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Showing papers on "Lie group published in 2016"


Book
12 Mar 2016
TL;DR: In this article, the authors provide a survey of quantization on compact Lie groups and homogeneous Lie groups, including pseudo-differential operators on the Heisenberg group, Schrodinger representations and Weyl quantization.
Abstract: Preface.- Introduction.- Notation and conventions.- 1 Preliminaries on Lie groups.- 2 Quantization on compact Lie groups.- 3 Homogeneous Lie groups.- 4 Rockland operators and Sobolev spaces.- 5 Quantization on graded Lie groups.- 6 Pseudo-differential operators on the Heisenberg group.- A Miscellaneous.- B Group C* and von Neumann algebras.- Schrodinger representations and Weyl quantization.- Explicit symbolic calculus on the Heisenberg group.- List of quantizations.- Bibliography.- Index.

268 citations


Journal ArticleDOI
TL;DR: In this article, a mathematical definition of the coordinate ring of the Coulomb branch is proposed, using the vanishing cycle cohomology group of a certain moduli space for a gauged σ-model on the $2$-sphere associated with a compact Lie group.
Abstract: Consider the $3$-dimensional $\mathcal{N}=4$ supersymmetric gauge theory associated with a compact Lie group $G$ and its quaternionic representation $\mathrm{M}$. Physicists study its Coulomb branch, which is a noncompact hyper-Kahler manifold, such as instanton moduli spaces on $\mathbb{R}^4 , \mathrm{SU}(2)$-monopole moduli spaces on $\mathbb{R}^3$, etc. In this paper and its sequel, we propose a mathematical definition of the coordinate ring of the Coulomb branch, using the vanishing cycle cohomology group of a certain moduli space for a gauged $\sigma$-model on the $2$-sphere associated with $(G, \mathrm{M})$. In this first part, we check that the cohomology group has the correct graded dimensions expected from the monopole formula proposed by Cremonesi, Hanany and Zaffaroni. A ring structure (on the cohomology of a modified moduli space) will be introduced in the sequel of this paper.

208 citations


Proceedings ArticleDOI
01 Jun 2016
TL;DR: This work uses rolling maps for recognizing human actions from 3D skeletal data and unwrap the action curves onto the Lie algebra so3 × ... × so3 (which is a vector space) by combining the logarithm map with rolling maps, and perform classification in the Liegebra.
Abstract: Recently, skeleton-based human action recognition has been receiving significant attention from various research communities due to the availability of depth sensors and real-time depth-based 3D skeleton estimation algorithms. In this work, we use rolling maps for recognizing human actions from 3D skeletal data. The rolling map is a welldefined mathematical concept that has not been explored much by the vision community. First, we represent each skeleton using the relative 3D rotations between various body parts. Since 3D rotations are members of the special orthogonal group SO3, our skeletal representation becomes a point in the Lie group SO3 × … × SO3, which is also a Riemannian manifold. Then, using this representation, we model human actions as curves in this Lie group. Since classification of curves in this non-Euclidean space is a difficult task, we unwrap the action curves onto the Lie algebra so3 × … × so3 (which is a vector space) by combining the logarithm map with rolling maps, and perform classification in the Lie algebra. Experimental results on three action datasets show that the proposed approach performs equally well or better when compared to state-of-the-art.

195 citations


Posted Content
TL;DR: In this paper, the Coulomb branch is defined as an affine algebraic variety with a singularity and a Coulomb action in the form Ω( √ √ N 2 ).
Abstract: Consider the $3$-dimensional $\mathcal N=4$ supersymmetric gauge theory associated with a compact Lie group $G_c$ and its quaternionic representation $\mathbf M$. Physicists study its Coulomb branch, which is a noncompact hyper-Kahler manifold with an $\mathrm{SU}(2)$-action, possibly with singularities. We give a mathematical definition of the Coulomb branch as an affine algebraic variety with $\mathbb C^\times$-action when $\mathbf M$ is of a form $\mathbf N\oplus\mathbf N^*$, as the second step of the proposal given in arXiv:1503.03676.

191 citations


Posted Content
TL;DR: Li et al. as mentioned in this paper incorporated the Lie group structure into a deep network architecture to learn more appropriate Lie group features for skeleton-based action recognition, and designed rotation mapping layers to transform the input Lie group feature into desirable ones, which are aligned better in the temporal domain.
Abstract: In recent years, skeleton-based action recognition has become a popular 3D classification problem. State-of-the-art methods typically first represent each motion sequence as a high-dimensional trajectory on a Lie group with an additional dynamic time warping, and then shallowly learn favorable Lie group features. In this paper we incorporate the Lie group structure into a deep network architecture to learn more appropriate Lie group features for 3D action recognition. Within the network structure, we design rotation mapping layers to transform the input Lie group features into desirable ones, which are aligned better in the temporal domain. To reduce the high feature dimensionality, the architecture is equipped with rotation pooling layers for the elements on the Lie group. Furthermore, we propose a logarithm mapping layer to map the resulting manifold data into a tangent space that facilitates the application of regular output layers for the final classification. Evaluations of the proposed network for standard 3D human action recognition datasets clearly demonstrate its superiority over existing shallow Lie group feature learning methods as well as most conventional deep learning methods.

170 citations


Arun Ram1
01 Jan 2016
TL;DR: The algebraic Bethe Ansatz as discussed by the authors is the essence of the quantum inverse scattering method, which emerges as a natural development of the following different directions in mathematical physics: the quantum theory of magnets [Bet1931], the method of commuting transfer-matrices in classical statistical mechanics [Bax1982], and factorizable scattering theory [Yan1967,Zam1979].
Abstract: The Algebraic Bethe Ansatz, which is the essence of the quantum inverse scattering method, emerges as a natural development of the following different directions in mathematical physics: the inverse scattering method for solving nonlinear equations of evolution [GGK1967], quantum theory of magnets [Bet1931], the method of commuting transfer-matrices in classical statistical mechanics [Bax1982]] and factorizable scattering theory [Yan1967,Zam1979]. It was formulated in our papers [STF1979,TFa1979,Fad1984]. Two simple algebraic formulas lie in the foundation or the method: RT1T2=T2 T1R (*) and R12R13R23= R23R13R12. (**) Their exact meaning will be explained in the next section. In the original context or the Algebraic Bethe Ansats T plays the role of the quantum monodromy matrix of the auxiliary linear problem and is a matrix with operator-valued entries whereas R is an ordinary "c-number" matrix. The second formula can be considered as a compatibility condition for the first one.

112 citations


Journal ArticleDOI
TL;DR: In this article, the authors consider the question of existence of lattices in solvable Lie groups up to dimension six and examine whether the corresponding solvmanifolds are formal, symplectic, Kaehler or (Hard) Lefschetz.
Abstract: A nilmanifold resp. solvmanifold is a compact homogeneous space of a connected and simply-connected nilpotent resp. solvable Lie group by a lattice, i.e. a discrete co-compact subgroup. There is an easy criterion for nilpotent Lie groups which enables one to decide whether there is a lattice or not. Moreover, it is easy to decide whether a nilmanifold is formal, Kaehlerian or (Hard) Lefschetz. The study of solvmanifolds meets with noticeably greater obstacles than the study of nilmanifolds. Even the construction of solvmanifolds is considerably more difficult than is the case for nilmanifolds. The reason is that there is no simple criterion for the existence of a lattice in a connected and simply-connected solvable Lie group. We consider the question of existence of lattices in solvable Lie groups up to dimension six and examine whether the corresponding solvmanifolds are formal, symplectic, Kaehler or (Hard) Lefschetz.

110 citations


Journal ArticleDOI
TL;DR: In this paper, a decentralized consensus control of a formation of rigid-body spacecraft is studied in the framework of geometric mechanics while accounting for a constant communication time delay between spacecraft, where relative position and attitude are represented on the Lie group SE(3) and the communication topology is modeled as a digraph.
Abstract: The decentralized consensus control of a formation of rigid-body spacecraft is studied in the framework of geometric mechanics while accounting for a constant communication time delay between spacecraft. The relative position and attitude (relative pose) are represented on the Lie group SE(3) and the communication topology is modeled as a digraph. The consensus problem is converted into a local stabilization problem of the error dynamics associated with the Lie algebra se(3) in the form of linear time-invariant delay differential equations with a single discrete delay in the case of a circular orbit, whereas it is in the form of linear time-periodic delay differential equations in the case of an elliptic orbit, in which the stability may be assessed using infinite-dimensional Floquet theory. The proposed technique is applied to the consensus control of four spacecraft in the vicinity of a Molniya orbit.

97 citations


Book
10 Sep 2016
TL;DR: In this article, a tensorial characterization of linear structures is presented, and a tensor characterization of partial linear structures, including vector bundles, is given for linear dynamical systems.
Abstract: Foreword: The birth and the long gestation of a project.- Some examples of linear and nonlinear physical systems and their dynamical equations.- Equations of the motion for evolution systems.- Linear systems with infinite degrees of freedom.- Constructing nonlinear systems out of linear ones.- The language of geometry and dynamical systems: the linearity paradigm.- Linear dynamical systems: The algebraic viewpoint.- From linear dynamical systems to vector fields.- Exterior differential calculus on linear spaces.- Exterior differential calculus on submanifolds.- A tensorial characterization of linear structures.- Partial linear structures: Vector bundles.- Covariant calculus.- Riemannian and Pseudo-Riemannian metrics on linear vector spaces.- Invariant geometric structures and the classical formulations of dynamics of Poisson, Jacobi, Hamilton and Lagrange.- Linear vector fields.- Additional invariant structures for linear vector fields.- Poisson structures.- The inverse problem for Poisson structures.- Symplectic structures.- Lagrangian structures.- Invariant Hermitean structures and the geometry of quantum systems.- Invariant Hermitean inner products.- Complex structures and complex exterior calculus.- Algebras associated with Hermitean structures.- The geometry of quantum dynamical evolution.- The Geometry of Quantum Mechanics and the GNS construction.- Alternative Hermitean structures for quantum systems.- Folding and unfolding Classical and Quantum systems.- Introduction: separable dynamics.- The geometrical description of reduction.- The algebraic description.- Reduction in Quantum Mechanics.- Integrable and superintegrable systems.- The geometrization of the notion of integrability.- The normal form of an integrable system.- Lax representation.- The Calogero system: inverse scattering.- Lie-Scheffers systems.- The inhomogeneous linear equation revisited.- Inhomogeneous linear systems.- Non-linear superposition rule.- Related maps.- Lie systems on Lie groups and homogeneous spaces.- Some examples of Lie systems.- Hamiltonian systems of Lie type.

94 citations


Journal ArticleDOI
TL;DR: In this paper, the notions of intrinsic graphs and of intrinsic Lipschitz graphs within Carnot groups are studied and both geometric and analytic characterizations and a clarifying relation between these graphs and Rumin's complex of differential forms are provided.
Abstract: A Carnot group is a connected, simply connected, nilpotent Lie group with stratified Lie algebra. We study the notions of intrinsic graphs and of intrinsic Lipschitz graphs within Carnot groups. Intrinsic Lipschitz graphs are the natural local analogue inside Carnot groups of Lipschitz submanifolds in Euclidean spaces, where “natural” emphasizes that the notion depends only on the structure of the algebra. Intrinsic Lipschitz graphs unify different alternative approaches through Lipschitz parameterizations or level sets. We provide both geometric and analytic characterizations and a clarifying relation between these graphs and Rumin’s complex of differential forms.

93 citations


01 Jan 2016
TL;DR: Differential geometry and lie groups for physicists for physicists is available in our book collection and an online access to it is set as public so that anyone can download it instantly as mentioned in this paper.
Abstract: differential geometry and lie groups for physicists is available in our book collection an online access to it is set as public so you can download it instantly. Our digital library hosts in multiple locations, allowing you to get the most less latency time to download any of our books like this one. Merely said, the differential geometry and lie groups for physicists is universally compatible with any devices to read.

Book ChapterDOI
TL;DR: In this paper, the equivalence of two a priori different methods of construction and description of a wide class of integrable models is demonstrated, and a unified approach for their investigation is proposed.
Abstract: The main idea of this work is to demonstrate the equivalence of two a priori different methods of construction and description of a wide class of integrable models, and thus, to propose a unified approach for their investigation In the first, well-known method [24], the phase space is taken as a quotient of double Bruhat cells of a Kac–Moody Lie group, with the Poisson structure defined by a classical r-matrix, and the integrals of motion are just the Ad-invariant functions The second method was suggested recently by A

Journal ArticleDOI
TL;DR: In this article, the median class of a non-elementary action by automorphisms does not vanish and to what extent it does vanish if the action is an elementary action.
Abstract: We define a bounded cohomology class, called the median class, in the second bounded cohomology, with appropriate coefficients, of the automorphism group of a finite-dimensional CAT(0) cube complex X. The median class of X behaves naturally with respect to taking products and appropriate subcomplexes and defines in turn the median class of an action by automorphisms of X. We show that the median class of a non-elementary action by automorphisms does not vanish and we show to what extent it does vanish if the action is elementary. We obtain as a corollary a superrigidity result and show, for example, that any irreducible lattice in the product of at least two locally compact connected groups acts on a finite-dimensional CAT(0) cube complex X with a finite orbit in the Roller compactification of X. In the case of a product of Lie groups, the appendix by Caprace allows us to deduce that the fixed point is in fact inside the complex X. In the course of the proof, we construct a Γequivariant measurable map from a Poisson boundary of Γ with values in the non-terminating ultrafilters on the Roller boundary of X.

Journal ArticleDOI
TL;DR: In this paper, the authors investigated the extent to which C(Γ) determines Γ when Γ is a group of geometric interest, and they proved that Γ 1 is a lattice in PSL(2, R) and Γ 2 is a connected Lie group.
Abstract: Let C(Γ) be the set of isomorphism classes of the finite groups that are quotients (homomorphic images) of Γ. We investigate the extent to which C(Γ) determines Γ when Γ is a group of geometric interest. If Γ1 is a lattice in PSL(2, R) and Γ2 is a lattice in any connected Lie group, then C(Γ1) = C(Γ2) implies that Γ1 ≅ Γ2. If F is a free group and Γ is a right-angled Artin group or a residually free group (with one extra condition), then C(F) = C(Γ) implies that F ≅ Γ. If Γ1 < PSL(2, C) and Γ2 < G are nonuniform arithmetic lattices, where G is a semisimple Lie group with trivial centre and no compact factors, then C(Γ1) = C(Γ2) implies that G ≅ PSL(2, C) and that Γ2 belongs to one of finitely many commensurability classes. These results are proved using the theory of profinite groups; we do not exhibit explicit finite quotients that distinguish among the groups in question. But in the special case of two non-isomorphic triangle groups, we give an explicit description of finite quotients that distinguish between them.

Journal ArticleDOI
TL;DR: In this work, the symmetry group and similarity reductions of the two-dimensional generalized Benney system are investigated by means of the geometric approach of an invariance group, which is equivalent to the classical Lie symmetry method.
Abstract: In this work, the symmetry group and similarity reductions of the two-dimensional generalized Benney system are investigated by means of the geometric approach of an invariance group, which is equivalent to the classical Lie symmetry method. Firstly, the vector field associated with the Lie group of transformation is obtained. Then the point transformations are proposed, which keep the solutions of the generalized Benney system invariant. Finally, the symmetry reductions and explicitly exact solutions of the generalized Benney system are derived by solving the corresponding symmetry equations.

Journal ArticleDOI
TL;DR: In this paper, the SU(2) theory was studied at finite temperature using a simple massive extension of standard background field methods, and it was shown that two-loop corrections yield improved values for the first-order transition temperature as compared to the one-loop result.
Abstract: We study the confinement-deconfinement phase transition of pure Yang-Mills theories at finite temperature using a simple massive extension of standard background field methods. We generalize our recent next-to-leading-order perturbative calculation of the Polyakov loop and of the related background field effective potential for the SU(2) theory to any compact and connex Lie group with a simple Lie algebra. We discuss in detail the SU(3) theory, where the two-loop corrections yield improved values for the first-order transition temperature as compared to the one-loop result. We also show that certain one-loop artifacts of thermodynamical observables disappear at two-loop order, as was already the case for the SU(2) theory. In particular, the entropy and the pressure are positive for all temperatures. Finally, we discuss the groups SU(4) and Sp(2) which shed interesting light, respectively, on the relation between the (de)confinement of static matter sources in the various representations of the gauge group and on the use of the background field itself as an order parameter for confinement. In both cases, we obtain first-order transitions, in agreement with lattice simulations and other continuum approaches.

01 Jan 2016
TL;DR: In this paper, the authors studied the structure of Kaehlerian homogeneous spaces of semisimple Lie groups and obtained a partial answer to the question of E. Cartan.
Abstract: Recently the results of E. Cartan on Hermitian symmetric space have been extended to the case of Kaehlerian homogeneous spaces by several authors. In particular the structure of compact Kaehlerian homogeneous spaces has been fully exploited. As for the non-conmpact case, there are few known results except in the case where the groups are semi-simple or reductive. A. Borel [1] and J. L. Koszul [6] have studied the structure of Kaehlerian homogeneous spaces of semi-simple Lie groups and have shown, independently of each other, an interesting result that a bounded domain in a unitary space admitting a transitive semi-simple Lie group of complex analytic homeomorphisms is symmetric. This result gives a partial answer to the problem of E. Cartan. In his imiportant paper in 1935 [2], E. Cartan raised the question whether a bounded homogeneous domain is always a symmetric bounded domain. Y. Matsushima [8] has st-udied the structure of Kaehlerian homogeneous spaces of reductive Lie groups and has shown that these spaces are the direct produict of a locally flat Kaehlerian space and a Kaehlerian homogeneous space of a semi-simple Lie group. The purpose of the present paper is to study the structure of EKaehlerian homogeneous spaces of a more general class of Lie groups, that is, those of unimodular Lie groups. Specifically, we shall deal with the following two cases. First we shall consider the case where the isotropy group is semisimple and obtain the following two theorems:

Proceedings ArticleDOI
16 May 2016
TL;DR: This work develops a visual inertial odometry system based on the Unscented Kalman Filter acting on the Lie group SE(3) to obtain an unique, singularity-free representation of a rigid body pose and presents experimental results to show the effectiveness of the proposed approach for state estimation of a quadrotor platform.
Abstract: The combination of on-board sensors measurements with different statistical characteristics can be employed in robotics for localization and control, especially in GPS-denied environments. In particular, most aerial vehicles are packaged with low cost sensors, important for aerial robotics, such as camera, a gyroscope, and an accelerometer. In this work, we develop a visual inertial odometry system based on the Unscented Kalman Filter (UKF) acting on the Lie group SE(3), such to obtain an unique, singularity-free representation of a rigid body pose. We model this pose with the Lie group SE(3) and model the noise on the corresponding Lie algebra. Moreover, we extend the concepts used in the standard UKF formulation, such as state uncertainty and modeling, to correctly incorporate elements that do not belong to an Euclidean space such as the Lie group members. In this analysis, we use the parallel transport, which requires us to explicitly consider SE(3) as representing rigid bodies though the use of the affine connection. We present experimental results to show the effectiveness of the proposed approach for state estimation of a quadrotor platform.

Journal ArticleDOI
TL;DR: In this article, the spectral gap property for dense subgroups generated by algebraic elements in any compact simple Lie group was established, generalizing earlier results of Bourgain and Gamburd for unitary groups.
Abstract: We establish the spectral gap property for dense subgroups generated by algebraic elements in any compact simple Lie group, generalizing earlier results of Bourgain and Gamburd for unitary groups.

Journal ArticleDOI
TL;DR: In this article, the generalized fifth order KdV equation does not admit nonclassical type symmetries, and the symmetry reductions and exact solutions to this equation are constructed on the basis of the optimal system.
Abstract: We study the generalized fifth order KdV equation using group methods and conservation laws. All of the geometric vector fields of the special fifth order KdV equation are presented. By using the nonclassical Lie group method, it is show that this equation does not admit nonclassical type symmetries. Then, on the basis of the optimal system, the symmetry reductions and exact solutions to this equation are constructed. For some special cases, we obtain additional nontrivial conservation laws and scaling symmetries.

Journal ArticleDOI
TL;DR: In this article, it was shown that the abnormal set lies in a proper analytic subvariety and a strong version of Sard's property holds for all step-2 Carnot groups and several other classes of Lie groups endowed with left-invariant distributions.
Abstract: In Carnot–Caratheodory or sub-Riemannian geometry, one of the major open problems is whether the conclusions of Sard's theorem holds for the endpoint map, a canonical map from an infinite-dimensional path space to the underlying finite-dimensional manifold. The set of critical values for the endpoint map is also known as abnormal set, being the set of endpoints of abnormal extremals leaving the base point. We prove that a strong version of Sard's property holds for all step-2 Carnot groups and several other classes of Lie groups endowed with left-invariant distributions. Namely, we prove that the abnormal set lies in a proper analytic subvariety. In doing so we examine several characterizations of the abnormal set in the case of Lie groups.

Journal ArticleDOI
TL;DR: In this paper, the authors studied controllability of linear systems on Lie groups by taking into account the eigenvalues of an associated derivation, i.e., the reachable set of the neutral element is open and the derivation has only pure imaginary eigen values.
Abstract: Linear systems on Lie groups are a natural generalization of linear systems on Euclidian spaces. For such systems, this paper studies controllability by taking into consideration the eigenvalues of an associated derivation ${\mathcal{D}}$. When the state space is a solvable connected Lie group, controllability of the system is guaranteed if the reachable set of the neutral element is open and the derivation ${\mathcal{D}}$ has only pure imaginary eigenvalues. For bounded systems on nilpotent Lie groups such conditions are also necessary.

Journal ArticleDOI
TL;DR: In this paper, the authors give a global characterisation of classes of ultradifferentiable functions and corresponding ultradistributions on a compact manifold X. The characterisation is given in terms of the eigenfunction expansion of an elliptic operator on X.
Abstract: In this paper we give a global characterisation of classes of ultradifferentiable functions and corresponding ultradistributions on a compact manifold X. The characterisation is given in terms of the eigenfunction expansion of an elliptic operator on X. This extends the result for analytic functions on compact manifolds by Seeley in 1969, and the characterisation of Gevrey functions and Gevrey ultradistributions on compact Lie groups and homogeneous spaces by the authors (2014).

Journal ArticleDOI
TL;DR: This paper presents some exact solutions for the drift-flux model of two-phase flows using Lie group analysis which provides new insights into the fundamental properties of weak discontinuities and helps one to understand better on existence of solutions.
Abstract: This paper presents some exact solutions for the drift-flux model of two-phase flows using Lie group analysis. The analysis involves an isentropic no-slip conservation of mass for each phase and the conservation of momentum for the mixture. The present analysis employs a complete Lie algebra of infinitesimal symmetries. Subsequent to these theoretical analysis a symmetry group is established. The symmetry generators are used for constructing similarity variables which reduce the model equations to a system of ordinary differential equations (ODEs). In particular, a general framework is discussed for solving the model equations analytically. As a consequence of this, new classes of exact group-invariant solutions are developed. This provides new insights into the fundamental properties of weak discontinuities and helps one to understand better on existence of solutions.

Posted Content
TL;DR: In this article, a classification of all compact Hermitian manifold with flat Bismut connection is given, and the universal cover of such a manifold is a Lie group equipped with a bi-invariant metric and a compatible left invariant complex structure.
Abstract: In this paper, we give a classification of all compact Hermitian manifolds with flat Bismut connection. We show that the torsion tensor of such a manifold must be parallel, thus the universal cover of such a manifold is a Lie group equipped with a bi-invariant metric and a compatible left invariant complex structure. In particular, isosceles Hopf surfaces are the only Bismut flat compact non-K\"ahler surfaces, while central Calabi-Eckmann threefolds are the only simply-connected compact Bismut flat threefolds.

Journal ArticleDOI
09 May 2016-EPL
TL;DR: In this paper, the invariance properties of the generalized time fractional Burgers equation were investigated for unidirectional propagation of weakly nonlinear acoustic waves through a gas-filled pipe.
Abstract: Under investigation in this work are the invariance properties of the generalized time fractional Burgers equation, which can be used to describe the physical processes of unidirectional propagation of weakly nonlinear acoustic waves through a gas-filled pipe. The Lie group analysis method is applied to consider its vector fields and symmetry reductions. Furthermore, based on the sub-equation method, a new type of explicit solutions for the equation is well constructed with a detailed analysis. By means of the power series theory, exact power series solutions of the equation are also constructed. Finally, by using the new conservation theorem, conservation laws of the equation are well constructed with a detailed derivation.

Journal ArticleDOI
TL;DR: In this article, the authors introduce a theory of homotopy moment maps, which is a morphism from the Lie algebra of the group into the observables which lifts the infinitesimal action.

Journal ArticleDOI
TL;DR: This paper develops a framework for shape analysis of curves in Lie groups for problems of computer animations and uses these methods to find cyclic approximations of non-cyclic character animations and interpolate between existing animations to generate new ones.
Abstract: Shape analysis methods have in the past few years become very popular, both for theoretical exploration as well as from an application point of view. Originally developed for planar curves, these methods have been expanded to higher dimensional curves, surfaces, activities, character motions and many other objects.   In this paper, we develop a framework for shape analysis of curves in Lie groups for problems of computer animations. In particular, we will use these methods to find cyclic approximations of non-cyclic character animations and interpolate between existing animations to generate new ones.

Journal ArticleDOI
TL;DR: In this article, the authors analyse how the Weyl quantisation for compact Lie groups presented in the second article of this series fits with the projective-phase space structure of loop quantum gravity-type models.
Abstract: In this article, the third of three, we analyse how the Weyl quantisation for compact Lie groups presented in the second article of this series fits with the projective-phase space structure of loop quantum gravity-type models. Thus, the proposed Weyl quantisation may serve as the main mathematical tool to implement the program of space adiabatic perturbation theory in such models. As we already argued in our first article, space adiabatic perturbation theory offers an ideal framework to overcome the obstacles that hinder the direct implementation of the conventional Born-Oppenheimer approach in the canonical formulation of loop quantum gravity.

Journal ArticleDOI
TL;DR: In this article, a framework for describing locally gauge invariant states on lattices using projected entangled pair states (PEPS) is presented, suitable for all combinations of matter and gauge fields in lattice gauge theories defined by either finite or compact Lie groups.
Abstract: Tensor network states, and in particular projected entangled pair states (PEPS), suggest an innovative approach for the study of lattice gauge theories, both from a pure theoretic point of view, and as a tool for the analysis of the recent proposals for quantum simulations of lattice gauge theories. In this paper we present a framework for describing locally gauge invariant states on lattices using PEPS. The PEPS constructed hereby shall include both bosonic and fermionic states, suitable for all combinations of matter and gauge fields in lattice gauge theories defined by either finite or compact Lie groups.