Showing papers on "Lie group published in 1998"

Lie groups and Lie algebras

Abstract: From the reviews of the French edition "This is a rich and useful volume. The material it treats has relevance well beyond the theory of Lie groups and algebras, ranging from the geometry of regular polytopes and paving problems to current work on finite simple groups having a (B,N)-pair structure, or "Tits systems". A historical note provides a survey of the contexts in which groups generated by reflections have arisen. A brief introduction includes almost the only other mention of Lie groups and algebras to be found in the volume. Thus the presentation here is really quite independent of Lie theory. The choice of such an approach makes for an elegant, self-contained treatment of some highly interesting mathematics, which can be read with profit and with relative ease by a very wide circle of readers (and with delight by many, if the reviewer is at all representative)." (G.B. Seligman in MathReviews)

Representations and invariants of the classical groups

Abstract: 1. Classical groups as linear algebraic groups 2. Basic structure of classical groups 3. Algebras and representations 4. Polynomials and tensor invariants 5. Highest weight theory 6. Spinors 7. Cohomology and characters 8. Branching laws 9. Tensor representations of GL(V) 10. Tensor represenations of O(V) and Sp(V) 11. Algebraic groups and homogeneous spaces 12. Representations on Aff(X) A. Algebraic geometry B. Linear and multilinear algebra C. Associative algebras and Lie algebras D. Manifolds and Lie groups.

Mathematical topics between classical and quantum mechanics

Abstract: Introductory Overview.- I. Observables and Pure States.- Observables.- Pure States.- From Pure States to Observables.- II. Quantization and the Classical Limit.- Foundations.- Quantization on Flat Space.- Quantization on Riemannian Manifolds.- III. Groups, Bundles, and Groupoids.- Lie Groups and Lie Algebras.- Internal Symmetries and External Gauge Fields.- Lie Groupoids and Lie Algebroids.- IV. Reduction and Induction.- Reduction.- Induction.- Applications in Relativistic Quantum Theory.- I Observables and Pure States.- 1 The Structure of Algebras of Observables.- 1.1 Jordan-Lie Algebras and C*-Algebras.- 1.2 Spectrum and Commutative C*-Algebras.- 1.3 Positivity, Order, and Morphisms.- 1.4 States.- 1.5 Representations and the GNS-Construction.- 1.6 Examples of C*-Algebras and State Spaces.- 1.7 Von Neumann Algebras.- 2 The Structure of Pure State Spaces.- 2.1 Pure States and Compact Convex Sets.- 2.2 Pure States and Irreducible Representations.- 2.3 Poisson Manifolds.- 2.4 The Symplectic Decomposition of a Poisson Manifold.- 2.5 (Projective) Hilbert Spaces as Symplectic Manifolds..- 2.6 Representations of Poisson Algebras.- 2.7 Transition Probability Spaces.- 2.8 Pure State Spaces as Transition Probability Spaces.- 3 From Pure States to Observables.- 3.1 Poisson Spaces with a Transition Probability.- 3.2 Identification of the Algebra of Observables.- 3.3 Spectral Theorem and Jordan Product.- 3.4 Unitarity and Leibniz Rule.- 3.5 Orthomodular Lattices.- 3.6 Lattices Associated with States and Observables.- 3.7 The Two-Sphere Property in a Pure State Space.- 3.8 The Poisson Structure on the Pure State Space.- 3.9 Axioms for the Pure State Space of a C*-Algebra.- II Quantization and the Classical Limit.- 1 Foundations.- 1.1 Strict Quantization of Observables.- 1.2 Continuous Fields of C*-Algebras.- 1.3 Coherent States and Berezin Quantization.- 1.4 Complete Positivity.- 1.5 Coherent States and Reproducing Kernels.- 2 Quantization on Flat Space.- 2.1 The Heisenberg Group and its Representations.- 2.2 The Metaplectic Representation.- 2.3 Berezin Quantization on Flat Space.- 2.4 Properties of Berezin Quantization on Flat Space.- 2.5 Weyl Quantization on Flat Space.- 2.6 Strict Quantization and Continuous Fields on Flat Space.- 2.7 The Classical Limit of the Dynamics.- 3 Quantization on Riemannian Manifolds.- 3.1 Some Affine Geometry.- 3.2 Some Riemannian Geometry.- 3.3 Hamiltonian Riemannian Geometry.- 3.4 Weyl Quantization on Riemannian Manifolds.- 3.5 Proof of Strictness.- 3.6 Commutation Relations on Riemannian Manifolds.- 3.7 The Quantum Hamiltonian and its Classical Limit.- III Groups, Bundles, and Groupoids.- 1 Lie Groups and Lie Algebras.- 1.1 Lie Algebra Actions and the Momentum Map.- 1.2 Hamiltonian Group Actions.- 1.3 Multipliers and Central Extensions.- 1.4 The (Twisted) Lie-Poisson Structure.- 1.5 Projective Representations.- 1.6 The Twisted Enveloping Algebra.- 1.7 Group C*-Algebras.- 1.8 A Generalized Peter-Weyl Theorem.- 1.9 The Group C* Algebra as a Strict Quantization.- 1.10 Representation Theory of Compact Lie Groups.- 1.11 Berezin Quantization of Coadjoint Orbits.- 2 Internal Symmetries and External Gauge Fields.- 2.1 Bundles.- 2.2 Connections.- 2.3 Cotangent Bundle Reduction.- 2.4 Bundle Automorphisms and the Gauge Group.- 2.5 Construction of Classical Observables.- 2.6 The Classical Wong Equations.- 2.7 The H-Connection.- 2.8 The Quantum Algebra of Observables.- 2.9 Induced Group Representations.- 2.10 The Quantum Wong Hamiltonian.- 2.11 From the Quantum to the Classical Wong Equations.- 2.12 The Dirac Monopole.- 3 Lie Groupoids and Lie Algebroids.- 3.1 Groupoids.- 3.2 Half-Densities on Lie Groupoids.- 3.3 The Convolution Algebra of a Lie Groupoid.- 3.4 Action *-Algebras.- 3.5 Representations of Groupoids.- 3.6 The C*-Algebra of a Lie Groupoid.- 3.7 Examples of Lie Groupoid C*-Algebras.- 3.8 Lie Algebroids.- 3.9 The Poisson Algebra of a Lie Algebroid.- 3.10 A Generalized Exponential Map.- 3.11 The Groupoid C*-Algebra as a Strict Quantization.- 3.12 The Normal Groupoid of a Lie Groupoid.- IV Reduction and Induction.- 1 Reduction.- 1.1 Basics of Constraints and Reduction.- 1.2 Special Symplectic Reduction.- 1.3 Classical Dual Pairs.- 1.4 The Classical Imprimitivity Theorem.- 1.5 Marsden-Weinstein Reduction.- 1.6 Kazhdan-Kostant-Sternberg Reduction.- 1.7 Proof of the Classical Transitive Imprimitivity Theorem.- 1.8 Reduction in Stages.- 1.9 Coadjoint Orbits of Nilpotent Groups.- 1.10 Coadjoint Orbits of Semidirect Products.- 1.11 Singular Marsden-Weinstein Reduction.- 2 Induction.- 2.1 Hilbert C*-Modules.- 2.2 Rieffel Induction.- 2.3 The C*-Algebra of a Hilbert C*-Module.- 2.4 The Quantum Imprimitivity Theorem.- 2.5 Quantum Marsden-Weinstein Reduction.- 2.6 Induction in Stages.- 2.7 The Imprimitivity Theorem for Gauge Groupoids.- 2.8 Covariant Quantization.- 2.9 The Quantization of Constrained Systems.- 2.10 Quantization of Singular Reduction.- 3 Applications in Relativistic Quantum Theory.- 3.1 Coadjoint Orbits of the Poincare Group.- 3.2 Orbits from Covariant Reduction.- 3.3 Representations of the Poincare Group.- 3.4 The Origin of Gauge Invariance.- 3.5 Quantum Field Theory of Photons.- 3.6 Classical Yang-Mills Theory on a Circle.- 3.7 Quantum Yang-Mills Theory on a Circle.- 3.8 Induction in Quantum Yang-Mills Theory on a Circle.- 3.9 Vacuum Angles in Constrained Quantization.- Notes.- I.- II.- III.- IV.- References.

Lie group valued moment maps

Abstract: We develop a theory of "quasi"-Hamiltonian G-spaces for which the moment map takes values in the group G itself rather than in the dual of the Lie algebra. The theory includes counterparts of Hamiltonian, reductions, the Guillemin-Sternberg symplectic cro

The structure of compact groups

Abstract: The theme of this book is the Structure Theory of compact groups. It contains a completely selfcontained introduction to linear Lie groups and a substantial body of material on compact Lie groups. The authors’ approach is distinctive in so far as they define a linear Lie group as a particular subgroup of the multiplicative group of a Banach algebra. Compact Lie groups are recognized at an early stage as being linear Lie groups. This approach avoids the use of machinery on manifolds. The text is written in a style to make it accessible to the beginning graduate student with a basic knowledge in analysis, algebra, and topology. At the same time the expert will find it an excellent and rich source of information on the general structure theory of compact groups.

Diffeomorphisms Groups and Pattern Matching in Image Analysis

Abstract: In a previous paper, it was proposed to see the deformations of a common pattern as the action of an infinite dimensional group. We show in this paper that this approac h can be applied numerically for pattern matching in image analysis of digital images. Using Lie group ideas, we construct a distance between deformations defined through a metric given the cost of infinitesimal deformations. Then we propose a numerical scheme to solve a variational problem involving this distance and leading to a sub-optimal gradient pattern matching. Its links with fluid models are established.

Moving coframes. I. A practical algorithm

Abstract: This is the first in a series of papers devoted to the development and applications of a new general theory of moving frames. In this paper, we formulate a practical and easy to implement explicit method to compute moving frames, invariant differential forms, differential invariants and invariant differential operators, and solve general equivalence problems for both finite-dimensional Lie group actions and infinite Lie pseudo-groups. A wide variety of applications, ranging from differential equations to differential geometry to computer vision are presented. The theoretical justifications for the moving coframe algorithm will appear in the next paper in this series.

Runge-Kutta methods on Lie groups

Abstract: We construct generalized Runge-Kutta methods for integration of differential equations evolving on a Lie group. The methods are using intrinsic operations on the group, and we are hence guaranteed that the numerical solution will evolve on the correct manifold. Our methods must satisfy two different criteria to achieve a given order. These tasks are completely independent, so once correction functions are found to the given order, we can turn any classical RK scheme into an RK method of the same order on any Lie group. The theory in this paper shows the tight connections between the algebraic structure of the order conditions of RK methods and the algebraic structure of the so called ‘universal enveloping algebra’ of Lie algebras. This may give important insight also into the classical RK theory.

Metrics on states from actions of compact groups

Abstract: Let a compact Lie group act ergodically on a unital $C^*$-algebra $A$. We consider several ways of using this structure to define metrics on the state space of $A$. These ways involve length functions, norms on the Lie algebra, and Dirac operators. The main thrust is to verify that the corresponding metric topologies on the state space agree with the weak-$*$ topology.

On the generation of smooth three-dimensional rigid body motions

Abstract: This paper addresses the problem of generating smooth trajectories between an initial and a final position and orientation in space. The main idea is to define a functional depending on velocity or its derivatives that measures smoothness of trajectories and find a trajectory that minimizes this functional. In order to ensure that the computed trajectories are independent of the parametrization of positions and orientations, we use the notions of Riemannian metric and covariant derivative from differential geometry and formulate the problem as a variational problem on the Lie group of spatial rigid body displacements. We show that by choosing an appropriate measure of smoothness, the trajectories can be made to satisfy boundary conditions on the velocities or higher order derivatives. Dynamically smooth trajectories can be obtained by incorporating the inertia of the system into the definition of the Riemannian metric. We state the necessary conditions for the shortest distance, minimum acceleration and minimum jerk trajectories.

Lie Symmetry Analysis of Differential Equations in Finance

Abstract: Lie group theory is applied to differential equations occurring as mathematical models in financial problems. We begin with the complete symmetry analysis of the one-dimensional Black–Scholes model and show that this equation is included in Sophus Lie's classification of linear second-order partial differential equations with two independent variables. Consequently, the Black–Scholes transformation of this model into the heat transfer equation follows directly from Lie's equivalence transformation formulas. Then we carry out the classification of the two-dimensional Jacobs–Jones model equations according to their symmetry groups. The classification provides a theoretical background for constructing exact (invariant) solutions, examples of which are presented.

Discrete decomposability of the restriction of Aq(λ) with respect to reductive subgroups III. Restriction of Harish-Chandra modules and associated varieties

Abstract: Let H⊂G be real reductive Lie groups and π an irreducible unitary representation of G. We introduce an algebraic formulation (discretely decomposable restriction) to single out the nice class of the branching problem (breaking symmetry in physics) in the sense that there is no continuous spectrum in the irreducible decomposition of the restriction π| H . This paper offers basic algebraic properties of discretely decomposable restrictions, especially for a reductive symmetric pair (G,H) and for the Zuckerman-Vogan derived functor module , and proves that the sufficient condition [Invent. Math. '94] is in fact necessary. A finite multiplicity theorem is established for discretely decomposable modules which is in sharp contrast to known examples of the continuous spectrum. An application to the restriction π| H of discrete series π for a symmetric space G/H is also given.

Homogeneous Fedosov Star Products on Cotangent Bundles I: Weyl and Standard Ordering with Differential Operator Representation

Abstract: In this paper we construct homogeneous star products of Weyl type on every cotangent bundle T*Q by means of the Fedosov procedure using a symplectic torsion-free connection on T*Q homogeneous of degree zero with respect to the Liouville vector field. By a fibrewise equivalence transformation we construct a homogeneous Fedosov star product of standard ordered type equivalent to the homogeneous Fedosov star product of Weyl type. Representations for both star product algebras by differential operators on functions on Q are constructed leading in the case of the standard ordered product to the usual standard ordering prescription for smooth complex-valued functions on T*Q polynomial in the momenta (where an arbitrary fixed torsion-free connection ∇0 on Q is used). Motivated by the flat case T*ℝn another homogeneous star product of Weyl type corresponding to the Weyl ordering prescription is constructed. The example of the cotangent bundle of an arbitrary Lie group is explicitly computed and the star product given by Gutt is rederived in our approach.

Universal Solutions of Quantum Dynamical Yang–Baxter Equations

Abstract: We construct a universal trigonometric solution of the Gervais–Neveu–Felder equation in the case of finite-dimensional simple Lie algebras and finite-dimensional contragredient simple Lie superalgebras.

Hilbert-Schmidt lower bounds for estimators on matrix lie groups for ATR

Abstract: Deformable template representations of observed imagery model the variability of target pose via the actions of the matrix Lie groups on rigid templates. In this paper, we study the construction of minimum mean squared error estimators on the special orthogonal group, SO(n), for pose estimation. Due to the nonflat geometry of SO(n), the standard Bayesian formulation of optimal estimators and their characteristics requires modifications. By utilizing Hilbert-Schmidt metric defined on GL(n), a larger group containing SO(n), a mean squared criterion is defined on SO(n). The Hilbert-Schmidt estimate (HSE) is defined to be a minimum mean squared error estimator, restricted to SO(n). The expected error associated with the HSE is shown to be a lower bound, called the Hilbert-Schmidt bound (HSB), on the error incurred by any other estimator. Analysis and algorithms are presented for evaluating the HSE and the HSB in cases of both ground-based and airborne targets.

The Maxwell–Vlasov equations in Euler–Poincaré form

Abstract: Low's well-known action principle for the Maxwell–Vlasov equations of ideal plasma dynamics was originally expressed in terms of a mixture of Eulerian and Lagrangian variables. By imposing suitable constraints on the variations and analyzing invariance properties of the Lagrangian, as one does for the Euler equations for the rigid body and ideal fluids, we first transform this action principle into purely Eulerian variables. Hamilton's principle for the Eulerian description of Low's action principle then casts the Maxwell–Vlasov equations into Euler–Poincare form for right invariant motion on the diffeomorphism group of position-velocity phase space, [openface R]6. Legendre transforming the Eulerian form of Low's action principle produces the Hamiltonian formulation of these equations in the Eulerian description. Since it arises from Euler–Poincare equations, this Hamiltonian formulation can be written in terms of a Poisson structure that contains the Lie–Poisson bracket on the dual of a semidirect product Lie algebra. Because of degeneracies in the Lagrangian, the Legendre transform is dealt with using the Dirac theory of constraints. Another Maxwell–Vlasov Poisson structure is known, whose ingredients are the Lie–Poisson bracket on the dual of the Lie algebra of symplectomorphisms of phase space and the Born–Infeld brackets for the Maxwell field. We discuss the relationship between these two Hamiltonian formulations. We also discuss the general Kelvin–Noether theorem for Euler–Poincare equations and its meaning in the plasma context.

Discrete decomposability of the restriction of A_q(λ)with respect to reductive subgroups II-micro-local analysis and asymptotic K-support

Abstract: Let G' c G be real reductive Lie groups. This paper offers a criterion on the triplet (G, G', ir) that the irreducible unitary representation ir of G splits into a discrete sum of irreducible unitary representations of a subgroup G' when restricted to G', each of finite multiplicity. Furthermore, we shall give an upper estimate of the multiplicity of an irreducible unitary representation of G' occurring in WrIG'

Classical lifting processes and multiplicative vector fields

Abstract: We extend the calculus of multiplicative vector fields and differential forms and their intrinsic derivatives from Lie groups to Lie groupoids; this generalization turns out to include also the classical process of complete lifting from arbitrary manifolds to tangent and cotangent bundles. Using this calculus we give a new description of the Lie bialgebroid structure associated with a Poisson groupoid.

Learning Lie Groups for Invariant Visual Perception

Abstract: One of the most important problems in visual perception is that of visual invariance: how are objects perceived to be the same despite undergoing transformations such as translations, rotations or scaling? In this paper, we describe a Bayesian method for learning invariances based on Lie group theory. We show that previous approaches based on first-order Taylor series expansions of inputs can be regarded as special cases of the Lie group approach, the latter being capable of handling in principle arbitrarily large transfonnations. Using a matrix-exponential based generative model of images, we derive an unsupervised algorithm for learning Lie group operators from input data containing infinitesimal transfonnations. The on-line unsupervised learning algorithm maximizes the posterior probability of generating the training data. We provide experimental results suggesting that the proposed method can learn Lie group operators for handling reasonably large 1-D translations and 2-D rotations.

The classical evaluation of relativistic spin networks

Abstract: The evaluation of a relativistic spin network for the classical case of the Lie group SU(2) is given by an integral formula over copies of SU(2). For the graph determined by a 4-simplex this gives the evaluation as an integral over a space of geometries for a 4-simplex.

Deformation of compact Clifford-Klein forms of indefinite-Riemannian homogeneous manifolds

Abstract: 1 .1 . Let G be a Lie group, and H a closed subgroup of G . If a discrete subgroup Γ of G acts properly discontinuously and freely on G/H , then the double coset space Γ\G/H carries naturally a manifold structure such that the quotient map G/H → Γ\G/H is locally diffeomorphic. The manifold Γ\G/H is said to be a Clifford-Klein form of G/H . If it is compact, then Γ is said to be a uniform lattice for G/H . A typical example is a compact Riemann surface Mg with genus g ≥ 2, which is biholomorphic to a compact Clifford-Klein form of the Poincare disk G/H ' SL(2,R)/SO(2) by the uniformization theorem. It is important from geometric view point that a Clifford-Klein form Γ\G/H inherits any G-invariant local geometric structure on G/H such as (indefinite)Riemannian metric, complex structure, symplectic structure, causal structure and so on.

A general theory of phase-space quasiprobability distributions

Abstract: We present a general theory of quasiprobability distributions on phase spaces of quantum systems whose dynamical symmetry groups are (finite-dimensional) Lie groups. The family of distributions on a phase space is postulated to satisfy the Stratonovich - Weyl correspondence with a generalized traciality condition. The corresponding family of the Stratonovich - Weyl kernels is constructed explicitly. In the presented theory we use the concept of generalized coherent states, that brings physical insight into the mathematical formalism.

On the classification of unitary representations of reductive Lie groups

Abstract: Each subset is identified conjecturally (Conjecture 0.6) with a collection of unitary representations of a certain subgroup G(λu) of G. (We will give strong evidence and partial results for this conjecture.) In this way the problem of classifying Πu(G) would be reduced (by induction on the dimension of G) to the case G(λu) = G. Before considering the general program in more detail, we describe it in the familiar case G = SL(2,R). (This example will be treated more com-

Regularity of the transfer map for cohomologous cocycles

Abstract: In this paper we obtain results about the regularity of the transfer map between two cocycles over an Anosov system, with values in either a diffeomorphism or a Lie group. We also explain how certain examples of de la Llave show that our results are essentially optimal.

The identity component of the isometry group of a compact Lorentz manifold

Abstract: This result may be compared with a Theorem of E. Ghys [Ghy] (see also [Bel]), asserting a similar conclusion, but assuming that M has dimension 3, and that the action is just volume preserving and locally free. The statement there, is that the action of AG may be extended to an action of a finite cover of PSL(2,R), or to an action of the solvable 3-dimensional Lie group SOL. Here we have another motivation. We want to understand the structure of Lie groups acting isometrically on compact Lorentz manifold. The first results in the subject are due to [Zim] and [Gro]. A “final” result is due to [A-S] and [Zeg1], independently. Necessary and sufficient conditions were given in order that a Lie group acts isometrically (and locally faithfully) on a compact Lorentz manifold. Note however, that if a group acts in such a fashion, then its subgroups also act in the same way. For instance, all known examples of isometric actions of AG are obtained by viewing it as a subgroup of SL(2,R). So a natural question is: what are the maximal (connected) Lie groups acting isometrically on a compact Lorentz manifold? Equivalently:

Finite theories and marginal operators on the brane

Abstract: We show how to use D and NS fivebranes in Type IIB superstring theory to construct large classes of finite = 1 supersymmetric four dimensional field theories. In this construction, the beta functions of the theories are directly related to the bending of branes; in finite theories the branes are not bent, and vice versa. Many of these theories have multiple dimensionless couplings. A group of duality transformations acts on the space of dimensionless couplings; for a large subclass of models, this group always includes an overall SL(2,) invariance. In addition, we find even larger classes of theories which, although not finite, also have one or more marginal operators.