Showing papers on "Lie group published in 2010"

Lie Groups and Algebras with Applications to Physics, Geometry, and Mechanics

Abstract: This is an introductory text on Lie groups and algebras and their roles in diverse areas of pure and applied mathematics and physics. The material is presented in a way that is at once intuitive, geometric, applications oriented, and, most of the time, mathematically rigorous. It is intended for students and researchers without an extensive background in physics, algebra, or geometry. In addition to an exposition of the fundamental machinery of the subject, there are many concrete examples that illustrate the role of Lie groups and algebras in various fields of mathematics and physics: elementary particle physics, Riemannian geometry, symmetries of differential equations, completely integrable systems, and bifurcation theory.

Integral Geometry and Radon Transforms

Abstract: 1.- The Radon Transformon on Rn 1.1- Introduction 1.2- The Radon Transform: The Support Theorem 1.3- The Inversion Formula: Injectivity Questions 1.4- The Plancherel Formula 1.5- Radon Transform of Distribution 1.6- Integration over d-planes: X-Ray Transforms 1.7- Applications 2.- A Duality in Integral Geometry 2.1- Homogeneous Spaces in Duality 2.2- The Radon Transform for the Double Fibration: Principal Problems 2.3- Orbital Integrals 2.4- Examples of Radon Transforms for Homogeneous Spaces in Duality 3.- The Radon Transform on Two-Point Homogeneous Spaces 3.1- Spaces of Constant Curvature: Inversion and Support Theorems 3.2- Compact Two-Point Homogeneous Spaces: Applications 3.3- Noncompact Two-Point Homogeneous Spaces 3.4- Support Theorems Relative to Horocycles 4.- The X-Ray Transform on a Symmetric Space 4.1- Compact Symmetric Spaces: Injectivity and Local Inversion: Support Theorem 4.2- Noncompact Symmetric Spaces: Global Inversion and General Support Theorem 4.3- Maximal Tori and Minimal Spheres in Compact Symmetric Spaces 5.- Orbital Integrals 5.1- Isotropic Spaces 5.2- Orbital Integrals 5.3- Generalized Riesz Potentials 5.4- Determination of a Function from its Integral over Lorentzian Spheres 5.5- Orbital Integrands and Huygens' Principle 6.- The Mean-Value Operator 6.1- An Injectivity Result 6.2- Asgeirsson's Mean-Value Theorem Generalized 6.3- John's Indentities 7.- Fourier Transforms and Distribution: A Rapid Course 7.1-The Topology of Spaces D(Rn), E(Rn) and S(Rn) 7.2- Distribution 7.3- Convolutions 7.4- The Fourier Transform 7.5- Differential Operators with Constant Coefficients 7.6- Riesz Potentials 8.- Lie Transformation Groups and Differential Operators 8.1- Manifolds and Lie Groups 8.2- Lie Transformation Groups and Radon Transforms 9.- Bibliography 10.- Notational Conventions 11.- Index.

Symmetry constraints on generalizations of Bjorken flow

Abstract: I explain a generalization of Bjorken flow where the medium has finite transverse size and expands both radially and along the beam axis. If one assumes that the equations of viscous hydrodynamics can be used, with p={epsilon}/3 and zero bulk viscosity, then the flow I describe can be developed into an exact solution of the relativistic Navier-Stokes equations. The local four-velocity in the flow is entirely determined by the assumption of symmetry under a subgroup of the conformal group.

Surface group representations with maximal Toledo invariant

Abstract: We develop the theory of maximal representations of the fundamental group π 1 (Σ) of a compact connected oriented surface Σ (possibly with boundary) into Lie groups G of Hermitian type. For any homomorphism p: π 1 (Σ) → G, we define the Toledo invariant T(Σ, p), a numerical invariant which has both topological and analytical interpretations. We establish important properties of T(Σ, ρ), among which continuity, uniform boundedness on the representation variety, additivity under connected sum of surfaces and congruence relations mod ℤ. We thus obtain information about the representation variety as well as striking geometric properties of maximal representations, that is representations whose Toledo invariant achieves the maximum value. Moreover we establish properties of boundary maps associated to maximal representations which generalize naturally monotonicity properties of semiconjugations of the circle. We define a rotation number function for general locally compact groups and study it in detail for groups of Hermitian type. Properties of the rotation number, together with the existence of boundary maps, lead to additional invariants for maximal representations and show that the subset of maximal representations is always real semialgebraic.

Ageing and dynamical scaling far from equilibrium

Abstract: Volume II: 1. Ageing Phenomena 1.1 Introduction 1.2 Phase-Ordering Kinetics 1.3 Phenomenology of Ageing 1.4 Scaling Behaviour of Intergrated Responses 1.5 Values of Non-Equilibrium Exponents 1.6 Global Persistence 2. Exactly Solvable Models 2.1 One-dimensional Glauber-Ising Model 2.2. A Non-Glauberian Kinetic Ising Model 2.3 The Free Random Walk 2.4 The Spherical Model 2.5 The Long-range Spherical Model 2.6 XY-Model in Spin-wave Approximation 2.7 OJK Approximation 2.8 Further Solvable Models 3. Simple Ageing: an Overview 3.1 Non-equilibrium Critical Dynamics 3.2 Ordered Initial States 3.3 Conserved Order-parameter (Model B) 3.4 Fully Frustrated Systems 3.5 Disordered Systems I: Ferromagnets 3.6 Disordered Systems II: Critical Glassy Systems 3.7 Surface Effects 3.8 Ageing with Absorbing Steady-states I 3.9 Ageing with Absorbing Steady-states II 3.10 Reversible Reaction-diffusion Systems 3.11 Growth processes 4. Local scale invariance I: z = 2 4.1 Introduction 4.2 The Schrodinger Group 4.3 From Schrodinger-invariane to Ageing 4.4 Conformal Invariance and Ageing 4.5 Galilei-invariance 4.6 Calculation of Two-time Response and Correlation Functions 4.7 Tests of Ageing and Conformal-invariance for z = 2 4.8 Nonrelativistic AdS/CFT Correspondence 5. Local scale invariance II: z 2 5.1 Axioms of Local Scale-invariance 5.2 Construction of the Infinitesimal Generators 5.3 Generalised Bargman Superselection Rule 5.4 Calculation of Two-time Responses 5.5 Calculation of Two-time Correlations 5.6 Tests of Local Scale-invariance with z 2 5.7 Global Time-reparametrisation-invariance 5.8 Concluding Remarks 6. Lifshitz Points 6.1 Phenomenology 6.2 Critical Exponents at Lifshitz Points 6.3 A Different Type of Local Scale-transformation 6.4 Application to Lifshitz Points 6.5 Conclusions Appendices: A. Equilibrium Models A.1 Potts Model A.2 Clock Model A.3 Turban Model A.4 Baxter-Wu Model A.5 Blume-Capel Model A.6 XY Model A.7 O(n) Model A.8 Double Exchange Model A.9 Hilhorst-van Leeuven Model A.10 Frustrated Spin Models A.11 Weakly random Spin Systems A.12 Logarithmic Sub-scaling Exponents A.13 Ising Spin Glasses A.14 Gauge Glasses D. Langevin Equations and Path Integrals I. Cluster Algorithms: Competing Interactions J. Fractional Derivatives J.1 Singular Fractional Derivatives J.2 Fractional Laplacians K. Conformally Invarioant Interacting Fields K.1 Conformal Invariance and Coupling Constants K.2 Conformally Conserved Currents L. Lie Groups and Lie Algebras: a Reminder L.1 Finite Groups L.2 Continuous Groups and Lie groups L.3 From Lie groups to Lie Algebras and Back L.4 Matrix Representations and the Cartan-Weyl Basis L.5 Function-space Representations L.6 Central Extensions Solutions. Frequently Used Symbols. Abbreviations. References. List of Tables. List of Figures. Index

An Invitation to Higher Gauge Theory

Abstract: In this easy introduction to higher gauge theory, we describe parallel transport for particles and strings in terms of 2-connections on 2-bundles. Just as ordinary gauge theory involves a gauge group, this generalization involves a gauge '2-group'. We focus on 6 examples. First, every abelian Lie group gives a Lie 2-group; the case of U(1) yields the theory of U(1) gerbes, which play an important role in string theory and multisymplectic geometry. Second, every group representation gives a Lie 2-group; the representation of the Lorentz group on 4d Minkowski spacetime gives the Poincar\'e 2-group, which leads to a spin foam model for Minkowski spacetime. Third, taking the adjoint representation of any Lie group on its own Lie algebra gives a 'tangent 2-group', which serves as a gauge 2-group in 4d BF theory, which has topological gravity as a special case. Fourth, every Lie group has an 'inner automorphism 2-group', which serves as the gauge group in 4d BF theory with cosmological constant term. Fifth, every Lie group has an 'automorphism 2-group', which plays an important role in the theory of nonabelian gerbes. And sixth, every compact simple Lie group gives a 'string 2-group'. We also touch upon higher structures such as the 'gravity 3-group' and the Lie 3-superalgebra that governs 11-dimensional supergravity.

Gradient-Like Observers for Invariant Dynamics on a Lie Group

Abstract: This paper proposes a design methodology for non-linear state observers for invariant kinematic systems posed on finite dimensional connected Lie groups, and studies the associated fundamental system structure. The concept of synchrony of two dynamical systems is specialized to systems on Lie groups. For invariant systems this leads to a general factorization theorem of a nonlinear observer into a synchronous (internal model) term and an innovation term. The synchronous term is fully specified by the system model. We propose a design methodology for the innovation term based on gradient-like terms derived from invariant or non-invariant cost functions. The resulting nonlinear observers have strong (almost) global convergence properties and examples are used to demonstrate the relevance of the proposed approach.

Maxwell strata in sub-Riemannian problem on the group of motions of a plane

Abstract: The left-invariant sub-Riemannian problem on the group of motions of a plane is considered. Sub-Riemannian geodesics are parameterized by Jacobi's functions. Discrete symmetries of the problem generated by reflections of pendulum are described. The corresponding Maxwell points are characterized, on this basis an upper bound on the cut time is obtained.

Coordinated Motion Design on Lie Groups

Abstract: The present paper proposes a unified geometric framework for coordinated motion on Lie groups It first gives a general problem formulation and analyzes ensuing conditions for coordinated motion Then, it introduces a precise method to design control laws in fully actuated and underactuated settings with simple integrator dynamics It thereby shows that coordination can be studied in a systematic way once the Lie group geometry of the configuration space is well characterized Applying the proposed general methodology to particular examples allows to retrieve control laws that have been proposed in the literature on intuitive grounds A link with Brockett's double bracket flows is also made The concepts are illustrated on SO(3) , SE(2) and SE(3)

Integrable Evolution Equations on Spaces of Tensor Densities and Their Peakon Solutions

Abstract: We study a family of equations defined on the space of tensor densities of weight λ on the circle and introduce two integrable PDE. One of the equations turns out to be closely related to the inviscid Burgers equation while the other has not been identified in any form before. We present their Lax pair formulations and describe their bihamiltonian structures. We prove local wellposedness of the corresponding Cauchy problem and include results on blow-up as well as global existence of solutions. Moreover, we construct “peakon” and “multi-peakon” solutions for all λ ≠ 0, 1, and “shock-peakons” for λ = 3. We argue that there is a natural geometric framework for these equations that includes other well-known integrable equations and which is based on V. Arnold’s approach to Euler equations on Lie groups.

On the Use of Lie Group Time Integrators in Multibody Dynamics

Abstract: This paper proposes a family of Lie group time integrators for the simulation of flexible multibody systems. The method provides an elegant solution to the rotation parametrization problem. As an extension of the classical generalized-a method for dynamic systems, it can deal with constrained equations of motion. Second-order accuracy is demonstrated in the unconstrained case. The performance is illustrated on several critical benchmarks of rigid body systems with high rotation speeds, and second-order accuracy is evidenced in all of them, even for constrained cases. The remarkable simplicity of the new algorithms opens some interesting perspectives for real-time applications, model-based control, and optimization of multibody systems.

Eisenstein series for higher-rank groups and string theory amplitudes

Abstract: Scattering amplitudes of superstring theory are strongly constrained by the requirement that they be invariant under dualities generated by discrete subgroups, E_n(Z), of simply-laced Lie groups in the E_n series (n<= 8). In particular, expanding the four-supergraviton amplitude at low energy gives a series of higher derivative corrections to Einstein's theory, with coefficients that are automorphic functions with a rich dependence on the moduli. Boundary conditions supplied by string and supergravity perturbation theory, together with a chain of relations between successive groups in the E_n series, constrain the constant terms of these coefficients in three distinct parabolic subgroups. Using this information we are able to determine the expressions for the first two higher derivative interactions (which are BPS-protected) in terms of specific Eisenstein series. Further, we determine key features of the coefficient of the third term in the low energy expansion of the four-supergraviton amplitude (which is also BPS-protected) in the E_8 case. This is an automorphic function that satisfies an inhomogeneous Laplace equation and has constant terms in certain parabolic subgroups that contain information about all the preceding terms.

Einstein solvmanifolds are standard

Abstract: We study Einstein manifolds admitting a transitive solvable Lie group of isometries (solvmanifolds) It is conjectured that these exhaust the class of noncompact homogeneous Einstein manifolds J Heber has shown that under a simple algebraic condition (he calls such a solvmanifold standard), Einstein solvmanifolds have many remarkable structural and uniqueness properties In this paper, we prove that any Einstein solvmanifold is standard, by applying a stratification procedure adapted from one in geometric invariant theory due to F Kirwan

The Geometrical Language of Continuum Mechanics

Abstract: Epstein presents the fundamental concepts of modern differential geometry within the framework of continuum mechanics. Divided into three parts of roughly equal length, the book opens with a motivational chapter to impress upon the reader that differential geometry is indeed the natural language of continuum mechanics or, better still, that the latter is a prime example of the application and materialisation of the former. In the second part, the fundamental notions of differential geometry are presented with rigor using a writing style that is as informal as possible. Differentiable manifolds, tangent bundles, exterior derivatives, Lie derivatives, and Lie groups are illustrated in terms of their mechanical interpretations. The third part includes the theory of fiber bundles, G-structures, and groupoids, which are applicable to bodies with internal structure and to the description of material inhomogeneity. The abstract notions of differential geometry are thus illuminated by practical and intuitively meaningful engineering applications.

Eisenstein series for higher-rank groups and string theory amplitudes

Abstract: Scattering amplitudes of superstring theory are strongly constrained by the requirement that they be invariant under dualities generated by discrete subgroups, En(Z), of simply-laced Lie groups in the En series (n � 8). In particular, expanding the four-supergraviton amplitude at low energy gives a series of higher derivative corrections to Einstein's theory, with coefficients that are automorphicfunctions with a rich dependence on the moduli. Boundary conditions supplied by string and supergravity perturbation theory, together with a chain of relations between successive groups in the En series, constrain the constant terms of these coefficients in three distinct parabolic subgroups. Using this information we are able to determine the expressions for the first two higher derivative interactions (which are BPS-protected) in terms of specific Eisenstein series. Further, we determine key features of the coefficient of the third term in the low energy expansionof the four-supergraviton amplitude (which is also BPS-protected) in the E8 case. This is an automorphic function that satisfies an inhomogeneous Laplace equation and has constant terms in certain parabolic subgroups that contain information about all the preceding terms.

A Practical Guide to the Invariant Calculus

Abstract: This book explains recent results in the theory of moving frames that concern the symbolic manipulation of invariants of Lie group actions. In particular, theorems concerning the calculation of generators of algebras of differential invariants, and the relations they satisfy, are discussed in detail. The author demonstrates how new ideas lead to significant progress in two main applications: the solution of invariant ordinary differential equations and the structure of Euler-Lagrange equations and conservation laws of variational problems. The expository language used here is primarily that of undergraduate calculus rather than differential geometry, making the topic more accessible to a student audience. More sophisticated ideas from differential topology and Lie theory are explained from scratch using illustrative examples and exercises. This book is ideal for graduate students and researchers working in differential equations, symbolic computation, applications of Lie groups and, to a lesser extent, differential geometry.

Pure Spinors on Lie groups

Abstract: For any manifold M, the direct sum TM oplus T*M carries a natural inner product given by the pairing of vectors and covectors. Differential forms on M may be viewed as spinors for the corresponding Clifford bundle, and in particular there is a notion of emph{pure spinor}. In this paper, we study pure spinors and Dirac structures in the case when M=G is a Lie group with a bi-invariant pseudo-Riemannian metric, e.g. G semi-simple. The applications of our theory include the construction of distinguished volume forms on conjugacy classes in G, and a new approach to the theory of quasi-Hamiltonian G-spaces.

Quantized Nambu–Poisson manifolds and n-Lie algebras

Abstract: We investigate the geometric interpretation of quantized Nambu–Poisson structures in terms of noncommutative geometries. We describe an extension of the usual axioms of quantization in which classical Nambu–Poisson structures are translated to n-Lie algebras at quantum level. We demonstrate that this generalized procedure matches an extension of Berezin–Toeplitz quantization yielding quantized spheres, hyperboloids, and superspheres. The extended Berezin quantization of spheres is closely related to a deformation quantization of n-Lie algebras as well as the approach based on harmonic analysis. We find an interpretation of Nambu–Heisenberg n-Lie algebras in terms of foliations of Rn by fuzzy spheres, fuzzy hyperboloids, and noncommutative hyperplanes. Some applications to the quantum geometry of branes in M-theory are also briefly discussed.

On two-dimensional holonomy

Abstract: We define the thin fundamental categorical group P2(M,�) of a based smooth manifold (M,�) as the categorical group whose objects are rank-1 homotopy classes of based loops on M, and whose morphisms are rank2 homotopy classes of homotopies between based loops on M. Here two maps are rank-n homotopic, when the rank of the differential of the homotopy between them equals n. Let C(G) be a Lie categorical group coming from a Lie crossed module G = (∂: E ! G, ⊲). We construct categorical holonomies, defined to be smooth morphisms P2(M,�) ! C(G), by using a notion of categorical connections, being a pair (ω, m), where ω is a connection 1-form on P, a principal G bundle over M, and m is a 2-form on P with values in the Lie algebra of E, with the pair (ω, m) satisfying suitable conditions. As a further result, we are able to define Wilson spheres in this context.

Lorentz Ricci Solitons on 3-dimensional Lie groups

Abstract: The three-dimensional Heisenberg group H 3 has three left-invariant Lorentzian metrics g 1, g 2, and g 3 as in Rahmani (J. Geom. Phys. 9(3), 295–302 (1992)). They are not isometric to each other. In this paper, we characterize the left-invariant Lorentzian metric g 1 as a Lorentz Ricci Soliton. This Ricci Soliton g 1 is a shrinking non-gradient Ricci Soliton. We also prove that the group E(2) of rigid motions of Euclidean 2-space and the group E(1, 1) of rigid motions of Minkowski 2-space have Lorentz Ricci Solitons.

A Lie group variational integrator for rigid body motion in SE(3) with applications to underwater vehicle dynamics

Abstract: The topic of variational integrators for mechanical systems whose dynamics evolve on nonlinear spaces has seen strong growth recently. Within this class of variational integrators is the subclass of Lie group variational integrators that can be used for mechanical systems whose dynamics evolve on Lie groups. This class of mechanical systems includes all systems that can be modeled as rigid bodies or connections of rigid bodies. In this paper, we present a Lie group variational integrator for the full (translation and orientation) motion of a rigid body under the possible influence of nonconservative forces and torques. We use a discretization scheme for such systems which is based on the discrete Lagrange-d'Alembert principle to obtain the Lie group variational integrator. We apply the composition of the Lie group variational integrator with its adjoint and a Crouch-Grossman method to the example of a conservative underwater system. We show numerically that with respect to energy these manifold methods, as expected, behave as a symplectic integrator and a nonsymplectic integrator, respectively.

Cech cocycles for differential characteristic classes -- An infinity-Lie theoretic construction

Abstract: What are called secondary characteristic classes in Chern-Weil theory are a refinement of ordinary characteristic classes of principal bundles from cohomology to differential cohomology We consider the problem of refining the construction of secondary characteristic classes from cohomology sets to cocycle spaces; and from Lie groups to higher connected covers of Lie groups by smooth infinity-groups, ie, by smooth groupal A-infinity-spaces Namely, we realize differential characteristic classes as morphisms from infinity-groupoids of smooth principal infinity-bundles with connections to infinity-groupoids of higher U(1)-gerbes with connections This allows us to study the homotopy fibers of the differential characteristic maps thus obtained and to show how these describe differential obstruction problems This applies in particular to the higher twisted differential spin structures called twisted differential string structures and twisted differential fivebrane structures

Multiplicative bundle gerbes with connection

Abstract: Multiplicative bundle gerbes are gerbes over a Lie group which are compatible with the group structure. In this article connections on such bundle gerbes are introduced and studied. It is shown that multiplicative bundle gerbes with connection furnish geometrical constructions of the following objects: smooth central extensions of loop groups, Chern–Simons actions for arbitrary gauge groups, and symmetric bi-branes for WZW models with topological defect lines.

Counting arithmetic lattices and surfaces

Abstract: We give estimates on the number AL H (x) of conjugacy classes of arithmetic lattices Γ of covolume at most x in a simple Lie group H. In particular, we obtain a first concrete estimate on the number of arithmetic 3-manifolds of volume at most x. Our main result is for the classical case H = PSL(2,ℝ) where we show that limx→∞ log AL H (X)/x log x = 1/2π. The proofs use several different techniques: geometric (bounding the number of generators of Γ as a function of its covolume), number theoretic (bounding the number of maximal such Γ) and sharp estimates on the character values of the symmetric groups (to bound the subgroup growth of Γ).

Locally homogeneous geometric manifolds

Abstract: Motivated by Felix Klein's notion that geometry is governed by its group of symmetry transformations, Charles Ehresmann initiated the study of geometric structures on topological spaces locally modeled on a homogeneous space of a Lie group. These locally homogeneous spaces later formed the context of Thurston's 3-dimensional geometrization program. The basic problem is for a given topology S and a geometry X = G/H, to classify all the possible ways of introducing the local geometry of G/H into S. For example, a sphere admits no local Euclidean geometry: there is no metrically accurate Euclidean atlas of the earth. One develops a space whose points are equivalence classes of geometric structures on S, which itself exhibits a rich geometry and symmetries arising from the topological symmetries of S. In this talk I will survey several examples of the classification of locally homogeneous geometric structures on manifolds in low dimension, and how it leads to a general study of surface group representations. In particular geometric structures are a useful tool in understanding local and global properties of deformation spaces of representations of fundamental groups.

Variational identities and Hamiltonian structures

Abstract: This report is concerned with Hamiltonian structures of classical and super soliton hierarchies. In the classical case, basic tools are variational identities associated with continuous and discrete matrix spectral problems, targeted to soliton equations derived from zero curvature equations over general Lie algebras, both semisimple and non‐semisimple. In the super case, a supertrace identity is presented for constructing Hamiltonian structures of super soliton equations associated with Lie superalgebras. We illustrate the general theories by the KdV hierarchy, the Volterra lattice hierarchy, the super AKNS hierarchy, and two hierarchies of dark KdV equations and dark Volterra lattices. The resulting Hamiltonian structures show the commutativity of each hierarchy discussed and thus the existence of infinitely many commuting symmetries and conservation laws.