19,576 research outputs found
Classical dynamical r-matrices and homogeneous Poisson structures on and
Let G be a finite dimensional simple complex group equipped with the standard
Poisson Lie group structure. We show that all G-homogeneous (holomorphic)
Poisson structures on , where is a Cartan subgroup, come
from solutions to the Classical Dynamical Yang-Baxter equations which are
classified by Etingof and Varchenko. A similar result holds for the maximal
compact subgroup K, and we get a family of K-homogeneous Poisson structures on
, where is a maximal torus of K. This family exhausts all
K-homogeneous Poisson structures on up to isomorphisms. We study some
Poisson geometrical properties of members of this family such as their
symplectic leaves, their modular classes, and the moment maps for the T-action
Hopf algebroids and quantum groupoids
We introduce the notion of Hopf algebroids, in which neither the total
algebras nor the base algebras are required to be commutative. We give a class
of Hopf algebroids associated to module algebras of the Drinfeld doubles of
Hopf algebras when the -matrices act properly. When this construction is
applied to quantum groups, we get examples of quantum groupoids, which are
semi-classical limits of Poisson groupoids. The example of quantum is
worked out in details.Comment: 30 pages, in Late
On a Dimension Formula for Twisted Spherical Conjugacy Classes in Semisimple Algebraic Groups
Let be a connected semisimple algebraic group over an algebraically
closed field of characteristic zero, and let be an automorphism of .
We give a characterization of -twisted spherical conjugacy classes in
by a formula for their dimensions in terms of certain elements in the Weyl
group of , generalizing a result of N. Cantarini, G. Carnovale, and M.
Costantini when is the identity automorphism. For simple and an
outer automorphism of , we also classify the Weyl group elements that appear
in the dimension formula.Comment: 8 page
Photonic Crystal Architecture for Room Temperature Equilibrium Bose-Einstein Condensation of Exciton-Polaritons
We describe photonic crystal microcavities with very strong light-matter
interaction to realize room-temperature, equilibrium, exciton-polariton
Bose-Einstein condensation (BEC). This is achieved through a careful balance
between strong light-trapping in a photonic band gap (PBG) and large exciton
density enabled by a multiple quantum-well (QW) structure with moderate
dielectric constant. This enables the formation of long-lived, dense 10~m
- 1~cm scale cloud of exciton-polaritons with vacuum Rabi splitting (VRS) that
is roughly 7\% of the bare exciton recombination energy. We introduce a
woodpile photonic crystal made of CdMgTe with a 3D PBG of 9.2\%
(gap to central frequency ratio) that strongly focuses a planar guided optical
field on CdTe QWs in the cavity. For 3~nm QWs with 5~nm barrier width the
exciton-photon coupling can be as large as \hbar\Ome=55~meV (i.e., vacuum
Rabi splitting 2\hbar\Ome=110~meV). The exciton recombination energy of
1.65~eV corresponds to an optical wavelength of 750~nm. For 106 QWs
embedded in the cavity the collective exciton-photon coupling per QW,
\hbar\Ome/\sqrt{N}=5.4~meV, is much larger than state-of-the-art value of
3.3~meV, for CdTe Fabry-P\'erot microcavity. The maximum BEC temperature is
limited by the depth of the dispersion minimum for the lower polariton branch,
over which the polariton has a small effective mass where
is the electron mass in vacuum. By detuning the bare exciton
recombination energy above the planar guided optical mode, a larger dispersion
depth is achieved, enabling room-temperature BEC
Mixed product Poisson structures associated to Poisson Lie groups and Lie bialgebras
We introduce and study some mixed product Poisson structures on product
manifolds associated to Poisson Lie groups and Lie bialgebras. For
quasitriangular Lie bialgebras, our construction is equivalent to that of
fusion products of quasi-Poisson G-manifolds introduced by Alekseev, Kosmann-
Schwarzbach, and Meinrenken. Our primary examples include four series of
holomorphic Poisson structures on products of flag varieties and related spaces
of complex semi-simple Lie groups.Comment: 37 pages, submitted to IMR
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