherland-Moser multi-particle system. In the affine Lie group case, we prove that the space of equivariant functions having a fixed homogeneity degree with respect to the action of the center of the group is finite-dimensional and spanned by weighted traces of intertwining operators. This space coincides with the space of Wess-Zumino-Witten conformal blocks on an elliptic curve. We compute the radial part of the second order Laplace operator on the affine Lie group acting on equivariant functions, and find that it is a certain parabolic partial differential operator, which degenerates to the elliptic Calogero-Sutherland-Moser hamiltonian as the central charge tends to minus the dual Coxeter number (the critical level). Quantum integrals of this hamiltonian are obtained as radial part of the higher Sugawara operators which are central at the critical level.

Non-linear finite $W$-symmetries and applications in elementary systems

Abstract: In this paper it is stressed that there is no {\em physical} reason for symmetries to be linear and that Lie group theory is therefore too restrictive. We illustrate this with some simple examples. Then we give a readable review on the theory finite $W$-algebras, which is an important class of non-linear symmetries. In particular, we discuss both the classical and quantum theory and elaborate on several aspects of their representation theory. Some new results are presented. These include finite $W$ coadjoint orbits, real forms and unitary representation of finite $W$-algebras and Poincare-Birkhoff-Witt theorems for finite $W$-algebras. Also we present some new finite $W$-algebras that are not related to $sl(2)$ embeddings. At the end of the paper we investigate how one could construct physical theories, for example gauge field theories, that are based on non-linear algebras.

Lagrangian Formulation of Symmetric Space sine-Gordon Models

Abstract: The symmetric space sine-Gordon models arise by conformal reduction of ordinary 2-dim $\sigma$-models, and they are integrable exhibiting a black-hole type metric in target space. We provide a Lagrangian formulation of these systems by considering a triplet of Lie groups $F \supset G \supset H$. We show that for every symmetric space $F/G$, the generalized sine-Gordon models can be derived from the $G/H$ WZW action, plus a potential term that is algebraically specified. Thus, the symmetric space sine-Gordon models describe certain integrable perturbations of coset conformal field theories at the classical level. We also briefly discuss their vacuum structure, Backlund transformations, and soliton solutions.

Lie symmetries of finite‐difference equations

Abstract: Discretizations of the Helmholtz, heat, and wave equations on uniform lattices are considered in various space–time dimensions. The symmetry properties of these finite‐difference equations are determined and it is found that they retain the same Lie symmetry algebras as their continuum limits. Solutions with definite transformation properties are obtained; identities and formulas for these functions are then derived using the symmetry algebra.

CHAPTER 19 – Homotopy Theory of Lie Groups

Abstract: This chapter discusses the homotopy theory of lie groups. M.S. Lie created the notion of Lie group, then called it a topological group. One of the motivations was to consider various geometries from the group theoretic viewpoint. A Lie group is a manifold with group structure, and locally it corresponds to a Lie algebra. A Lie group is compact or connected if the underlying manifold is compact or connected. Two Lie groups are locally isomorphic if there exists a homeomorphism between two neighborhoods of the identities compatible with the product. A Lie group G is orientable as a manifold. In fact, an orientation at the identity can be translated to an arbitrary point by left translation. Compact connected Lie groups are locally isomorphic to direct products of tori and simple nonabelian Lie groups. Thus, the classification problem of such groups reduces to that of simple groups.