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Showing papers on "Lie group published in 2009"


Book
16 Jun 2009
TL;DR: Lie groups and algebraic groups have been studied in a wide range of contexts, e.g. in this article, where the structure of classical groups and their representation on Spaces of Regular Functions are discussed.
Abstract: Lie Groups and Algebraic Groups.- Structure of Classical Groups.- Highest-Weight Theory.- Algebras and Representations.- Classical Invariant Theory.- Spinors.- Character Formulas.- Branching Laws.- Tensor Representations of GL(V).- Tensor Representations of O(V) and Sp(V).- Algebraic Groups and Homogeneous Spaces.- Representations on Spaces of Regular Functions.

559 citations


Book
26 Oct 2009
TL;DR: In this article, the basic formalism of Probability theory has been used for the quantization of spin-coherent states in the context of quantum information and quantum physics.
Abstract: Part I: Coherent States 1. Introduction 2. The Standard Coherent States: The Basics 3. The Standard Coherent States: The (Elementary) Mathematics 4. Coherent States in Quantum Information: An Example of Experimental Manipulation 5. Coherent States: A General Construction 6. The Spin Coherent States 7. Selected Pieces of Applications of Standard and Spin Coherent States 8. SU(1,1) or SL(2,R)Coherent States 9. Another Family of SU(1,1) Coherent States for Quantum Systems 10. Squeezed States and their SU(1,1) Content 11. Fermionic Coherent States Part II: Coherent State Quantization 12. Standard Coherent Quantization: The Klauder-Berezin Approach 13. Coherent State or Frame Quantization 14. CS Quantization of Finite Set, Unit Interval, and Circle 15. CS Quantization of Motions on Circle, Interval, and Others 16. Quantization of the Motion on the Torus 17. Fuzzy Geometries: Sphere and Hyperboloid 18. Conclusion and Outlook Appendices A. The Basic Formalism of Probability Theory B. The Basics of Lie Algebra, Lie Groups, and their Representation C. SU(2)-Material D. Wigner-Eckart Theorem for CS quantized Spin Harmonics E. Symmetrization of the Commutator Bibliography

448 citations


Book
23 Nov 2009
TL;DR: In this paper, Fourier analysis on Compact Lie Group (CLG) and Fourier Analysis on SU(2) is used to analyze pseudo-differential operators on SU (2).
Abstract: Preface.- Introduction.- Part I Foundations of Analysis.- A Sets, Topology and Metrics.- B Elementary Functional Analysis.- C Measure Theory and Integration.- D Algebras.- Part II Commutative Symmetries.- 1 Fourier Analysis on Rn.- 2 Pseudo-differential Operators on Rn.- 3 Periodic and Discrete Analysis.- 4 Pseudo-differential Operators on Tn.- 5 Commutator Characterisation of Pseudo-differential Operators.- Part III Representation Theory of Compact Groups.- 6 Groups.- 7 Topological Groups.- 8 Linear Lie Groups.- 9 Hopf Algebras.- Part IV Non-commutative Symmetries.- 10 Pseudo-differential Operators on Compact Lie Groups.- 11 Fourier Analysis on SU(2).- 12 Pseudo-differential Operators on SU(2).- 13 Pseudo-differential Operators on Homogeneous Spaces.- Bibliography.- Notation.- Index.

356 citations


Journal ArticleDOI
TL;DR: In this paper, the authors generalize the Deligne groupoid to a functor γ from L ∞ -algebras concentrated in degree > -n to n-groupoids.
Abstract: The Deligne groupoid is a functor from nilpotent differential graded Lie algebras concentrated in positive degrees to groupoids; in the special case of Lie algebras over a field of characteristic zero, it gives the associated simply connected Lie group. We generalize the Deligne groupoid to a functor γ from L ∞ -algebras concentrated in degree > -n to n-groupoids. (We actually construct the nerve of the n-groupoid, which is an enriched Kan complex.) The construction of gamma is quite explicit (it is based on Dupont's proof of the de Rham theorem) and yields higher dimensional analogues of holonomy and of the Campbell-Hausdorff formula. In the case of abelian L ∞ algebras (i.e., chain complexes), the functor γ is the Dold-Kan simplicial set.

244 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that under suitable assumptions, there is a one-to-one correspondence between classical groups and free quantum groups, in the compact orthogonal case.

232 citations


BookDOI
01 Jan 2009
TL;DR: In this article, infinite-dimensional Lie groups are used for topological and Holomorphic Gauge Theories, and applications of groups are discussed. But they do not cover the application of Lie groups in topological topology.
Abstract: Preliminaries.- Infinite-Dimensional Lie Groups: Their Geometry, Orbits, and Dynamical Systems.- Applications of Groups: Topological and Holomorphic Gauge Theories.

218 citations


Journal ArticleDOI
TL;DR: In this paper, a geometrical framework for the design of observers on finite-dimensional Lie groups for systems which possess some specific symmetries is presented, and the design and the error (between true and estimated state) equation are explicit and intrinsic.
Abstract: In this technical note, we give a geometrical framework for the design of observers on finite-dimensional Lie groups for systems which possess some specific symmetries. The design and the error (between true and estimated state) equation are explicit and intrinsic. We consider also a particular case: left-invariant systems on Lie groups with right equivariant output. The theory yields a class of observers such that the error equation is autonomous. The observers converge locally around any trajectory, and the global behavior is independent from the trajectory, which is reminiscent of the linear stationary case.

193 citations


Journal ArticleDOI
TL;DR: In this article, a Lagrangian-based 2+1 dimensional gauge theory with scale invariance and N=8 supersymmetry is presented. But it does not admit any tunable coupling constant.
Abstract: Based on recent developments, in this letter we find 2+1 dimensional gauge theories with scale invariance and N=8 supersymmetry. The gauge theories are defined by a Lagrangian and are based on an infinite set of 3-algebras, constructed as an extension of ordinary Lie algebras. Recent no-go theorems on the existence of 3-algebras are circumvented by relaxing the assumption that the invariant metric is positive definite. The gauge group is non compact, and its maximally compact subgroup can be chosen to be any ordinary Lie group, under which the matter fields are adjoints or singlets. The theories are parity invariant and do not admit any tunable coupling constant. In the case of SU(N) the moduli space of vacua contains a branch of the form (R^8)^N/S_N. These properties are expected for the field theory living on a stack of M2 branes.

192 citations


Journal ArticleDOI
TL;DR: This paper generalizes the original mean shift algorithm to data points lying on Riemannian manifolds to extend mean shift based clustering and filtering techniques to a large class of frequently occurring non-vector spaces in vision.
Abstract: The original mean shift algorithm is widely applied for nonparametric clustering in vector spaces. In this paper we generalize it to data points lying on Riemannian manifolds. This allows us to extend mean shift based clustering and filtering techniques to a large class of frequently occurring non-vector spaces in vision. We present an exact algorithm and prove its convergence properties as opposed to previous work which approximates the mean shift vector. The computational details of our algorithm are presented for frequently occurring classes of manifolds such as matrix Lie groups, Grassmann manifolds, essential matrices and symmetric positive definite matrices. Applications of the mean shift over these manifolds are shown.

182 citations


Journal ArticleDOI
TL;DR: In this article, an invariant definition of the hypoelliptic Laplacian on sub-Riemannian structures with constant growth vector, using the Popp's volume form introduced by Montgomery, was presented.

181 citations


Journal ArticleDOI
TL;DR: In this paper, the authors describe non-Abelian generalizations of the Kuramoto model for any classical compact Lie group and identify their main properties, including the nonlinear evolution equations maintaining the unitarity of all variables which therefore evolve on the compact manifold of U(n).
Abstract: We describe non-Abelian generalizations of the Kuramoto model for any classical compact Lie group and identify their main properties. These models may be defined on any complex network where the variable at each node is an element of the unitary group U(n), or a subgroup of U(n). The nonlinear evolution equations maintain the unitarity of all variables which therefore evolve on the compact manifold of U(n). Synchronization of trajectories with phase locking occurs as for the Kuramoto model, for values of the coupling constant larger than a critical value, and may be measured by various order and disorder parameters. Limit cycles are characterized by a frequency matrix which is independent of the node and is determined by minimizing a function which is quadratic in the variables. We perform numerical computations for n = 2, for which the SU(2) group manifold is S3, for a range of natural frequencies and all-to-all coupling, in order to confirm synchronization properties. We also describe a second generalization of the Kuramoto model which is formulated in terms of real m-vectors confined to the (m − 1)-sphere for any positive integer m, and investigate trajectories numerically for the S2 model. This model displays a variety of synchronization phenomena in which trajectories generally synchronize spatially but are not necessarily phase-locked, even for large values of the coupling constant.

01 Jan 2009
TL;DR: In this article, the stereographic projection from the north and south poles is used to obtain the coordinate maps of the unit sphere of a m−1-dimensional manifold with two charts.
Abstract: Definition 1.1. [Ol] An m-dimensional manifold M is a topological space covered by a collection of open subsets Wα ⊂M (coordinate charts) and maps Xα : Wα → Vα ⊂ R one-to-one and onto, where Vα is an open, connected subset of R. (Wα,Xα) define coordinates on M. M is a smooth manifold if the maps Xαβ = Xβ ◦X−1 α , are smooth where they are defined, i.e. on Xα(Wα ∩Wβ) to Xβ(Wα ∩Wβ). Example 1.2. R is a m-dimensional manifold covered with a single chart. Example 1.3. The unit sphere Sm−1 := {x ∈ R | ∑m i=1 x 2 i = 1} is a m−1dimensional manifold covered with two charts obtained by omitting the north and south poles respectively. The coordinate maps are obtained considering the stereographic projection from the north and south pole respectively.

Journal ArticleDOI
TL;DR: Numerical analysis of structure-preserving time-integrators for rigid body-type mechanical systems are derived from a discrete Hamilton–Pontryagin variational principle and a novel class of variational partitioned Runge–Kutta methods on Lie groups are derived.
Abstract: In this paper, structure-preserving time-integrators for rigid body-type mechanical systems are derived from a discrete Hamilton–Pontryagin variational principle. From this principle, one can derive a novel class of variational partitioned Runge–Kutta methods on Lie groups. Included among these integrators are generalizations of symplectic Euler and Stormer–Verlet integrators from flat spaces to Lie groups. Because of their variational design, these integrators preserve a discrete momentum map (in the presence of symmetry) and a symplectic form. In a companion paper, we perform a numerical analysis of these methods and report on numerical experiments on the rigid body and chaotic dynamics of an underwater vehicle. The numerics reveal that these variational integrators possess structure-preserving properties that methods designed to preserve momentum (using the coadjoint action of the Lie group) and energy (for example, by projection) lack.

Journal ArticleDOI
TL;DR: In this article, the divergence functions of a metric space estimate the length of a path connecting two points A, B at distance 3, S is a finite set of valuations of a number field K including all infinite valuations, and O s is the corresponding ring of S-integers.
Abstract: Divergence functions of a metric space estimate the length of a path connecting two points A, B at distance 3, S is a finite set of valuations of a number field K including all infinite valuations, and O s is the corresponding ring of S-integers.

Journal ArticleDOI
TL;DR: In this paper, an introductory course on control theory on Lie groups is presented, where controlability and optimal control for left-invariant problems on Lie group are addressed, accompanied by concrete examples.
Abstract: Lecture notes of an introductory course on control theory on Lie groups. Controllability and optimal control for left-invariant problems on Lie groups are addressed. A general theory is accompanied by concrete examples. The course is intended for graduate students; no preliminary knowledge of control theory or Lie groups is assumed.

Journal ArticleDOI
TL;DR: In this article, the authors study viscous hydrodynamics of hot conformal field theory plasma with multiple/non-Abelian symmetries in the framework of AdS/CFT correspondence, using a recently proposed method of directly solving bulk gravity in derivative expansion of local plasma parameters.
Abstract: We study viscous hydrodynamics of hot conformal field theory plasma with multiple/non-Abelian symmetries in the framework of AdS/CFT correspondence, using a recently proposed method of directly solving bulk gravity in derivative expansion of local plasma parameters. Our motivation is to better describe the real QCD plasma produced at RHIC, incorporating its U(1)Nf flavor symmetry as well as SU(2)I non-Abelian iso-spin symmetry. As concrete examples, we choose to study the STU model for multiple U(1)3 symmetries, which is a sub-sector of 5D N=4 gauged SUGRA dual to N=4 Super Yang-Mills theory, capturing Cartan U(1)3 dynamics inside the full R-symmetry. For SU(2), we analyze the minimal 4D N=3 gauged SUGRA whose bosonic action is simply an Einstein-Yang-Mills system, which corresponds to SU(2) R-symmetry dynamics on M2-branes at a Hyper-Kahler cone. By generalizing the bosonic action to arbitrary dimensions and Lie groups, we present our analysis and results for any non-Abelian plasma in arbitrary dimensions.

Journal ArticleDOI
TL;DR: In this article, the Bagger-Lambert action for any Lie 3-algebra is rewritten as a standard Chern-Simons action coupled to matter. But the non-renormalization of the coupling constant comes as a direct consequence of the Lie 3algebra structure underlying the Lie algebra.

Journal ArticleDOI
TL;DR: In this article, the Lie symmetry analysis is performed for the general Burgers' equation and the exact solutions and similarity reductions generated from the symmetry transformations are provided, some new method and techniques are employed simultaneously.


Journal ArticleDOI
TL;DR: In this paper, a family of equations defined on the space of tensor densities of weight on the circle was studied and two integrable PDEs were introduced, one closely related to the inviscid Burgers equation while the other has not been identified in any form before.
Abstract: We study a family of equations defined on the space of tensor densities of weight $\lambda$ 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 $\lambda eq 0,1$, and "shock-peakons" for $\lambda = 3$ 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

Journal ArticleDOI
TL;DR: In this article, the authors investigate the interplay arising between max algebra, convexity and scaling problems, and describe such scalings by means of the max algebraic subeigenvectors and Kleene stars of nonnegative matrices.

Journal ArticleDOI
TL;DR: In this paper, the Ricci curvatures, the scalar curvature and the sectional curvatures were obtained as functions of left invariant metrics on the three-dimensional Lie groups.
Abstract: For each simply connected three-dimensional Lie group we determine the automorphism group, classify the left invariant Riemannian metrics up to automorphism, and study the extent to which curvature can be altered by a change of metric. Thereby we obtain the principal Ricci curvatures, the scalar curvature and the sectional curvatures as functions of left invariant metrics on the three-dimensional Lie groups. Our results improve a bit of Milnor's results of [7] in the three-dimensional case, and Kowalski and Nikvcevic's results [6, Theorems 3.1 and 4.1] (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Journal ArticleDOI
TL;DR: In this article, the character of every finite dimensional simple module over a Lie superalgebra is computed using geometrical methods adapted from the Borel-Weil-Bott theory.
Abstract: In this paper, we use geometrical methods adapted from the Borel-Weil-Bott theory to compute the character of every finite dimensional simple module over a basic classical Lie superalgebra.

Journal ArticleDOI
TL;DR: In this paper, a unification of gravity with Yang-Mills fields based on a simple extension of the Plebanski action to a Lie group G which contains the local Lorentz group is studied.
Abstract: We study a unification of gravity with Yang-Mills fields based on a simple extension of the Plebanski action to a Lie group G which contains the local Lorentz group. The Coleman-Mandula theorem is avoided because the dynamics has no global spacetime symmetry. This may be applied to Lisi's proposal of an E8 unified theory, giving a fully E8 invariant action. The extended form of the Plebanski action suggests a new class of spin foam models.

Journal ArticleDOI
TL;DR: In this paper, the authors studied the subelliptic heat kernel on the Lie group SU(2) and some related functional inequalities and showed that the heat kernel can be computed in a three-dimensional model of a positively curved sub elliptic space.
Abstract: The Lie group SU(2) endowed with its canonical subriemannian structure appears as a three-dimensional model of a positively curved subelliptic space. The goal of this work is to study the subelliptic heat kernel on it and some related functional inequalities.

Journal ArticleDOI
TL;DR: The present paper describes an algorithm to compute averages on matrix Lie groups over the case of averaging over the special orthogonal group of matrices, the unitary group ofMatrices and the group of symmetric positive-definite matrices.
Abstract: Averaging is a common way to alleviate errors and random fluctuations in measurements and to smooth out data. Averaging also provides a way to merge structured data in a smooth manner. The present paper describes an algorithm to compute averages on matrix Lie groups. In particular, we discuss the case of averaging over the special orthogonal group of matrices, the unitary group of matrices and the group of symmetric positive-definite matrices.

Journal ArticleDOI
TL;DR: In this article, an algebraic methodology for designing exactly solvable Lie model Hamiltonians is presented, which consists in looking at the algebra generated by bond operators, and can be applied to solve numerous problems of current interest in the context of topological quantum order.
Abstract: We present an algebraic methodology for designing exactly solvable Lie model Hamiltonians. The idea consists in looking at the algebra generated by bond operators. We illustrate how this method can be applied to solve numerous problems of current interest in the context of topological quantum order. These include Kitaev's well-known toric code and honeycomb models as well as new models: a vector-exchange model and a Clifford $\ensuremath{\gamma}$ model in a triangular lattice.

Book ChapterDOI
01 Jan 2009
TL;DR: In this paper, the authors compare geometries of two different local Lie groups in a Carnot-Caratheodory space, and obtain quantitative estimates of their difference, based on Gromov's Theorem on nilpotentization of vector fields for which they give new and simple proof.
Abstract: We compare geometries of two different local Lie groups in a Carnot-Caratheodory space, and obtain quantitative estimates of their difference. This result is extended to Carnot-Caratheodory spaces with C1,α-smooth basis vector fields, α ∈ [0, 1], and the dependence of the estimates on α is established. From here we obtain the similar estimates for comparing geometries of a Carnot-Caratheodory space and a local Lie group. These results base on Gromov’s Theorem on nilpotentization of vector fields for which we give new and simple proof. All the above imply basic results of the theory: Gromov type Local Approximation Theorems, and for α > 0 Rashevskiǐ-Chow Theorem and Ball-Box Theorem, etc. We apply the obtained results for proving hc-differentiability of mappings of Carnot-Caratheodory spaces with continuous horizontal derivatives. The latter is used in proving the coarea formula for smooth contact mappings of Carnot-Caratheodory spaces, and the area formula for Lipschitz (with respect to sub-Riemannian metrics) mappings of Carnot-Caratheodory spaces.

Journal ArticleDOI
M. Hassan1
TL;DR: In this paper, the Darboux transformation on matrix solutions to the generalized coupled dispersionless integrable system based on a non-Abelian Lie group is studied, and the solutions are shown to be expressed in terms of quasideterminants.
Abstract: The Darboux transformation on matrix solutions to the generalized coupled dispersionless integrable system based on a non-Abelian Lie group is studied, and the solutions are shown to be expressed in terms of quasideterminants. As an explicit example, the Darboux transformation on scalar solutions to the system based on the Lie group SU(2) is discussed in detail, and the solutions are shown to be expressed as ratios of determinants.

Journal ArticleDOI
TL;DR: In this article, the big bounce transition within a flat Friedmann-Robertson-Walker model is analyzed in the setting of loop geometry underlying the loop cosmology, and the authors solve the constraint of the theory at the classical level to identify physical phase space and find the Lie algebra of the Dirac observables.
Abstract: The big bounce (BB) transition within a flat Friedmann-Robertson-Walker model is analyzed in the setting of loop geometry underlying the loop cosmology. We solve the constraint of the theory at the classical level to identify physical phase space and find the Lie algebra of the Dirac observables. We express energy density of matter and geometrical functions in terms of the observables. It is the modification of classical theory by the loop geometry that is responsible for BB. The classical energy scale specific to BB depends on a parameter that should be fixed either by cosmological data or determined theoretically at quantum level, otherwise the energy scale stays unknown.