Let $(R^{vee},R)$ be a dual pair of Hopf algebras in the category of Yetter-Drinfeld modules over a Hopf algebra $H$ with bijective antipode. We show that there is a braided monoidal isomorphism between rational left Yetter-Drinfeld modules over the
bosonizations of $R$ and of $R^{vee}$, respectively. As an application of this very general category isomorphism we obtain a natural proof of the existence of reflections of Nichols algebras of semisimple Yetter-Drinfeld modules over $H$. Key words: Hopf algebras, quantum groups, Weyl groupoid
We study graded right coideal subalgebras of Nichols algebras of semisimple Yetter-Drinfeld modules. Assuming that the Yetter-Drinfeld module admits all reflections and the Nichols algebra is decomposable, we construct an injective order preserving a
nd order reflecting map between morphisms of the Weyl groupoid and graded right coideal subalgebras of the Nichols algebra. Here morphisms are ordered with respect to right Duflo order and right coideal subalgebras are ordered with respect to inclusion. If the Weyl groupoid is finite, then we prove that the Nichols algebra is decomposable and the above map is bijective. In the special case of the Borel part of quantized enveloping algebras our result implies a conjecture of Kharchenko. Key words: Hopf algebra, quantum group, root system, Weyl group
We report the discovery of very shallow (DF/F = 3.4 10-4), periodic dips in the light curve of an active V = 11.7 G9V star observed by the CoRoT satellite, which we interpret as due to the presence of a transiting companion. We describe the 3-colour
CoRoT data and complementary ground-based observations that support the planetary nature of the companion. Methods. We use CoRoT color information, good angular resolution ground-based photometric observations in- and out- of transit, adaptive optics imaging, near-infrared spectroscopy and preliminary results from Radial Velocity measurements, to test the diluted eclipsing binary scenarios. The parameters of the host star are derived from optical spectra, which were then combined with the CoRoT light curve to derive parameters of the companion. We examine carefully all conceivable cases of false positives, and all tests performed support the planetary hypothesis. Blends with separation larger than 0.40 arcsec or triple systems are almost excluded with a 8 10-4 risk left. We conclude that, as far as we have been exhaustive, we have discovered a planetary companion, named CoRoT-7b, for which we derive a period of 0.853 59 +/- 3 10-5 day and a radius of Rp = 1.68 +/- 0.09 REarth. Analysis of preliminary radial velocity data yields an upper limit of 21 MEarth for the companion mass, supporting the finding. CoRoT-7b is very likely the first Super-Earth with a measured radius.
The physical characterization of exoplanets will require to take spectra at several orbital positions. For that purpose, a direct imaging capability is necessary. Direct imaging requires an efficient stellar suppression mechanism, associated with an
ultrasmooth telescope. We show that before future large space missions (interferometer, 4-8 m class coronograph, external occulter or Fresnel imager), direct imaging of giant planets and close-by super-Earth are at the cross-road of a high scientific interest and a reasonable feasibility. The scientific interest lies in the fact that super-Earths share common geophysical attributes with Earths. They already begin to be detected by radial velocity (RV) and, together with giant planets, they have a larger area than Earths, making them detectable with a 1.5-2 m class telescope in reflected light. We propose such a (space) telescope be a first step before large direct imaging missions.
The Super-Earth Explorer is an Off-Axis Space Telescope (SEE-COAST) designed for high contrast imaging. Its scientific objective is to make the physico-chemical characterization of exoplanets possibly down to 2 Earth radii >. For that purpose it will
analyze the spectral and polarimetric properties of the parent starlight reflected by the planets, in the wavelength range 400-1250 nm
