Classical dynamical r-matrices and homogeneous Poisson structures on G/HG/H and K/TK/T

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 $G/H$, where $H \subset G$ 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 $K/T$, where $T = K \cap H$ is a maximal torus of K. This family exhausts all K-homogeneous Poisson structures on $K/T$ 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 $R$-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 $sl(2)$ is worked out in details.Comment: 30 pages, in Late

On a Dimension Formula for Twisted Spherical Conjugacy Classes in Semisimple Algebraic Groups

Let $G$ be a connected semisimple algebraic group over an algebraically closed field of characteristic zero, and let $\th$ be an automorphism of $G$. We give a characterization of $\th$-twisted spherical conjugacy classes in $G$ by a formula for their dimensions in terms of certain elements in the Weyl group of $G$, generalizing a result of N. Cantarini, G. Carnovale, and M. Costantini when $\th$ is the identity automorphism. For $G$ simple and $\th$ an outer automorphism of $G$, 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~$\mu$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 Cd$_{0.6}$Mg$_{0.4}$Te 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 $N=$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 $\sim 10^{-5}m_0$ where $m_0$ 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