Showing papers on "Lie group published in 2011"

Structure and Geometry of Lie Groups

Abstract: Matrix Groups.- Concrete Matrix Groups. The Matrix Exponential Function. Linear Lie Groups. Lie Algebras.- Elementary Structure Theory of Lie Algebras. Root Decomposition. Representation Theory of Lie Algebras.

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 Poincare 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.

Stable group theory and approximate subgroups

Abstract: We note a parallel between some ideas of stable model theory and certain topics in finite combinatorics related to the sum-product phenomenon. For a simple linear group G, we show that a finite subset X with |X X \^{-1} X |/ |X| bounded is close to a finite subgroup, or else to a subset of a proper algebraic subgroup of G. We also find a connection with Lie groups, and use it to obtain some consequences suggestive of topological nilpotence. Combining these methods with Gromov's proof, we show that a finitely generated group with an approximate subgroup containing any given finite set must be nilpotent-by-finite. Model-theoretically we prove the independence theorem and the stabilizer theorem in a general first-order setting.

The theory of manipulations of pure state asymmetry I: basic tools and equivalence classes of states under symmetric operations

Abstract: the nal state can only break the symmetry in ways in which it was broken by the initial state, and its measure of asymmetry can be no greater than that of the initial state. It follows that for the purpose of understanding the consequences of symmetries of dynamics, in particular, complicated and open-system dynamics, it is useful to introduce the notion of a state’s asymmetry properties, which includes the type and measure of its asymmetry. We demonstrate and exploit the fact that the asymmetry properties of a state can also be understood in terms of information-theoretic concepts, for instance in terms of the state’s ability to encode information about an element of the symmetry group. We show that the asymmetry properties of a pure state relative to the symmetry groupG are completely specied by the characteristic function of the state, dened as (g) h jU(g)j i where g 2 G and U is the unitary representation of interest. Among other results, we show that for a symmetry described by a compact Lie group G, two pure states can be reversibly interconverted one to the other by symmetric dynamics if and only if their characteristic functions are equal up to a 1-dimensional representation of the group. PACS numbers:

Loop groups and twisted K-theory I

Abstract: This is the first in a series of papers investigating the relationship between the twisted equivariant K-theory of a compact Lie group G and the “Verlinde ring” of its loop group. In this paper we set up the foundations of twisted equivariant K-groups, and more generally twisted K-theory of groupoids. We establish enough basic properties to make effective computations. We determine the twisted equivariant K-groups of a compact connected Lie group G with torsion free fundamental group. We relate this computation to the representation theory of the loop group at a level related to the twisting.

Lectures on integrability of Lie brackets

Abstract: These notes are made up of five lectures. In Lecture 1, we introduce Lie groupoids and their infinitesimal versions. In Lecture 2, we introduce the abstract notion of a Lie algebroid and explain how many notions of differential geometry can be described in this language. Lectures 1 and 2 should help you in becoming familiar with the language of Lie groupoids and Lie algebroids. Lectures 3 and 4 are concerned with the various aspects of the integrability problem. These two lectures form the core material of this course and contain a detailed description of the integrability obstructions. In Lecture 5 we consider, as an example, aspects of integrability related to Poisson geometry. At

Stationary solutions for nonlinear dispersive Schrödinger’s equation

Abstract: This paper carries out the integration of the nonlinear dispersive Schrodinger’s equation by the aid of Lie group analysis. The stationary solutions are obtained. The two types of nonlinearity that are studied in this paper are power law and dual-power law so that the cases of Kerr law and parabolic law nonlinearity fall out as special cases.

Ricci soliton solvmanifolds

Abstract: All known examples of nontrivial homogeneous Ricci solitons are left-invariant metrics on simply connected solvable Lie groups whose Ricci operator is a multiple of the identity modulo derivations (called solsolitons, and nilsolitons in the nilpotent case). The tools from geometric invariant theory used to study Einstein solvmanifolds, turned out to be useful in the study of solsolitons as well. We prove that, up to isometry, any solsoliton can be obtained via a very simple construction from a nilsoliton together with any abelian Lie algebra of symmetric derivations of its metric Lie algebra. The following uniqueness result is also obtained: a given solvable Lie group can admit at most one solsoliton up to isometry and scaling. As an application, solsolitons of dimension at most 4 are classified.

Generalized complex structures and Lie brackets

Abstract: We remark that the equations underlying the notion of generalized complex structure have simple geometric meaning when passing to Lie algebroids/groupoids.

Discrete Geometric Optimal Control on Lie Groups

Abstract: We consider the optimal control of mechanical systems on Lie groups and develop numerical methods that exploit the structure of the state space and preserve the system motion invariants. Our approach is based on a coordinate-free variational discretization of the dynamics that leads to structure-preserving discrete equations of motion. We construct necessary conditions for optimal trajectories that correspond to discrete geodesics of a higher order system and develop numerical methods for their computation. The resulting algorithms are simple to implement and converge to a solution in very few iterations. A general software implementation is provided and applied to two example systems: an underactuated boat and a satellite with thrusters.

Multi-Galileons, solitons, and Derrick’s theorem

Abstract: The field theory Galilean symmetry, which was introduced in the context of modified gravity, gives a neat way to construct Lorentz-covariant theories of a scalar field, such that the equations of motion contain at most second-order derivatives. Here we extend the analysis to an arbitrary number of scalars, and examine the restrictions imposed by an internal symmetry, focussing in particular on SU(N) and SO(N). This therefore extends the possible gradient terms that may be used to stabilise topological objects such as sigma model lumps.

The Geometry of Complex Domains

Abstract: Preface -- 1 Preliminaries -- 2 Riemann Surfaces and Covering Spaces -- 3 The Bergman Kernel and Metric -- 4 Applications of Bergman Geometry -- 5 Lie Groups Realized as Automorphism Groups -- 6 The Significance of Large Isotropy Groups -- 7 Some Other Invariant Metrics -- 8 Automorphism Groups and Classification of Reinhardt Domains -- 9 The Scaling Method, I -- 10 The Scaling Method, II -- 11 Afterword -- Bibliography -- Index.

Asymptotic isoperimetry on groups and uniform embeddings into Banach spaces

Abstract: We characterize the possible asymptotic behaviors of the compression associated to a uniform embedding into some Lp-space, with 1 < p < ∞, for a large class of groups including connected Lie groups with exponential growth and word-hyperbolic finitely generated groups. In particular, the Hilbert compression rate of these groups is equal to 1. This also provides new and optimal estimates for the compression of a uniform embedding of the infinite 3-regular tree into some Lp-space. The main part of the paper is devoted to the explicit construction of affine isometric actions of amenable connected Lie groups on Lp-spaces whose compressions are asymptotically optimal. These constructions are based on an asymptotic lower bound of the Lp-isoperimetric profile inside balls. We compute the asymptotic of this profile for all amenable connected Lie groups and for all 1 ≤ p < ∞, providing new geometric invariants of these groups. We also relate the Hilbert compression rate with other asymptotic quantities such as volume growth and probability of return of random walks.

Observer design on the Special Euclidean group SE(3)

Abstract: This paper proposes a nonlinear pose observer designed directly on the Lie group structure of the Special Euclidean group SE(3). We use a gradient-based observer design approach and ensure that the derived observer innovation can be implemented from position measurements. We prove local exponential stability of the error and instability of the non-zero critical points. Simulations indicate that the observer is indeed almost globally stable as would be expected.

A discrete mechanics approach to the Cosserat rod theory-Part 1: static equilibria

Abstract: A theory of discrete Cosserat rods is formulated in the language of discrete Lagrangian mechanics. By exploiting Kirchhoff's kinetic analogy, the potential energy density of a rod is a function on the tangent bundle of the configuration manifold and thus formally corresponds to the Lagrangian function of a dynamical system. The equilibrium equations are derived from a variational principle using a formulation that involves null-space matrices. In this formulation, no Lagrange multipliers are necessary to enforce orthonormality of the directors. Noether's theorem relates first integrals of the equilibrium equations to Lie group actions on the configuration bundle, so-called symmetries. The symmetries relevant for rod mechanics are frame-indifference, isotropy, and uniformity. We show that a completely analogous and self-contained theory of discrete rods can be formulated in which the arc-length is a discrete variable ab initio. In this formulation, the potential energy density is defined directly on pairs of points along the arc-length of the rod, in analogy to Veselov's discrete reformulation of Lagrangian mechanics. A discrete version of Noether's theorem then identifies exact first integrals of the discrete equilibrium equations. These exact conservation properties confer the discrete solutions accuracy and robustness, as demonstrated by selected examples of application.

Analysis and Design of Nonlinear Control Systems

Abstract: Topological Space.- Differentiable Manifold.- Algebra, Lie Group and Lie Algebra.- Controllability and Observability.- Global Controllability of Affine Control Systems.- Stability and Stabilization.- Decoupling.- Input-Output Structure.- Linearization of Nonlinear Systems.- Design of Center Manifold.- Output Regulation.- Dissipative Systems.- L 2 -Gain Synthesis.- Switched Systems.- Discontinuous Dynamical Systems.

The fundamental Gray 3-groupoid of a smooth manifold and local 3-dimensional holonomy based on a 2-crossed module

Abstract: We define the thin fundamental Gray 3-groupoid S3(M) of a smooth manifold M and define (by using differential geometric data) 3-dimensional holonomies, to be smooth strict Gray 3-groupoid maps S3(M) ! C(H), where H is a 2-crossed module of Lie groups and C(H) is the Gray 3groupoid naturally constructed from H. As an application, we define Wilson 3-sphere observables.

The Hom–Yang–Baxter equation and Hom–Lie algebras

Abstract: Motivated by recent work on Hom–Lie algebras, a twisted version of the Yang–Baxter equation, called the Hom–Yang–Baxter equation (HYBE), was introduced by Yau [J. Phys. A 42, 165202 (2009)]. In this paper, several more classes of solutions of the HYBE are constructed. Some of the solutions of the HYBE are closely related to the quantum enveloping algebra of sl(2), the Jones–Conway polynomial, and Yetter-Drinfel'd modules. Under some invertibility conditions, we construct a new infinite sequence of solutions of the HYBE from a given one.

Homogeneous paracontact metric three-manifolds

Abstract: The complete classification of three-dimensional homogeneous paracontact metric manifolds is obtained. In the symmetric case, such a manifold is either flat or of constant sectional curvature −1. In the non-symmetric case, it is a Lie group equipped with a left-invariant paracontact metric structure.

The Lie Algebraic Significance of Symmetric Informationally Complete Measurements

Abstract: Examples of symmetric informationally complete positive operator-valued measures (SIC-POVMs) have been constructed in every dimension ⩽67. However, it remains an open question whether they exist in all finite dimensions. A SIC-POVM is usually thought of as a highly symmetric structure in quantum state space. However, its elements can equally well be regarded as a basis for the Lie algebra gl (d,C). In this paper we examine the resulting structure constants, which are calculated from the traces of the triple products of the SIC-POVM elements and which, it turns out, characterize the SIC-POVM up to unitary equivalence. We show that the structure constants have numerous remarkable properties. In particular we show that the existence of a SIC-POVM in dimension d is equivalent to the existence of a certain structure in the adjoint representation of gl (d,C). We hope that transforming the problem in this way, from a question about quantum state space to a question about Lie algebras, may help to make the existence problem tractable.

Lie crossed modules and gauge-invariant actions for 2-BF theories

Abstract: We generalize the BF theory action to the case of a general Lie crossed module (∂ : H → G, ⊲), where G and H are non-abelian Lie groups. Our construction requires the existence of G-invariant non-degenerate bilinear forms on the Lie algebras of G and H and we show that there are many examples of such Lie crossed modules by using the construction of crossed modules provided by short chain complexes of vector spaces. We also generalize this construction to an arbitrary chain complex of vector spaces, of finite type. We construct two gauge-invariant actions for 2-flat and fake-flat 2-connections with auxiliary fields. The first action is of the same type as the BFCG action introduced by Girelli, Pfeiffer and Popescu for a special class of Lie crossed modules, where H is abelian. The second action is an extended BFCG action which contains an additional auxiliary field. However, these two actions are related by a field redefinition. We also construct a three-parameter deformation of the extended BFCG action, which we believe to be relevant for the construction of non-trivial invariants of knotted surfaces embedded in the four-sphere. This author was supported by CMA, through Financiamento Base 2009 ISFL-1-297 from FCT/MCTES/PT. Work supported by the FCT grants PTDC/MAT/101503/2008 and PTDC/MAT/098770/2008. Member of the Mathematical Physics Group, University of Lisbon. Work supported by the FCT grants PTDC/MAT/099880/2008 and PTDC/MAT/69635/2006.

Automorphic Forms on SL2 (R)

Abstract: This book provides an introduction to some aspects of the analytic theory of automorphic forms on G=SL2(R) or the upper-half plane X, with respect to a discrete subgroup G of G of finite covolume. The point of view is inspired by the theory of infinite dimensional unitary representations of G; this is introduced in the last sections, making this connection explicit. The topics treated include the construction of fundamental domains, the notion of automorphic form on G\"G and its relationship with the classical automorphic forms on X, Poincare series, constant terms, cusp forms, finite dimensionality of the space of automorphic forms of a given type, compactness of certain convolution operators, Eisenstein series, unitary representations of G, and the spectral decomposition of L2 (G\"G). The main prerequisites are some results in functional analysis (reviewed, with references) and some familiarity with the elementary theory of Lie groups and Lie algebras. Graduate students and researchers in analytic number theory will find much to interest them in this book.

Protection of quantum systems by nested dynamical decoupling

Abstract: Based on a theorem we establish on dynamical decoupling of time-dependent systems, we present a scheme of nested Uhrig dynamical decoupling (NUDD) to protect multiqubit systems in generic quantum baths to arbitrary decoupling orders, using only single-qubit operations. The number of control pulses in NUDD increases polynomially with the decoupling order. For general multilevel systems, this scheme can preserve a set of unitary Hermitian system operators which mutually either commute or anticommute, and hence all operators in the Lie algebra generated from this set of operators, generating an effective symmetry group for the system up to a given order of precision. NUDD can be implemented with pulses of finite amplitude, up to an error in the second order of the pulse durations.

Approximate groups. I The torsion-free nilpotent case

Abstract: We describe the structure of ``K-approximate subgroups'' of torsion-free nilpotent groups, paying particular attention to Lie groups. Three other works, by Fisher-Katz-Peng, Sanders and Tao, have appeared which independently address related issues. We comment briefly on some of the connections between these papers.