Showing papers on "Lie group published in 2015"

Introduction to Arithmetic Groups

Abstract: This book provides a gentle introduction to the study of arithmetic subgroups of semisimple Lie groups. This means that the goal is to understand the group SL(n,Z) and certain of its subgroups. Among the major results discussed in the later chapters are the Mostow Rigidity Theorem, the Margulis Superrigidity Theorem, Ratner's Theorems, and the classification of arithmetic subgroups of classical groups. As background for the proofs of these theorems, the book provides primers on lattice subgroups, arithmetic groups, real rank and Q-rank, ergodic theory, unitary representations, amenability, Kazhdan's property (T), and quasi-isometries. Numerous exercises enhance the book's usefulness both as a textbook for a second-year graduate course and for self-study. In addition, notes at the end of each chapter have suggestions for further reading. (Proofs in this book often consider only an illuminating special case.) Readers are expected to have some acquaintance with Lie groups, but appendices briefly review the prerequisite background.

Hořava-Lifshitz gravity from dynamical Newton-Cartan geometry

Abstract: Recently it has been established that torsional Newton-Cartan (TNC) geometry is the appropriate geometrical framework to which non-relativistic field theories couple. We show that when these geometries are made dynamical they give rise to Hořava-Lifshitz (HL) gravity. Projectable HL gravity corresponds to dynamical Newton-Cartan (NC) geometry without torsion and non-projectable HL gravity corresponds to dynamical NC geometry with twistless torsion (hypersurface orthogonal foliation). We build a precise dictionary relating all fields (including the scalar khronon), their transformations and other properties in both HL gravity and dynamical TNC geometry. We use TNC invariance to construct the effective action for dynamical twistless torsional Newton-Cartan geometries in 2+1 dimensions for dynamical exponent 1 < z ≤ 2 and demonstrate that this exactly agrees with the most general forms of the HL actions constructed in the literature. Further, we identify the origin of the U(1) symmetry observed by Hořava and Melby-Thompson as coming from the Bargmann extension of the local Galilean algebra that acts on the tangent space to TNC geometries. We argue that TNC geometry, which is manifestly diffeomorphism covariant, is a natural geometrical framework underlying HL gravity and discuss some of its implications.

Conservation laws for time-fractional subdiffusion and diffusion-wave equations

Abstract: A new technique for constructing conservation laws for fractional differential equations not having a Lagrangian is proposed. The technique is based on the methods of Lie group analysis and employs the concept of nonlinear self-adjointness which is enhanced to the certain class of fractional evolution equations. The proposed approach is demonstrated on subdiffusion and diffusion-wave equations with the Riemann–Liouville and Caputo time-fractional derivatives. It is shown that these equations are nonlinearly self-adjoint, and therefore, desired conservation laws can be calculated using the appropriate formal Lagrangians. The explicit forms of fractional generalizations of the Noether operators are also proposed for the equations with the Riemann–Liouville and Caputo time-fractional derivatives of order $$\alpha \in (0,2)$$ . Using these operators and formal Lagrangians, new conservation laws are constructed for the linear and nonlinear time-fractional subdiffusion and diffusion-wave equations by their Lie point symmetries.

Lie Groups and Geometric Aspects of Isometric Actions

Abstract: 1: Basic results on Lie groups.- 2: Lie groups with bi-invariant metrics.- 3: Proper and isometric acions.- 4: Adjoint and conjugation actions.- 5: Polar foliations.- 6: Low cohomogeneity actions and positive curvature.- Appendix: Rudiments of smooth manifolds.

Continuous-Discrete Extended Kalman Filter on Matrix Lie Groups Using Concentrated Gaussian Distributions

Abstract: In this paper we generalize the continuous-discrete extended Kalman filter (CD-EKF) to the case where the state and the observations evolve on connected unimodular matrix Lie groups. We propose a new assumed density filter called continuous-discrete extended Kalman filter on Lie groups (CD-LG-EKF). It is built upon a geometrically meaningful modeling of the concentrated Gaussian distribution on Lie groups. Such a distribution is parametrized by a mean and a covariance matrix defined on the Lie group and in its associated Lie algebra respectively. Our formalism yields tractable equations for both non-linear continuous time propagation and discrete update of the distribution parameters under the assumption that the posterior distribution of the state is a concentrated Gaussian. As a side effect, we contribute to the derivation of the first and second order differential of the matrix Lie group logarithm using left connection. We also show that the CD-LG-EKF reduces to the usual CD-EKF if the state and the observations evolve on Euclidean spaces. Our approach leads to a systematic methodology for the design of filters, which is illustrated by the application to a camera pose filtering problem with observations on Lie group. In this application, the CD-LG-EKF significantly outperforms two constrained non-linear filters (one based on a linearization technique and the other on the unscented transform) applied on the embedding space of the Lie group.

Anosov representations and proper actions

Abstract: We establish several characterizations of Anosov representations of word hyperbolic groups into real reductive Lie groups, in terms of a Cartan projection or Lyapunov projection of the Lie group. Using a properness criterion of Benoist and Kobayashi, we derive applications to proper actions on homogeneous spaces of reductive groups.

An Abstract Nash–Moser Theorem and Quasi-Periodic Solutions for NLW and NLS on Compact Lie Groups and Homogeneous Manifolds

Abstract: We prove an abstract implicit function theorem with parameters for smooth operators defined on scales of sequence spaces, modeled for the search of quasi-periodic solutions of PDEs. The tame estimates required for the inverse linearised operators at each step of the iterative scheme are deduced via a multiscale inductive argument. The Cantor-like set of parameters where the solution exists is defined in a non inductive way. This formulation completely decouples the iterative scheme from the measure theoretical analysis of the parameters where the small divisors non-resonance conditions are verified. As an application, we deduce the existence of quasi-periodic solutions for forced NLW and NLS equations on any compact Lie group or manifold which is homogeneous with respect to a compact Lie group, extending previous results valid only for tori. A basic tool of harmonic analysis is the highest weight theory for the irreducible representations of compact Lie groups.

Symmetry broken and restored coupled-cluster theory: I. Rotational symmetry and angular momentum

Abstract: We extend coupled-cluster (CC) theory performed on top of a Slater determinant breaking rotational symmetry to allow for the exact restoration of the angular momentum at any truncation order. The main objective relates to the description of near-degenerate finite quantum systems with an open-shell character. As such, the newly developed many-body formalism offers a wealth of potential applications and further extensions dedicated to the ab initio description of, e.g., doubly open-shell atomic nuclei and molecule dissociation. The formalism, which encompasses both single-reference CC theory and projected Hartree–Fock theory as particular cases, permits the computation of usual sets of connected diagrams while consistently incorporating static correlations through the highly non-perturbative restoration of rotational symmetry. Interestingly, the yrast spectroscopy of the system, i.e. the lowest energy associated with each angular momentum, is accessed within a single calculation. A key difficulty presently overcome relates to the necessity to handle generalized energy and norm kernels for which naturally terminating CC expansions could be eventually obtained. The present work focuses on SU(2) but can be extended to any (locally) compact Lie group and to discrete groups, such as most point groups. In particular, the formalism will be soon generalized to U(1) symmetry associated with particle number conservation. This is relevant to Bogoliubov CC theory that was recently applied to singly open-shell nuclei.

A family of wave-breaking equations generalizing the Camassa-Holm and Novikov equations

Abstract: A 4-parameter polynomial family of equations generalizing the Camassa-Holm and Novikov equations that describe breaking waves is introduced. A classification of low-order conservation laws, peaked travelling wave solutions, and Lie symmetries is presented for this family. These classifications pick out a 1-parameter equation that has several interesting features: it reduces to the Camassa-Holm and Novikov equations when the polynomial has degree two and three; it has a conserved H1 norm and it possesses N-peakon solutions when the polynomial has any degree; and it exhibits wave-breaking for certain solutions describing collisions between peakons and anti-peakons in the case N = 2.

L p Fourier multipliers on compact Lie groups

Abstract: In this paper we prove L p Fourier multiplier theorems for invariant and also non- invariantoperatorsoncompactLiegroupsinthespiritofthewell-knownHormander-Mikhlin theorem on R n and its variants on tori T n We also give applications to a-priori estimates for non-hypoelliptic operators Already in the case of tori we get an interesting refinement of the classical multiplier theorem

Fractional Lie group method of the time-fractional Boussinesq equation

Abstract: Finding the symmetries of the nonlinear fractional differential equations is a topic which has many applications in various fields of science and engineering. In this manuscript, firstly, we are interested in finding the Lie point symmetries of the time-fractional Boussinesq equation. After that, by using the infinitesimal generators, we determine their corresponding invariant solutions.

Split Orthogonal Group: A Guiding Principle for Sign-Problem-Free Fermionic Simulations.

Abstract: We present a guiding principle for designing fermionic Hamiltonians and quantum Monte Carlo (QMC) methods that are free from the infamous sign problem by exploiting the Lie groups and Lie algebras that appear naturally in the Monte Carlo weight of fermionic QMC simulations Specifically, rigorous mathematical constraints on the determinants involving matrices that lie in the split orthogonal group provide a guideline for sign-free simulations of fermionic models on bipartite lattices This guiding principle not only unifies the recent solutions of the sign problem based on the continuous-time quantum Monte Carlo methods and the Majorana representation, but also suggests new efficient algorithms to simulate physical systems that were previously prohibitive because of the sign problem

Mixing of frame flow for rank one locally symmetric spaces and measure classification

Abstract: Let G be a connected simple linear Lie group of rank one, and let Γ < G be a discrete Zariski dense subgroup admitting a finite Bowen-Margulis-Sullivan measure m BMS. We show that the right translation action of the one-dimensional diagonalizable subgroup is mixing on (Γ\G, m BMS). Together with the work of Roblin, this proves ergodicity of the Burger-Roblin measure under the horospherical group N, establishes a classification theorem for N invariant Radon measures on Γ\G, and provides precise asymptotics for the Haar measure matrix coefficients.

Group theoretic, lie algebraic and Jordan algebraic formulations of the sic existence problem

Abstract: Although symmetric informationally complete positive operator valued measures (SIC POVMs, or SICs for short) have been constructed in every dimension up to 67, a general existence proof remains elusive. The purpose of this paper is to show that the SIC existence problem is equivalent to three other, on the face of it quite different problems. Although it is still not clear whether these reformulations of the problem will make it more tractable, we believe that the fact that SICs have these connections to other areas of mathematics is of some intrinsic interest. Specifically, we reformulate the SIC problem in terms of (1) Lie groups, (2) Lie algebras and (3) Jordan algebras (the second result being a greatly strengthened version of one previously obtained by Appleby, Flammia and Fuchs). The connection between these three reformulations is non-trivial: It is not easy to demonstrate their equivalence directly, without appealing to their common equivalence to SIC existence. In the course of our analysis we obtain a number of other results which may be of some independent interest.

Representations of classical Lie groups and quantized free convolution

Abstract: We study the decompositions into irreducible components of tensor products and restrictions of irreducible representations for all series of classical Lie groups as the rank of the group goes to infinity. We prove the Law of Large Numbers for the random counting measures describing the decomposition. This leads to two operations on measures which are deformations of the notions of the free convolution and the free projection. We further prove that if one replaces counting measures with others coming from the work of Perelomov and Popov on the higher order Casimir operators for classical groups, then the operations on the measures turn into the free convolution and projection themselves. We also explain the relation between our results and limit shape theorems for uniformly random lozenge tilings with and without axial symmetry.

Invariant Solutions to the Strominger System on Complex Lie Groups and Their Quotients

Abstract: Using canonical 1-parameter family of Hermitian connections on the tangent bundle, we provide invariant solutions to the Strominger system on certain complex Lie groups and their quotients. Both flat and non-flat cases are discussed in detail. This paper answers a question proposed by Andreas and Garcia-Fernandez in Comm Math Phys 332(3):1381–1383, 2014.

A classifying space for commutativity in Lie groups

Abstract: In this article we consider a space BcomG assembled from commuting elements in a Lie group G first defined by Adem, Cohen and Torres-Giese. We describe homotopy-theoretic properties of these spaces using homotopy colimits, and their role as a classifying space for transitionally commutative bundles. We prove that Z BcomU is a loop space and define a notion of commutative K‐theory for bundles over a finite complex X , which is isomorphic to aX;Z BcomUc. We compute the rational cohomology of BcomG for G equal to any of the classical groups SU.r/, U.q/ and Sp.k/, and exhibit the rational cohomologies of BcomU , BcomSU and BcomSp as explicit polynomial rings. 22E99; 55R35

Higher-order Lie symmetries in identifiability and predictability analysis of dynamic models

Abstract: Parameter estimation in ordinary differential equations (ODEs) has manifold applications not only in physics but also in the life sciences. When estimating the ODE parameters from experimentally observed data, the modeler is frequently concerned with the question of parameter identifiability. The source of parameter nonidentifiability is tightly related to Lie group symmetries. In the present work, we establish a direct search algorithm for the determination of admitted Lie group symmetries. We clarify the relationship between admitted symmetries and parameter nonidentifiability. The proposed algorithm is applied to illustrative toy models as well as a data-based ODE model of the NFκB signaling pathway. We find that besides translations and scaling transformations also higher-order transformations play a role. Enabled by the knowledge about the explicit underlying symmetry transformations, we show how models with nonidentifiable parameters can still be employed to make reliable predictions.

Geometric flows and their solitons on homogeneous spaces

Abstract: We develop a general approach to study geometric flows on homogeneous spaces. Our main tool will be a dynamical system defined on the variety of Lie algebras called the bracket flow, which coincides with the original geometric flow after a natural change of variables. The advantage of using this method relies on the fact that the possible pointed (or Cheeger-Gromov) limits of solutions, as well as self-similar solutions or soliton structures, can be much better visualized. The approach has already been worked out in the Ricci flow case and for general curvature flows of almost-hermitian structures on Lie groups. This paper is intended as an attempt to motivate the use of the method on homogeneous spaces for any flow of geometric structures under minimal natural assumptions. As a novel application, we find a closed G2-structure on a nilpotent Lie group which is an expanding soliton for the Laplacian flow and is not an eigenvector.

On the Chern‐Ricci flow and its solitons for Lie groups

Abstract: This paper is concerned with Chern-Ricci flow evolution of left-invariant hermitian structures on Lie groups. We study the behavior of a solution, as t is approaching the first time singularity, by rescaling in order to prevent collapsing and obtain convergence in the pointed (or Cheeger-Gromov) sense to a Chern-Ricci soliton. We give some results on the Chern-Ricci form and the Lie group structure of the pointed limit in terms of the starting hermitian metric and, as an application, we obtain a complete picture for the class of solvable Lie groups having a codimension one normal abelian subgroup. We have also found a Chern-Ricci soliton hermitian metric on most of the complex surfaces which are solvmanifolds, including an unexpected shrinking soliton example.

Discrete variational Lie group formulation of geometrically exact beam dynamics

Abstract: The goal of this paper is to derive a structure preserving integrator for geometrically exact beam dynamics, by using a Lie group variational integrator. Both spatial and temporal discretization are implemented in a geometry preserving manner. The resulting scheme preserves both the discrete momentum maps and symplectic structures, and exhibits almost-perfect energy conservation. Comparisons with existing numerical schemes are provided and the convergence behavior is analyzed numerically.

Deformation Quantization for Actions of Kahlerian Lie Groups

Abstract: Introduction Notations and conventions Oscillatory integrals Tempered pairs for Kahlerian Lie groups Non-formal star-products Deformation of Frechet algebras Quantization of polarized symplectic symmetric spaces Quantization of Kahlerian Lie groups Deformation of $C^*$-algebras Bibliography

On the Hamiltonian integrability of the bi-Yang-Baxter sigma-model

Abstract: The bi-Yang-Baxter sigma-model is a certain two-parameter deformation of the principal chiral model on a real Lie group G for which the left and right G-symmetries of the latter are both replaced by Poisson-Lie symmetries. It was introduced by C. Klimcik who also recently showed it admits a Lax pair, thereby proving it is integrable at the Lagrangian level. By working in the Hamiltonian formalism and starting from an equivalent description of the model as a two-parameter deformation of the coset sigma-model on G x G / G_diag, we show that it also admits a Lax matrix whose Poisson bracket is of the standard r/s-form characterised by a twist function which we determine. A number of results immediately follow from this, including the identification of certain complex Poisson commuting Kac-Moody currents as well as an explicit description of the q-deformed symmetries of the model. Moreover, the model is also shown to fit naturally in the general scheme recently developed for constructing integrable deformations of sigma-models. Finally, we show that although the Poisson bracket of the Lax matrix still takes the r/s-form after fixing the G_diag gauge symmetry, it is no longer characterised by a twist function.