Bouloc D, Miranda E, Zung N T
Institut de Mathématiques de Toulouse, UMR5219, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse, France
Laboratory of Geometry and Dynamical Systems-EPSEB, Department of Mathematics-UPC and BGSMath, Universitat Politècnica de Catalunya, Avinguda del Doctor Marañon 44-50, 08028 Barcelona, Spain.
Philos Trans A Math Phys Eng Sci. 2018 Sep 17;376(2131):20170423. doi: 10.1098/rsta.2017.0423.
In this paper, we show that every singular fibre of the Gelfand-Cetlin system on co-adjoint orbits of unitary groups is a smooth isotropic submanifold which is diffeomorphic to a two-stage quotient of a compact Lie group by free actions of two other compact Lie groups. In many cases, these singular fibres can be shown to be homogeneous spaces or even diffeomorphic to compact Lie groups. We also give a combinatorial formula for computing the dimensions of all singular fibres, and give a detailed description of these singular fibres in many cases, including the so-called (multi-)diamond singularities. These (multi-)diamond singular fibres are degenerate for the Gelfand-Cetlin system, but they are Lagrangian submanifolds diffeomorphic to direct products of special unitary groups and tori. Our methods of study are based on different ideas involving complex ellipsoids, Lie groupoids and also general ideas coming from the theory of singularities of integrable Hamiltonian systems.This article is part of the theme issue 'Finite dimensional integrable systems: new trends and methods'.
在本文中，我们证明了酉群余伴随轨道上格尔范德 - 采特林系统的每一个奇异纤维都是一个光滑各向同性子流形，它与一个紧致李群被另外两个紧致李群的自由作用所得到的两阶段商微分同胚。在许多情况下，这些奇异纤维可被证明是齐性空间，甚至与紧致李群微分同胚。我们还给出了一个用于计算所有奇异纤维维数的组合公式，并在许多情形下对这些奇异纤维进行了详细描述，包括所谓的（多）菱形奇点。这些（多）菱形奇异纤维对于格尔范德 - 采特林系统是退化的，但它们是与特殊酉群和环面的直积微分同胚的拉格朗日子流形。我们的研究方法基于涉及复椭球、李群胚的不同思想，以及来自可积哈密顿系统奇点理论的一般思想。本文是主题为“有限维可积系统：新趋势与方法”的一部分。
