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Poisson Lie groups and their relations to quantum groups

Janusz Grabowski (1995)

Banach Center Publications

The notion of Poisson Lie group (sometimes called Poisson Drinfel'd group) was first introduced by Drinfel'd [1] and studied by Semenov-Tian-Shansky [7] to understand the Hamiltonian structure of the group of dressing transformations of a completely integrable system. The Poisson Lie groups play an important role in the mathematical theories of quantization and in nonlinear integrable equations. The aim of our lecture is to point out the naturality of this notion and to present basic facts about...

Poisson structures on certain moduli spaces for bundles on a surface

Johannes Huebschmann (1995)

Annales de l'institut Fourier

Let Σ be a closed surface, G a compact Lie group, with Lie algebra g , and ξ : P → Σ a principal G -bundle. In earlier work we have shown that the moduli space N ( ξ ) of central Yang-Mills connections, with reference to appropriate additional data, is stratified by smooth symplectic manifolds and that the holonomy yields a homeomorphism from N ( ξ ) onto a certain representation space Rep ξ ( Γ , G ) , in fact a diffeomorphism, with reference to suitable smooth structures C ∞ ( N ( ξ ) ) and C ∞ Rep ξ ( Γ , G ) , where Γ denotes the universal central extension of...

Propagation des singularités pour les opérateurs différentiels de type principal localement résolubles à coefficients analytiques en dimension 2

Paul Godin (1979)

Annales de l'institut Fourier

Sur une variété analytique paracompacte de dimension 2, on considère un opérateur différentiel P à symbole principal p m analytique vérifiant la condition ( 𝒫 ) de Nirenberg et Treves. En ajoutant une nouvelle variable et en utilisant des estimations a priori de type Carleman, on montre qu’il y a propagation des singularités pour P , dans p m - 1 ( 0 ) , le long des feuilles intégrales du système différentiel engendré par les champs hamiltoniens de Re p m et Im p m .

Reduction by group symmetry of second order variational problems on a semidirect product of Lie groups with positive definite riemannian metric

Claudio Altafini (2004)

ESAIM: Control, Optimisation and Calculus of Variations

For a riemannian structure on a semidirect product of Lie groups, the variational problems can be reduced using the group symmetry. Choosing the Levi-Civita connection of a positive definite metric tensor, instead of any of the canonical connections for the Lie group, simplifies the reduction of the variations but complicates the expression for the Lie algebra valued covariant derivatives. The origin of the discrepancy is in the semidirect product structure, which implies that the riemannian exponential...

Reduction by group symmetry of second order variational problems on a semidirect product of Lie groups with positive definite Riemannian metric

Claudio Altafini (2010)

ESAIM: Control, Optimisation and Calculus of Variations

For a Riemannian structure on a semidirect product of Lie groups, the variational problems can be reduced using the group symmetry. Choosing the Levi-Civita connection of a positive definite metric tensor, instead of any of the canonical connections for the Lie group, simplifies the reduction of the variations but complicates the expression for the Lie algebra valued covariant derivatives. The origin of the discrepancy is in the semidirect product structure, which implies that the Riemannian exponential...

Relativistic stability of matter (I).

Charles L. Fefferman, Rafael de la Llave (1986)

Revista Matemática Iberoamericana

In this article, we study the quantum mechanics of N electrons and M nuclei interacting by Coulomb forces. Motivated by an important idea of Chandrasekhar and following Herbst [H], we modify the usual kinetic energy -∆ to take into account an effect from special relativity. As a result, the system can implode for unfavorable values of the nuclear charge Z and the fine structure constant α. This is analogous to the gravitational collapse of a heavy star. Our goal here is to find those values of α...

Remarks on the Lichnerowicz-Poisson cohomology

Izu Vaisman (1990)

Annales de l'institut Fourier

The paper begins with some general remarks which include the Mayer-Vietoris exact sequence, a covariant version of the Lichnerowicz-Poisson cohomology, and the definition of an associated Serre-Hochshild spectral sequence. Then we consider the regular case, and we discuss the Poisson cohomology by using a natural bigrading of the Lichnerowicz cochain complex. Furthermore, if the symplectic foliation of the Poisson manifold is either transversally Riemannian or transversally symplectic, the spectral...

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