Hall algebra approach to Drinfeld's presentation of quantum loop algebras

The quantum loop algebra $U_{v}(\mathcal{L}\mathfrak{g})$ was defined as a generalization of the Drinfeld's new realization of the quantum affine algebra to the loop algebra of any Kac-Moody algebra $\mathfrak{g}$. It has been shown by Schiffmann that the Hall algebra of the category of coherent sheaves on a weighted projective line is closely related to the quantum loop algebra $U_{v}(\mathcal{L}\mathfrak{g})$, for some $\mathfrak{g}$ with a star-shaped Dynkin diagram. In this paper we study Drinfeld's presentation of $U_{v}(\mathcal{L}\mathfrak{g})$ in the double Hall algebra setting, based on Schiffmann's work. We explicitly find out a collection of generators of the double composition algebra \mathbf{DC}(\Coh(\mathbb{X})) and verify that they satisfy all the Drinfeld relations.Comment: 31 pages, revised versio

High-contrast coronagraph for ground-based imaging of Jupiter-like planets

We propose a high-contrast coronagraph for direct imaging of young Jupiter-like planets orbiting nearby bright stars. The coronagraph employs a step-transmission filter in which the intensity is apodized with a finite number of steps of identical transmission in each step. It should be installed on a large ground-based telescope equipped with state-of-the-art adaptive optics systems. In that case, contrast ratios around 10^-6 should be accessible within 0.1 arc seconds of the central star. In recent progress, a coronagraph with circular apodizing filter has been developing, which can be used for a ground-based telescope with central obstruction and spider structure. It is shown that ground-based direct imaging of Jupiter-like planets is promising with current technology