Quadratic algebras related to elliptic curves

We construct quadratic finite-dimensional Poisson algebras and their quantum versions related to rank N and degree one vector bundles over elliptic curves with n marked points. The algebras are parameterized by the moduli of curves. For N=2 and n=1 they coincide with the Sklyanin algebras. We prove that the Poisson structure is compatible with the Lie-Poisson structure on the direct sum of n copies of sl(N). The derivation is based on the Poisson reduction from the canonical brackets on the affine space over the cotangent bundle to the groups of automorphisms of vector bundles.Comment: 21 page

Bruhat Order in the Full Symmetric sln\mathfrak{sl}_n Toda Lattice on Partial Flag Space

In our previous paper [Comm. Math. Phys. 330 (2014), 367-399] we described the asymptotic behaviour of trajectories of the full symmetric $\mathfrak{sl}_n$ Toda lattice in the case of distinct eigenvalues of the Lax matrix. It turned out that it is completely determined by the Bruhat order on the permutation group. In the present paper we extend this result to the case when some eigenvalues of the Lax matrix coincide. In that case the trajectories are described in terms of the projection to a partial flag space where the induced dynamical system verifies the same properties as before: we show that when $t\to\pm\infty$ the trajectories of the induced dynamical system converge to a finite set of points in the partial flag space indexed by the Schubert cells so that any two points of this set are connected by a trajectory if and only if the corresponding cells are adjacent. This relation can be explained in terms of the Bruhat order on multiset permutations

Gimbals Drive and Control Electronics Design, Development and Testing of the LRO High Gain Antenna and Solar Array Systems

Launched June 18, 2009 on an Atlas V rocket, NASA's Lunar Reconnaissance Orbiter (LRO) is the first step in NASA's Vision for Space Exploration program and for a human return to the Moon. The spacecraft (SC) carries a wide variety of scientific instruments and provides an extraordinary opportunity to study the lunar landscape at resolutions and over time scales never achieved before. The spacecraft systems are designed to enable achievement of LRO's mission requirements. To that end, LRO's mechanical system employed two two-axis gimbal assemblies used to drive the deployment and articulation of the Solar Array System (SAS) and the High Gain Antenna System (HGAS). This paper describes the design, development, integration, and testing of Gimbal Control Electronics (GCE) and Actuators for both the HGAS and SAS systems, as well as flight testing during the on-orbit commissioning phase and lessons learned