Showing papers on "Lie group published in 2014"

Human Action Recognition by Representing 3D Skeletons as Points in a Lie Group

Abstract: Recently introduced cost-effective depth sensors coupled with the real-time skeleton estimation algorithm of Shotton et al. [16] have generated a renewed interest in skeleton-based human action recognition. Most of the existing skeleton-based approaches use either the joint locations or the joint angles to represent a human skeleton. In this paper, we propose a new skeletal representation that explicitly models the 3D geometric relationships between various body parts using rotations and translations in 3D space. Since 3D rigid body motions are members of the special Euclidean group SE(3), the proposed skeletal representation lies in the Lie group SE(3)×…×SE(3), which is a curved manifold. Using the proposed representation, human actions can be modeled as curves in this Lie group. Since classification of curves in this Lie group is not an easy task, we map the action curves from the Lie group to its Lie algebra, which is a vector space. We then perform classification using a combination of dynamic time warping, Fourier temporal pyramid representation and linear SVM. Experimental results on three action datasets show that the proposed representation performs better than many existing skeletal representations. The proposed approach also outperforms various state-of-the-art skeleton-based human action recognition approaches.

Cluster algebras

Abstract: Cluster algebras were conceived by Fomin and Zelevinsky (1) in the spring of 2000 as a tool for studying dual canonical bases and total positivity in semisimple Lie groups. However, the theory of cluster algebras has since taken on a life of its own, as connections and applications have been discovered in diverse areas of mathematics, including representation theory of quivers and finite dimensional algebras, cf., for example, refs. 2⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓–15; Poisson geometry (16⇓⇓–19); Teichmuller theory (20⇓⇓⇓–24); string theory (25⇓⇓⇓⇓⇓–31); discrete dynamical systems and integrability (6, 32⇓⇓⇓⇓⇓–38); and combinatorics (39⇓⇓⇓⇓⇓⇓⇓–47). Quite remarkably, cluster algebras provide a unifying algebraic and combinatorial framework for a wide variety of phenomena in these and other settings. We refer the reader to the survey papers (36, 48⇓⇓⇓⇓–53) and to the cluster algebras portal (www.math.lsa.umich.edu/~fomin/cluster.html) for various introductions to cluster algebras and their links with other subjects in mathematics (and physics). In brief, a cluster algebra A of rank k is a subring of an ambient field ℱ of rational functions in k variables, say x 1, …, x k . Unlike most commutative rings, a cluster algebra is not presented at the outset via a complete set of generators and relations. Instead, from the data of the initial seed — which includes the k initial cluster variables x 1, …, x k , plus an exchange matrix — one uses an iterative procedure called “mutation” to produce the rest of the cluster variables. In particular, each new cluster … [↵][1]1To whom correspondence should be addressed. Email: williams{at}math.berkeley.edu. [1]: #xref-corresp-1-1

Box graphs and singular fibers

Abstract: We determine the higher codimension fibers of elliptically fibered Calabi-Yau fourfolds with section by studying the three-dimensional $ \mathcal{N} $ = 2 supersymmetric gauge theory with matter which describes the low energy effective theory of M-theory compactified on the associated Weierstrass model, a singular model of the fourfold. Each phase of the Coulomb branch of this theory corresponds to a particular resolution of the Weierstrass model, and we show that these have a concise description in terms of decorated box graphs based on the representation graph of the matter multiplets, or alternatively by a class of convex paths on said graph. Transitions between phases have a simple interpretation as “flopping” of the path, and in the geometry correspond to actual flop transitions. This description of the phases enables us to enumerate and determine the entire network between them, with various matter representations for all reductive Lie groups. Furthermore, we observe that each network of phases carries the structure of a (quasi-)minuscule representation of a specific Lie algebra. Interpreted from a geometric point of view, this analysis determines the generators of the cone of effective curves as well as the network of flop transitions between crepant resolutions of singular elliptic Calabi-Yau fourfolds. From the box graphs we determine all fiber types in codimensions two and three, and we find new, non-Kodaira, fiber types for E 6, E7 and E 8.

Determination of the Identifiable Parameters in Robot Calibration Based on the POE Formula

Abstract: This paper presents an analytical approach to determine and eliminate the redundant model parameters in serial-robot kinematic calibration based on the product of exponentials formula. According to the transformation principle of the Lie algebra se(3) between different frames, the connection between the joints' twist errors and the links' geometric ones is established. Identifiability analysis shows that the redundant errors are simply equivalent to the commutative elements of the robot's joint twists. Using the Lie bracket operation of se(3), a linear partitioning operator can be constructed to analytically separate the identifiable parameters from the system error vector. Then, error models satisfying the completeness, minimality, and model continuity requirements can be obtained for any serial robot with all combinations and configurations of revolute and prismatic joints. The conventional conclusion that the maximum number of independent parameters is 4r + 2p + 6 in a generic serial robot with r revolute and p prismatic joints is verified. Using the quotient manifold of the Lie group SE(3), the links' geometric errors and the joints' offset errors can be integrated as a whole, such that all these errors can be identified simultaneously. To verify the effectiveness of the proposed method, calibration simulations and experiments are conducted on an industrial six-degree-of-freedom (DoF) serial robot.

Hilbert's Fifth Problem and Related Topics

Abstract: Hilbert's fifth problem: Introduction Lie groups, Lie algebras, and the Baker-Campbell-Hausdorff formula Building Lie structure from representations and metrics Haar measure, the Peter-Weyl theorem, and compact or abelian groups Building metrics on groups, and the Gleason-Yamabe theorem The structure of locally compact groups Ultraproducts as a bridge between hard analysis and soft analysis Models of ultra approximate groups The microscopic structure of approximate groups Applications of the structural theory of approximate groups Related articles: The Jordan-Schur theorem Nilpotent groups and nilprogressions Ado's theorem Associativity of the Baker-Campbell-Hausdorff-Dynkin law Local groups Central extensions of Lie groups, and cocycle averaging The Hilbert-Smith conjecture The Peter-Weyl theorem and nonabelian Fourier analysis Polynomial bounds via nonstandard analysis Loeb measure and the triangle removal lemma Two notes on Lie groups Bibliography Index

Relative quantum field theory

Abstract: We highlight the general notion of a relative quantum field theory, which occurs in several contexts. One is in gauge theory based on a compact Lie algebra, rather than a compact Lie group. This is relevant to the maximal superconformal theory in six dimensions.

Invariant analysis and exact solutions of nonlinear time fractional Sharma–Tasso–Olver equation by Lie group analysis

Abstract: This paper is concerned with the time fractional Sharma–Tasso–Olver (FSTO) equation, Lie point symmetries of the FSTO equation with the Riemann–Liouville derivatives are considered. By using the Lie group analysis method, the invariance properties of the FSTO equation are investigated. In the sense of point symmetry, the vector fields of the FSTO equation are presented. And then, the symmetry reductions are provided. By making use of the obtained Lie point symmetries, it is shown that this equation can transform into a nonlinear ordinary differential equation of fractional order with the new independent variable ξ=xt −α/3. The derivative is an Erdelyi–Kober derivative depending on a parameter α. At last, by means of the sub-equation method, some exact and explicit solutions to the FSTO equation are given.

Heat kernel based decomposition of spaces of distributions in the framework of Dirichlet spaces

Abstract: Classical and nonclassical Besov and Triebel-Lizorkin spaces with complete range of indices are developed in the general setting of Dirichlet space with a doubling measure and local scale-invariant Poincare inequality. This leads to a heat kernel with small time Gaussian bounds and Holder continuity, which play a central role in this article. Frames with band limited elements of sub-exponential space localization are developed, and frame and heat kernel characterizations of Besov and Triebel-Lizorkin spaces are established. This theory, in particular, allows the development of Besov and Triebel-Lizorkin spaces and their frame and heat kernel characterization in the context of Lie groups, Riemannian manifolds, and other settings.

Geometry of Locally Compact Groups of Polynomial Growth and Shape of Large Balls

Abstract: We generalize Pansu’s thesis [27] about asymptotic cones of finitely generated nilpotent groups to arbitrary locally compact groups G of polynomial growth. We show that any such G is weakly commensurable to some simply connected solvable Lie group S, the Lie shadow of G and that balls in any reasonable left invariant metric on G admit a well-defined asymptotic shape. By-products include a formula for the asymptotics of the volume of large balls and an application to ergodic theory, namely that the Ergodic Theorem holds for all ball averages. Along the way we also answer negatively a question of Burago and Margulis [7] on asymptotic word metrics and give a geometric proof of some results of Stoll [33] of the rationality of growth series of Heisenberg groups.

Learning the Irreducible Representations of Commutative Lie Groups

Abstract: We present a new probabilistic model of compact commutative Lie groups that produces invariant-equivariant and disentangled representations of data. To define the notion of disentangling, we borrow a fundamental principle from physics that is used to derive the elementary particles of a system from its symmetries. Our model employs a newfound Bayesian conjugacy relation that enables fully tractable probabilistic inference over compact commutative Lie groups -- a class that includes the groups that describe the rotation and cyclic translation of images. We train the model on pairs of transformed image patches, and show that the learned invariant representation is highly effective for classification.

Morse actions of discrete groups on symmetric space

Abstract: We study the geometry and dynamics of discrete infinite covolume subgroups of higher rank semisimple Lie groups. We introduce and prove the equivalence of several conditions, capturing "rank one behavior'' of discrete subgroups of higher rank Lie groups. They are direct generalizations of rank one equivalents to convex cocompactness. We also prove that our notions are equivalent to the notion of Anosov subgroup, for which we provide a closely related, but simplified and more accessible reformulation, avoiding the geodesic flow of the group. We show moreover that the Anosov condition can be relaxed further by requiring only non-uniform unbounded expansion along the (quasi)geodesics in the group. A substantial part of the paper is devoted to the coarse geometry of these discrete subgroups. A key concept which emerges from our analysis is that of Morse quasigeodesics in higher rank symmetric spaces, generalizing the Morse property for quasigeodesics in Gromov hyperbolic spaces. It leads to the notion of Morse actions of word hyperbolic groups on symmetric spaces,i.e. actions for which the orbit maps are Morse quasiisometric embeddings, and thus provides a coarse geometric characterization for the class of subgroups considered in this paper. A basic result is a local-to-global principle for Morse quasigeodesics and actions. As an application of our techniques we show algorithmic recognizability of Morse actions and construct Morse "Schottky subgroups'' of higher rank semisimple Lie groups via arguments not based on Tits' ping-pong. Our argument is purely geometric and proceeds by constructing equivariant Morse quasiisometric embeddings of trees into higher rank symmetric spaces.

The Farrell-Jones Conjecture for cocompact lattices in virtually connected Lie groups

Abstract: 1.1. Motivation and summary. The algebraic K-theory and L-theory of group rings has gained a lot of attention in the last decades, in particular since they play a prominent role in the classification of manifolds. Computations are very hard and here the Farrell-Jones Conjecture comes into play. It identifies the algebraic K-theory and L-theory of group rings with the evaluation of an equivariant homology theory on the classifying space for the family of virtually cyclic subgroups. This is the analogue of classical results in the representation theory of finite groups such as the induction theorem of Artin or Brauer, where the value of a functor for finite groups is computed in terms of its values on a smaller family, for instance of cyclic or hyperelementary subgroups; in the Farrell-Jones setting the reduction is to virtually cyclic groups. The point is that this equivariant homology theory is much more accessible than the algebraic Kand L-groups themselves. Actually, most of all computations for infinite groups in the literature use the Farrell-Jones Conjecture and concentrate on the equivariant homology side. The Farrell-Jones Conjecture is not only important for calculations, but also gives structural insight, since the isomorphism occurring in its formulation has also geometric interpretations. This has the consequence that the Farrell-Jones Conjecture implies a variety of other well-known conjectures such as the ones due to Bass, Borel, Kaplansky, Novikov, and Serre which concern character theory for infinite groups, algebraic topology, the classification of manifolds, the ring structure of group rings, and group theory. We will discuss them in more detail in Subsection 2.2. The main result of this paper is to prove the Farrell-Jones Conjecture for a new prominent classes of groups, namely, cocompact lattices in almost connected Lie groups. We mention that the operator theoretic analog of the Farrell-Jones Conjecture, the Baum-Connes Conjecture, is known only for a few groups in this class. With the exception of the Novikov Conjecture, the conjectures listed above have not been known for this class so that our result presents also new contributions to them. Since we address a general version of the Farrell-Jones Conjectures, where one allows coefficients in additive categories, very powerful inheritance properties are valid which we will describe in Subsection 2.3. For instance, if this general version of the Farrell-Jones Conjecture holds for a group, it holds automatically for

Geometric quantization for proper moment maps: the Vergne conjecture

Abstract: We establish an analytic interpretation for the index of certain transversally elliptic symbols on non-compact manifolds. By using this interpretation, we establish a geometric quantization formula for a Hamiltonian action of a compact Lie group acting on a non-compact symplectic manifold with proper moment map. In particular, we present a solution to a conjecture of Michele Vergne in her ICM 2006 plenary lecture.

A new AdS 4/CFT 3 dual with extended SUSY and a spectral flow

Abstract: We construct a new AdS 4 background in Type IIB supergravity by means of a non-Abelian T-duality transformation on the Type IIA dual of ABJM. The analysis of probe and particle-like branes suggests a dual CFT in which each of the gauge groups is doubled. A common feature of non-Abelian T-duality is that in the absence of any global information coming from String Theory it gives rise to non-compact dual backgrounds, with coordinates living in the Lie algebra of the Lie group involved in the dualization. In backgrounds with CFT duals this poses obvious problems to the CFTs. In this paper we show that for the new AdS 4 background the gauge groups of the associated dual CFT undergo a spectral flow as the non-compact internal direction runs from 0 to infinity, which resembles Seiberg duality in $$ \mathcal{N} $$ = 1. This phenomenon, very reminiscent of the cascade, provides an interpretation in the CFT for the running of the non-compact coordinate, and suggests that at the end of the flow the extra charges disappear and the dual CFT is described by a 2-node quiver very similar to ABJM, albeit with reduced supersymmetry.

Koszul Information Geometry and Souriau Geometric Temperature/Capacity of Lie Group Thermodynamics

Abstract: The Francois Massieu 1869 idea to derive some mechanical and thermal properties of physical systems from “Characteristic Functions”, was developed by Gibbs and Duhem in thermodynamics with the concept of potentials, and introduced by Poincare in probability. This paper deals with generalization of this Characteristic Function concept by Jean-Louis Koszul in Mathematics and by Jean-Marie Souriau in Statistical Physics. The Koszul-Vinberg Characteristic Function (KVCF) on convex cones will be presented as cornerstone of “Information Geometry” theory, defining Koszul Entropy as Legendre transform of minus the logarithm of KVCF, and Fisher Information Metrics as hessian of these dual functions, invariant by their automorphisms. In parallel, Souriau has extended the Characteristic Function in Statistical Physics looking for other kinds of invariances through co-adjoint action of a group on its momentum space, defining physical observables like energy, heat and momentum as pure geometrical objects. In covariant Souriau model, Gibbs equilibriums states are indexed by a geometric parameter, the Geometric (Planck) Temperature, with values in the Lie algebra of the dynamical Galileo/Poincare groups, interpreted as a space-time vector, giving to the metric tensor a null Lie derivative. Fisher Information metric appears as the opposite of the derivative of Mean “Moment map” by geometric temperature, equivalent to a Geometric Capacity or Specific Heat. We will synthetize the analogies between both Koszul and Souriau models, and will reduce their definitions to the exclusive Cartan “Inner Product”. Interpreting Legendre transform as Fourier transform in (Min,+) algebra, we conclude with a definition of Entropy given by a relation mixing Fourier/Laplace transforms: Entropy = (minus) Fourier(Min,+) o Log o Laplace(+,X).

Eigenvalue Distributions of Reduced Density Matrices

Abstract: Given a random quantum state of multiple distinguishable or indistinguishable particles, we provide an effective method, rooted in symplectic geometry, to compute the joint probability distribution of the eigenvalues of its one-body reduced density matrices. As a corollary, by taking the distribution’s support, which is a convex moment polytope, we recover a complete solution to the one-body quantum marginal problem. We obtain the probability distribution by reducing to the corresponding distribution of diagonal entries (i.e., to the quantitative version of a classical marginal problem), which is then determined algorithmically. This reduction applies more generally to symplectic geometry, relating invariant measures for the coadjoint action of a compact Lie group to their projections onto a Cartan subalgebra, and can also be quantized to provide an efficient algorithm for computing bounded height Kronecker and plethysm coefficients.

Multiple Hamiltonian Structures for Toda-type systems

Abstract: Results on the finite nonperiodic Toda lattice are extended to some generalizations of the system: The relativistic Toda lattice, the generalized Toda lattice associated with simple Lie groups and the full Kostant-Toda lattice. The areas investigated, include master symmetries, recursion operators, higher Poisson brackets, invariants, and group symmetries for the systems. A survey of previous work on the classical Toda lattice is also included.

Gauge theory and mirror symmetry

Abstract: Author(s): Teleman, C | Abstract: Outlined here is a description of equivariance in the world of 2-dimensional extended topological quantum field theories, under a topological action of compact Lie groups. In physics language, I am gauging the theories-coupling them to a principal bundle on the surface world-sheet. I describe the data needed to gauge the theory, as well as the computation of the gauged theory, the result of integrating over all bundles. The relevant theories are 'A-models', such as arise from the Gromov-Witten theory of a symplectic manifold with Hamiltonian group action, and the mathematical description starts with a group action on the generating category (the Fukaya category, in this example) which is factored through the topology of the group. Their mirror description involves holomorphic symplectic manifolds and Lagrangians related to the Langlands dual group. An application recovers the complex mirrors of flag varieties proposed by Rietsch.

Stochastic Euler-Poincaré reduction

Abstract: We prove a Euler-Poincare reduction theorem for stochastic processes taking values on a Lie group, which is a generalization of the reduction argument for the deterministic case [J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, (Texts in Applied Mathematics). (Springer, 2003)]. We also show examples of its application to SO(3) and to the group of diffeomorphisms, which includes the Navier-Stokes equation on a bounded domain and the Camassa-Holm equation.

A new AdS_4/CFT_3 Dual with Extended SUSY and a Spectral Flow

Abstract: We construct a new AdS_4 background in Type IIB supergravity by means of a non-Abelian T-duality transformation on the Type IIA dual of ABJM. The analysis of probe and particle-like branes suggests a dual CFT in which each of the gauge groups is doubled. A common feature of non-Abelian T-duality is that in the absence of any global information coming from String Theory it gives rise to non-compact dual backgrounds, with coordinates living in the Lie algebra of the Lie group involved in the dualization. In backgrounds with CFT duals this poses obvious problems to the CFTs. In this paper we show that for the new AdS_4 background the gauge groups of the associated dual CFT undergo a spectral flow as the non-compact internal direction runs from 0 to infinity, which resembles Seiberg duality in N=1. This phenomenon, very reminiscent of the cascade, provides an interpretation in the CFT for the running of the non-compact coordinate, and suggests that at the end of the flow the extra charges disappear and the dual CFT is described by a 2-node quiver very similar to ABJM, albeit with reduced supersymmetry.

Gevrey functions and ultradistributions on compact Lie groups and homogeneous spaces

Abstract: In this paper we give global characterisations of Gevrey–Roumieu and Gevrey–Beurling spaces of ultradifferentiable functions on compact Lie groups in terms of the representation theory of the group and the spectrum of the Laplace–Beltrami operator. Furthermore, we characterise their duals, the spaces of corresponding ultradistributions. For the latter, the proof is based on first obtaining the characterisation of their α-duals in the sense of Kothe and the theory of sequence spaces. We also give the corresponding characterisations on compact homogeneous spaces.

Probability on Compact Lie Groups

Abstract: Introduction.- 1.Lie Groups.- 2.Representations, Peter-Weyl Theory and Weights.- 3.Analysis on Compact Lie Groups.- 4.Probability Measures on Compact Lie Groups.- 5.Convolution Semigroups of Measures.- 6.Deconvolution Density Estimation.- Appendices.- Index.- Bibliography.