论文标题
分层梯度哈密顿矢量场和集体可集成系统
Stratified Gradient Hamiltonian Vector Fields and Collective Integrable Systems
论文作者
论文摘要
对于标准的Lie-Poisson结构，我们在任何紧凑的Lie Group $ k $的双重代数上构建了完全可集成的系统。 These systems generalize key properties of Gelfand-Zeitlin systems: A) the pullback to any Hamiltonian $K$-manifold defines a Hamiltonian torus action on an open dense subset, B) if the $K$-manifold is multiplicity-free, then the resulting torus action is \textit{completely} integrable, and C) the collective moment map has convexity and fiber connectedness properties.这些系统概括了Gelfand-Zeitlin Systems与Gelfand-Zeitlin规范基础之间通过实际极化量化的关系。 为了构建这些系统，我们通过光滑的投影品种将田径化的构造概括为田园和卡维的整合系统，以构造奇异的准标记品种。在某些条件下，我们表明，这种变性的分层梯度哈密顿矢量场是定义的。
We construct completely integrable systems on the dual of the Lie algebra of any compact Lie group $K$ with respect to the standard Lie-Poisson structure. These systems generalize key properties of Gelfand-Zeitlin systems: A) the pullback to any Hamiltonian $K$-manifold defines a Hamiltonian torus action on an open dense subset, B) if the $K$-manifold is multiplicity-free, then the resulting torus action is \textit{completely} integrable, and C) the collective moment map has convexity and fiber connectedness properties. These systems generalize the relationship between Gelfand-Zeitlin systems and Gelfand-Zeitlin canonical bases via geometric quantization by a real polarization. To construct these systems, we generalize Harada and Kaveh's construction of integrable systems by toric degeneration on smooth projective varieties to singular quasi-projective varieties. Under certain conditions, we show that the stratified-gradient Hamiltonian vector field of such a degeneration, which is defined piece-wise, has a flow whose limit exists and defines continuous degeneration map.
