Showing papers on "Lie group published in 2012"

Group Invariant Scattering

Abstract: This paper constructs translation-invariant operators on L 2 .R d /, which are Lipschitz-continuous to the action of diffeomorphisms. A scattering propagator is a path-ordered product of nonlinear and noncommuting operators, each of which computes the modulus of a wavelet transform. A local integration defines a windowed scattering transform, which is proved to be Lipschitz-continuous to the action of C 2 diffeomorphisms. As the window size increases, it converges to a wavelet scattering transform that is translation invariant. Scattering coefficients also provide representations of stationary processes. Expected values depend upon high-order moments and can discriminate processes having the same power spectrum. Scattering operators are extended on L 2 .G/, where G is a compact Lie group, and are invariant under the action of G. Combining a scattering on L 2 .R d / and on L 2 .SO.d// defines a translation- and rotation-invariant scattering on L 2 .R d /. © 2012 Wiley Periodicals, Inc.

Seiberg-Witten geometry of four dimensional N=2 quiver gauge theories

Abstract: Seiberg-Witten geometry of mass deformed N=2 superconformal ADE quiver gauge theories in four dimensions is determined. We solve the limit shape equations derived from the gauge theory and identify the space M of vacua of the theory with the moduli space of the genus zero holomorphic (quasi)maps to the moduli space of holomorphic G-bundles on a (possibly degenerate) elliptic curve defined in terms of the microscopic gauge couplings, for the corresponding simple ADE Lie group G. The integrable systems underlying, or, rather, overlooking the special geometry of M are identified. The moduli spaces of framed G-instantons on R^2xT^2, of G-monopoles with singularities on R^2xS^1, the Hitchin systems on curves with punctures, as well as various spin chains play an important role in our story. We also comment on the higher dimensional theories. In the companion paper the quantum integrable systems and their connections to the representation theory of quantum affine algebras will be discussed

Stochastic Models, Information Theory, and Lie Groups, Volume 2

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The structure of approximate groups

Abstract: Let K⩾1 be a parameter. A K-approximate group is a finite set A in a (local) group which contains the identity, is symmetric, and such that A⋅A is covered by K left translates of A. The main result of this paper is a qualitative description of approximate groups as being essentially finite-by-nilpotent, answering a conjecture of H. Helfgott and E. Lindenstrauss. This may be viewed as a generalisation of the Freiman-Ruzsa theorem on sets of small doubling in the integers to arbitrary groups. We begin by establishing a correspondence principle between approximate groups and locally compact (local) groups that allows us to recover many results recently established in a fundamental paper of Hrushovski. In particular we establish that approximate groups can be approximately modeled by Lie groups. To prove our main theorem we apply some additional arguments essentially due to Gleason. These arose in the solution of Hilbert’s fifth problem in the 1950s. Applications of our main theorem include a finitary refinement of Gromov’s theorem, as well as a generalized Margulis lemma conjectured by Gromov and a result on the virtual nilpotence of the fundamental group of Ricci almost nonnegatively curved manifolds.

Čech cocycles for differential characteristic classes: an ∞-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 $\infty$-groups, i.e., by smooth groupal $A_\infty$- spaces. Namely, we realize differential characteristic classes as morphisms from $\infty$-groupoids of smooth principal $\infty$-bundles with connections to $\infty$-groupoids of higher $U(1)$-gerbes with connections. This allows us to study the homotopy fibres 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.

On the growth of $L^2$-invariants for sequences of lattices in Lie groups

Abstract: We study the asymptotic behaviour of Betti numbers, twisted torsion and other spectral invariants of sequences of locally symmetric spaces. Our main results are uniform versions of the DeGeorge--Wallach Theorem, of a theorem of Delorme and various other limit multiplicity theorems. A basic idea is to adapt the notion of Benjamini--Schramm convergence (BS-convergence), originally introduced for sequences of finite graphs of bounded degree, to sequences of Riemannian manifolds, and analyze the possible limits. We show that BS-convergence of locally symmetric spaces implies convergence, in an appropriate sense, of the associated normalized relative Plancherel measures. This then yields convergence of normalized multiplicities of unitary representations, Betti numbers and other spectral invariants. On the other hand, when the corresponding Lie group $G$ is simple and of real rank at least two, we prove that there is only one possible BS-limit, i.e. when the volume tends to infinity, locally symmetric spaces always BS-converge to their universal cover $G/K$. This leads to various general uniform results. When restricting to arbitrary sequences of congruence covers of a fixed arithmetic manifold we prove a strong quantitative version of BS-convergence which in turn implies upper estimates on the rate of convergence of normalized Betti numbers in the spirit of Sarnak--Xue. An important role in our approach is played by the notion of Invariant Random Subgroups. For higher rank simple Lie groups $G$, we exploit rigidity theory, and in particular the Nevo--Stuck--Zimmer theorem and Kazhdan's property (T), to obtain a complete understanding of the space of IRSs of $G$.

Homogeneous Finsler Spaces

Abstract: In this chapter we study homogeneous Finsler spaces. In Sect. 4.1, we define the notions of Minkowski Lie pairs and Minkowski Lie algebras to give an algebraic description of invariant Finsler metrics on homogeneous manifolds and bi-invariant Finsler metrics on Lie groups. Then in Sect. 4.2, we present a sufficient and necessary condition for a coset space to have invariant non-Riemannian Finsler metrics. In Sect. 4.3, we study homogeneous Finsler spaces of negative curvature and prove that every homogeneous Finsler space with nonpositive flag curvature and negative Ricci scalar must be simply connected. In Sect. 4.4, we apply our result to study the degree of symmetry of closed manifolds. In particular, we prove that if a closed manifold is not diffeomorphic to a rank-one Riemannian symmetric space, then its degree of symmetry can be realized by a non-Riemannian Finsler metric. Finally, in Sect. 4.5, we study fourth-root homogeneous Finsler metrics. As an explicit example, we give a classification of all invariant fourth-root Finsler metrics on Grassmannian manifolds.

Sub-Riemannian structures on 3D lie groups

Abstract: We give a complete classification of left-invariant sub-Riemannian structures on three-dimensional Lie groups in terms of the basic differential invariants. As a consequence, we explicitly find a sub-Riemannian isometry between the nonisomorphic Lie groups SL(2) and A +( $ \mathbb{R} $ )?×?S 1, where A +( $ \mathbb{R} $ ) denotes the group of orientation preserving affine maps on the real line.

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.

Lie bodies: a manifold representation of 3D human shape

Abstract: Three-dimensional object shape is commonly represented in terms of deformations of a triangular mesh from an exemplar shape. Existing models, however, are based on a Euclidean representation of shape deformations. In contrast, we argue that shape has a manifold structure: For example, summing the shape deformations for two people does not necessarily yield a deformation corresponding to a valid human shape, nor does the Euclidean difference of these two deformations provide a meaningful measure of shape dissimilarity. Consequently, we define a novel manifold for shape representation, with emphasis on body shapes, using a new Lie group of deformations. This has several advantages. First we define triangle deformations exactly, removing non-physical deformations and redundant degrees of freedom common to previous methods. Second, the Riemannian structure of Lie Bodies enables a more meaningful definition of body shape similarity by measuring distance between bodies on the manifold of body shape deformations. Third, the group structure allows the valid composition of deformations. This is important for models that factor body shape deformations into multiple causes or represent shape as a linear combination of basis shapes. Finally, body shape variation is modeled using statistics on manifolds. Instead of modeling Euclidean shape variation with Principal Component Analysis we capture shape variation on the manifold using Principal Geodesic Analysis. Our experiments show consistent visual and quantitative advantages of Lie Bodies over traditional Euclidean models of shape deformation and our representation can be easily incorporated into existing methods.

Invariant Higher-Order Variational Problems

Abstract: We investigate higher-order geometric k-splines for template matching on Lie groups. This is motivated by the need to apply diffeomorphic template matching to a series of images, e.g., in longitudinal studies of Computational Anatomy. Our approach formulates Euler-Poincare theory in higher-order tangent spaces on Lie groups. In particular, we develop the Euler-Poincare formalism for higher-order variational problems that are invariant under Lie group transformations. The theory is then applied to higher-order template matching and the corresponding curves on the Lie group of transformations are shown to satisfy higher-order Euler-Poincare equations. The example of SO(3) for template matching on the sphere is presented explicitly. Various cotangent bundle momentum maps emerge naturally that help organize the formulas. We also present Hamiltonian and Hamilton-Ostrogradsky Lie-Poisson formulations of the higher-order Euler-Poincare theory for applications on the Hamiltonian side.

Infinitely many N=2 SCFT with ADE flavor symmetry

Abstract: We present evidence that for each ADE Lie group G there is an infinite tower of 4D N=2 SCFTs, which we label as D(G,s) (with s a positive integer), having (at least) flavor symmetry G. For G=SU(2), D(SU(2),s) coincides with the Argyres--Douglas model of type D_{s+1}, while for larger flavor groups the models are new (but for a few previously known examples). When its flavor symmetry G is gauged, D(G,s) contributes to the Yang-Mills beta-function as s/[2(s+1)] adjoint hypermultiplets. The argument is based on a combination of Type IIB geometric engineering and the categorical deconstruction of arXiv:1203.6743. One first engineers a class of N=2 models which, trough the analysis of their category of quiver representations, are identified as asymptotically-free gauge theories with gauge group G coupled to some conformal matter system. Taking the limit g->0 one isolates the matter SCFT which is our D(G,s).

Dual pairing of symmetry and dynamical groups in physics

Abstract: This article reviews many manifestations and applications of dual representations of pairs of groups, primarily in atomic and nuclear physics. Examples are given to show how such paired representations are powerful aids in understanding the dynamics associated with shell-model coupling schemes and in identifying the physical situations for which a given scheme is most appropriate. In particular, they suggest model Hamiltonians that are diagonal in the various coupling schemes. The dual pairing of group representations has been applied profitably in mathematics to the study of invariant theory. We show that parallel applications to the theory of symmetry and dynamical groups in physics are equally valuable. In particular, the pairing of the representations of a discrete group with those of a continuous Lie group or those of a compact Lie with those of a non-compact Lie group makes it possible to infer many properties of difficult groups from those of simpler groups. This review starts with the representations of the symmetric and unitary groups, which are used extensively in the many-particle quantum mechanics of bosonic and fermionic systems. It gives a summary of the many solutions and computational techniques for solving problems that arise in applications of symmetry methods in physics and which result from the famous Schur-Weyl duality theorem for the pairing of these representations. It continues to examine many chains of symmetry groups and dual chains of dynamical groups associated with the several coupling schemes in atomic and nuclear shell models and the valuable insights and applications that result from this examination.

Counting lattice points

Abstract: For a locally compact second countable group G and a lattice subgroup Γ, we give an explicit quantitative solution of the lattice point counting problem in general domains in G, provided that i) G has finite upper local dimension, and the domains satisfy a basic regularity condition, ii) the mean ergodic theorem for the action of G on G/Γ holds, with a rate of convergence. The error term we establish matches the best current result for balls in symmetric spaces of simple higher-rank Lie groups, but holds in much greater generality. A significant advantage of the ergodic theoretic approach we use is that the solution to the lattice point counting problem is uniform over families of lattice subgroups provided they admit a uniform spectral gap. In particular, the uniformity property holds for families of finite index subgroups satisfying a quantitative variant of property τ . We discuss a number of applications, including: counting lattice points in general domains in semisimple S-algebraic groups, counting rational points on group varieties with respect to a height function, and quantitative angular (or conical) equidistribution of lattice points in symmetric spaces and in affine symmetric varieties. We note that the mean ergodic theorems which we establish are based on spectral methods, including the spectral transfer principle and the Kunze-Stein phenomenon. We formulate and prove appropriate analogues of both of these results in the set-up of adele groups, and they constitute a necessary step in our proof of quantitative results in counting rational points.

Heat kernel generated frames in the setting of Dirichlet spaces

Abstract: Wavelet bases and frames consisting of band limited functions of nearly exponential localization on Rd are a powerful tool in harmonic analysis by making various spaces of functions and distributions more accessible for study and utilization, and providing sparse representation of natural function spaces (e.g. Besov spaces) on Rd. Such frames are also available on the sphere and in more general homogeneous spaces, on the interval and ball. The purpose of this article is to develop band limited well-localized frames in the general setting of Dirichlet spaces with doubling measure and a local scale-invariant Poincar\'e inequality which lead to heat kernels with small time Gaussian bounds and H\"older continuity. As an application of this construction, band limited frames are developed in the context of Lie groups or homogeneous spaces with polynomial volume growth, complete Riemannian manifolds with Ricci curvature bounded from below and satisfying the volume doubling property, and other settings. The new frames are used for decomposition of Besov spaces in this general setting.

Symmetry preserving parameterization schemes

Abstract: Methods for the design of physical parameterization schemes that possess certain invariance properties are discussed. These methods are based on different techniques of group classification and provide means to determine expressions for unclosed terms arising in the course of averaging of nonlinear differential equations. The demand that the averaged equation is invariant with respect to a subalgebra of the maximal Lie invariance algebra of the unaveraged equation leads to a problem of inverse group classification which is solved by the description of differential invariants of the selected subalgebra. Given no prescribed symmetry group, the direct group classification problem is relevant. Within this framework, the algebraic method or direct integration of determining equations for Lie symmetries can be applied. For cumbersome parameterizations, a preliminary group classification can be carried out. The methods presented are exemplified by parameterizing the eddy vorticity flux in the averaged vorticity ...

Entanglement entropy of highly degenerate States and fractal dimensions.

Abstract: We consider the bipartite entanglement entropy of ground states of extended quantum systems with a large degeneracy. Often, as when there is a spontaneously broken global Lie group symmetry, basis elements of the lowest-energy space form a natural geometrical structure. For instance, the spins of a spin-1/2 representation, pointing in various directions, form a sphere. We show that for subsystems with a large number m of local degrees of freedom, the entanglement entropy diverges as d/2log⁡m, where d is the fractal dimension of the subset of basis elements with nonzero coefficients. We interpret this result by seeing d as the (not necessarily integer) number of zero-energy Goldstone bosons describing the ground state. We suggest that this result holds quite generally for largely degenerate ground states, with potential applications to spin glasses and quenched disorder.

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

Abstract: Classical and non classical 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 Poincar\'e inequality. This leads to Heat kernel with small time Gaussian bounds and H\"older 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 to develop Besov and Triebel-Lizorkin spaces and their frame and heat kernel characterization in the context of Lie groups, Riemannian manifold, and other settings.

A new type of non-Noether exact invariants and adiabatic invariants of generalized Hamiltonian systems

Abstract: For a generalized Hamiltonian system with the action of small forces of perturbation, the Lie symmetries, symmetrical perturbation, and adiabatic invariants is presented. Based on the invariance of equations of motion for the system under general infinitesimal transformation of Lie groups, the Lie symmetrical determining equations, and exact invariants of the system are given. Then the determining equations of Lie symmetrical perturbation and adiabatic invariants of the disturbed systems are obtained. Furthermore, in the special infinitesimal transformations, two deductions are given. At the end of the paper, one example is given to illustrate the application of the method and result.

Twisted K-Homology and Group-Valued Moment Maps

Abstract: Let G be a compact, simply connected Lie group. We develop a ‘quantization functor’ from prequantized quasi-Hamiltonian G-spaces (M,ω,Φ) at level k to the fusion ring (Verlinde algebra) R k (G). The quantization is defined as a push-forward in twisted equivariant K-homology. It may be computed by a fixed point formula, similar to the equivariant index theorem for Spin c -Dirac operators. Using the formula, we calculate in several examples.

Invariant Higher-Order Variational Problems II

Abstract: Motivated by applications in computational anatomy, we consider a second-order problem in the calculus of variations on object manifolds that are acted upon by Lie groups of smooth invertible transformations. This problem leads to solution curves known as Riemannian cubics on object manifolds that are endowed with normal metrics. The prime examples of such object manifolds are the symmetric spaces. We characterize the class of cubics on object manifolds that can be lifted horizontally to cubics on the group of transformations. Conversely, we show that certain types of non-horizontal geodesic on the group of transformations project to cubics. Finally, we apply second-order Lagrange–Poincare reduction to the problem of Riemannian cubics on the group of transformations. This leads to a reduced form of the equations that reveals the obstruction for the projection of a cubic on a transformation group to again be a cubic on its object manifold.

Polynomial Regression on Riemannian Manifolds

Abstract: In this paper we develop the theory of parametric polynomial regression in Riemannian manifolds and Lie groups. We show application of Riemannian polynomial regression to shape analysis in Kendall shape space. Results are presented, showing the power of polynomial regression on the classic rat skull growth data of Bookstein as well as the analysis of the shape changes associated with aging of the corpus callosum from the OASIS Alzheimer's study.

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.

Einstein metrics on compact Lie groups which are not naturally reductive

Abstract: The study of left-invariant Einstein metrics on compact Lie groups which are naturally reductive was initiated by D’Atri and Ziller (Mem Am Math Soc 18, (215) 1979). In 1996 the second author obtained non-naturally reductive Einstein metrics on the Lie group SU(n) for n ≥ 6, by using a method of Riemannian submersions. In the present work we prove existence of non-naturally reductive Einstein metrics on the compact simple Lie groups SO(n) (n ≥ 11), Sp(n) (n ≥ 3), E6, E7, and E8.

The cascade of orthogonal roots and the coadjoint structure of the nilradical of a Borel subgroup of a semisimple Lie group

Abstract: Let $G$ be a semisimple Lie group and let $\g = _- +\hh + $ be a triangular decomposition of $\g= \hbox{Lie}\,G$. Let $\b =\hh + $ and let $H,N,B$ be Lie subgroups of $G$ corresponding respectively to $\hh, $ and $\b$. We may identify $ _-$ with the dual space to $ $. The coadjoint action of $N$ on $ _-$ extends to an action of $B$ on $ _-$. There exists a unique nonempty Zariski open orbit $X$ of $B$ on $ _-$. Any $N$-orbit in $X$ is a maximal coadjoint orbit of $N$ in $ _-$. The cascade of orthogonal roots defines a cross-section $\r_-^{\times}$ of the set of such orbits leading to a decomposition $$X = N/R\times \r_-^{\times}.$$ This decomposition, among other things, establishes the structure of $S( )^{ }$ as a polynomial ring generated by the prime polynomials of $H$-weight vectors in $S( )^{ }$. It also leads tothe multiplicity 1 of $H$ weights in $S( )^{ }$.