Lie Group Research Papers - Academia.edu (original) (raw)

The concept of broken symmetry is used to study stability of equilibrium and time doubly-periodic bifurcating solutions of the complex nonresonant Lorenz model as a function of the fre-quency detuning on the basis of modiied Hopf theory.... more

The concept of broken symmetry is used to study stability of equilibrium and time doubly-periodic bifurcating solutions of the complex nonresonant Lorenz model as a function of the fre-quency detuning on the basis of modiied Hopf theory. By contrast to the well-known real Lorenz equations, the system in question is invariant under the action of Lie group transformations (ro-tations in complex planes) and an invariant set of stationary points is found to bifurcate into an invariant torus, which is stable under the detuning exceeding its critical value. If the detuning then goes downward numerical analysis reveals that after a cascade of period-doublings the strange Lorenz attractor is formed in the vicinity of zero detuning.

The aim of this paper is to use the so-called Cayley transform to compute the LS category of Lie groups and homogeneous spaces by giving explicit categorical open coverings. When applied to U(n), U(2n)/Sp(n)U(2n)/Sp(n)U(2n)/Sp(n) and U(n)/O(n)U(n)/O(n)U(n)/O(n) this method... more

The aim of this paper is to use the so-called Cayley transform to compute the LS category of Lie groups and homogeneous spaces by giving explicit categorical open coverings. When applied to U(n), U(2n)/Sp(n)U(2n)/Sp(n)U(2n)/Sp(n) and U(n)/O(n)U(n)/O(n)U(n)/O(n) this method is simpler than those formerly known. We also show that the Cayley transform is related to height functions in Lie groups, allowing to give a local linear model of the set of critical points. As an application we give an explicit covering of Sp(2)Sp(2)Sp(2) by categorical open sets. The obstacles to generalize these results to Sp(n)Sp(n)Sp(n) are discussed.

Given a semisimple, compact, connected Lie group G with complexification G^c, we show there is a stable range in the homotopy type of the universal moduli space of flat connections on a principal G-bundle on a closed Riemann surface, and... more

Given a semisimple, compact, connected Lie group G with complexification G^c, we show there is a stable range in the homotopy type of the universal moduli space of flat connections on a principal G-bundle on a closed Riemann surface, and equivalently, the universal moduli space of semistable holomorphic G^c-bundles. The stable range depends on the genus of the surface. We then identify the homology of this moduli space in the stable range in terms of the homology of an explicit infinite loop space. Rationally this says that the stable cohomology of this moduli space is generated by the Mumford-Morita-Miller kappa-classes, and the ring of characteristic classes of principal G-bundles, H^*(BG). We then identify the homotopy type of the category of one-manifolds and surface cobordisms, each equipped with a flat G-bundle. We also explain how these results may be generalized to arbitrary compact connected Lie groups. Our methods combine the classical techniques of Atiyah and Bott, with the new techniques coming out of Madsen and Weiss's proof of Mumford's conjecture on the stable cohomology of the moduli space of Riemann surfaces.

L’objet de l’exposé concerne les structures géométriques élémentaires de l’apprentissage machine, fondées sur le « gradient naturel » de la « Géométrie de l’Information », qui rend invariant par changement de paramétrisation le gradient... more

L’objet de l’exposé concerne les structures géométriques élémentaires de l’apprentissage machine, fondées sur le « gradient naturel » de la « Géométrie de l’Information », qui rend invariant par changement de paramétrisation le gradient d’apprentissage dans les réseaux de neurones par le biais de la matrice de Fisher. Après cette introduction exposant le « gradient naturel » [1] et ses extensions récentes aux réseaux de neurones profonds [2], nous développons de nouvelles mé-thodes pour étendre l’approche à des espaces plus abstraits et en particulier les groupes de Lie (matriciels).
L’apprentissage profond a été étendu récemment avec succès aux graphes, mais le thème émergent « (Matrix) Lie Group Machine Learning » [14][18][15][16][20][21] est une extension particulièrement intéressante pour les applications indus-trielles : reconnaissance de mouvements/cinématiques (série temporelle d’éléments du groupes SE(3)), reconnaissances de postures/gestes articulés [3](série temporelle de vecteurs d’éléments du groupe SO(3)), reconnaissance micro-Doppler [12](série temporelle d’éléments du groupe SU(1,1)) et en robotique (éléments de sous-groupes du groupe affine Aff(n)).
Nous exposerons l’extension de la notion de métrique de Fisher par le mathématicien Jean-Louis Koszul [4][6][17] sur les cônes convexes saillants. Pour l’extension de l’apprentissage machine aux groupes de Lie, nous présenterons les outils issues de la physique statistique à travers le modèle du physicien Jean-Marie Souriau de la « Thermodynamique des groupes de Lie » [7][8][19][22] basé sur la géométrie symplectique (application moment, 2 forme KKS « Kirillov-Kostant-Souriau » dans le cas non-équivariant [5], le cocycle symplectique de Souriau, les méthodes des orbites coadjointes [9] issues de la théorie des représentations de Kirillov des groupes de Lie [10][11]).
Nous terminerons par une illustration d’apprentissage machine pour les exemples canoniques de groupes de Lie matri-ciels classiques tels que les groupes SU(1,1)[13][23] (cas équivariant à cohomologie nulle), et le groupe SE(3) (cas non-équivariant à cohomologie non-nulle)[13].

We develop a Chern-Weil theory for compact Lie group action whose generic stabilizers are finite in the framework of equivariant cohomology. This provides a method of changing an equivariant closed form within its cohomological class to a... more

We develop a Chern-Weil theory for compact Lie group action whose generic stabilizers are finite in the framework of equivariant cohomology. This provides a method of changing an equivariant closed form within its cohomological class to a form more suitable to yield localization results. This work is motivated by our work on reproving wall crossing formulas in Seiberg-Witten theory, where

This work studies the dynamic modeling method for a service robot with Omni-directional Mobile ManipulatorS configuration. Based on screw theory, Lie group notations, reciprocal product of twist and wrench, and Jourdain principle, the... more

This work studies the dynamic modeling method for a service robot with Omni-directional Mobile ManipulatorS configuration. Based on screw theory, Lie group notations, reciprocal product of twist and wrench, and Jourdain principle, the robot’s motion equations including the whole body manipulation are formulated with left invariant representation. A legible and canonical dynamic model representing the relation between the inputs and the generalized dynamic load wrenches is presented. Considering the tradeoff between the symbolic concision, the modularization in code realization and the computation load, the dynamic model is decomposed into succinct block factorizations, and the basic computation unites are boiled down to the adjoint map corresponding to each joint. The traditional Lie bracket operation is extended to a generalized form. Computation efficiency, for the coefficient matrixes of the system motion equation, is discussed based on its special representation form. The generalization of the modeling method with Lie group and algebra tool is also summarized.

This article is a contribution to the domain of (convergent) deformation quantization of symmetric spaces by use of Lie groups representation theory. We realize the regular representation of SL(2,R)SL(2,\R)SL(2,R) on the space of smooth functions on... more

This article is a contribution to the domain of (convergent) deformation quantization of symmetric spaces by use of Lie groups representation theory. We realize the regular representation of SL(2,R)SL(2,\R)SL(2,R) on the space of smooth functions on the Poincar\'e disc as a sub-representation of SL(2,R)SL(2,\R)SL(2,R) in the Weyl-Moyal star product algebra on R2\R^2R2. We indicate how it is possible to extend

In this article we consider the Euler-Lagrange method associated to a suitable bilagrangian to study biharmonic curves of a Riemannian manifold. We apply this method to characterize biharmonic curves of the three-dimensional Lie group... more

In this article we consider the Euler-Lagrange method associated to a suitable bilagrangian to study biharmonic curves of a Riemannian manifold. We apply this method to characterize biharmonic curves of the three-dimensional Lie group Sol. We also classify, using a geometric method, the biharmonic curves of the three-dimensional Cartan-Vranceanu manifolds.

« La Physique mathématique, en incorporant à sa base la notion de groupe, marque la suprématie rationnelle…Chaque géométrie-et sans doute plus généralement chaque organisation mathématique de l'expérience-est caractérisée par un groupe... more

« La Physique mathématique, en incorporant à sa base la notion de groupe, marque la suprématie rationnelle…Chaque géométrie-et sans doute plus généralement chaque organisation mathématique de l'expérience-est caractérisée par un groupe spécial de transformations…. Le groupe apporte la preuve d'une mathématique fermée sur elle-même. Sa découverte clôt l'ère des conventions, plus ou moins indépendantes, plus ou moins cohérentes »-Gaston Bachelard, Le nouvel esprit scientifique, 1934 Tout mathématicien sait qu'il est impossible de comprendre un cours élémentaire en thermodynamique.

We study the problem of L^p-boundedness (1 < p < \infty) of operators of the form m(L_1,...,L_n) for a commuting system of self-adjoint left-invariant differential operators L_1,...,L_n on a Lie group G of polynomial growth, which... more

We study the problem of L^p-boundedness (1 < p < \infty) of operators of the form m(L_1,...,L_n) for a commuting system of self-adjoint left-invariant differential operators L_1,...,L_n on a Lie group G of polynomial growth, which generate an algebra containing a weighted subcoercive operator. In particular, when G is a homogeneous group and L_1,...,L_n are homogeneous, we prove analogues of the Mihlin-H\"ormander and Marcinkiewicz multiplier theorems.

This article is concerned with an extensive study of a infinite-dimensional Lie algebra sv, introduced in (14) in the context of non-equilibrium statistical physics, containing as subalgebras both the Lie algebra of invariance of the free... more

This article is concerned with an extensive study of a infinite-dimensional Lie algebra sv, introduced in (14) in the context of non-equilibrium statistical physics, containing as subalgebras both the Lie algebra of invariance of the free Schrodinger equation and the central charge-free Virasoro algebra Vect(S1). We call sv the Schrodinger-Virasoro Lie algebra. We choose to present sv from a Newtonian

Theta series for exceptional groups have been suggested as a possible description of the eleven-dimensional quantum supermembrane. We present explicit formulae for these automorphic forms whenever the underlying Lie group G is split (or... more

Theta series for exceptional groups have been suggested as a possible description of the eleven-dimensional quantum supermembrane. We present explicit formulae for these automorphic forms whenever the underlying Lie group G is split (or complex) and simply laced. Specifically, we review and construct explicitly the minimal representation of G, generalizing the Schrödinger representation of symplectic groups. We compute the spherical vector in this representation, i.e. the wave function invariant under the maximal compact subgroup, which plays the rôle of the summand in the automorphic theta series. We also determine the spherical vector over the complex field. We outline how the spherical vector over the p-adic number fields provides the summation measure in the theta series, postponing its determination to a sequel of this work. The simplicity of our result is suggestive of a new Born–Infeld-like description of the membrane where U-duality is realized non-linearly. Our results may also be used in constructing quantum mechanical systems with spectrum generating symmetries.

We compute Haar ensemble averages of ratios of random characteristic polynomials for the classical Lie groups K = O(N), SO(N), and USp(N). To that end, we start from the Clifford-Weyl algebera in its canonical realization on the complex... more

We compute Haar ensemble averages of ratios of random characteristic polynomials for the classical Lie groups K = O(N), SO(N), and USp(N). To that end, we start from the Clifford-Weyl algebera in its canonical realization on the complex of holomorphic differential forms for a C-vector space V. From it we construct the Fock representation of an orthosymplectic Lie superalgebra osp associated to V. Particular attention is paid to defining Howe's oscillator semigroup and the representation that partially exponentiates the Lie algebra representation of sp in osp. In the process, by pushing the semigroup representation to its boundary and arguing by continuity, we provide a construction of the Shale-Weil-Segal representation of the metaplectic group. To deal with a product of n ratios of characteristic polynomials, we let V = C^n \otimes C^N where C^N is equipped with its standard K-representation, and focus on the subspace of K-equivariant forms. By Howe duality, this is a highest-...