标题： 从广义哈密顿环面作用的泊松约化中得到的可积系统 显示英文标题

标题： Integrable systems from Poisson reductions of generalized Hamiltonian torus actions

摘要： 我们开发了一组充分条件，以保证在流形$M$上具有对称群$K$的可积系统可以下降到商泊松空间$M/K$的稠密开子集上的可积系统。 高维相空间$M$携带一个双向量$P_M$，在$C^\infty(M)$上产生一个括号，使得$C^\infty(M)^K$是一个泊松代数。 不可约系统在$M$上应具有“作用变量”，这些变量生成一个形式为$\mathrm{U}(1)^{\ell_1} \times \mathbb{R}^{\ell_2}$的群的适当且有效的动作，并下降为不可约系统的动作变量。 鉴于群的形式，且$P_M$可能是一个准泊松双向量，我们说我们处理的是广义哈密顿环面作用。 由于$\mathrm{U}(1)^{\ell_1} \times \mathbb{R}^{\ell_2}$的适当哈密顿作用的大量不变量，不可约系统通常是超可积的。 我们给出几个例子，并应用我们的构造来解决之前通过紧致李群的双重系统的约化得到的系统的可积性方面的开放问题：切丛、海森堡双重和准泊松双重。 此外，我们提供了众多应用，用于研究位于平坦连接模空间上的可积系统，使用准泊松方法。 显示英文摘要

摘要： We develop a set of sufficient conditions for guaranteeing that an integrable system with a symmetry group $K$ on a manifold $M$ descends to an integrable system on a dense open subset of the quotient Poisson space $M/K$. The higher dimensional phase space $M$ carries a bivector $P_M$ yielding a bracket on $C^\infty(M)$ such that $C^\infty(M)^K$ is a Poisson algebra. The unreduced system on $M$ is supposed to possess `action variables' that generate a proper, effective action of a group of the form $\mathrm{U}(1)^{\ell_1} \times \mathbb{R}^{\ell_2}$ and descend to action variables of the reduced system. In view of the form of the group and since $P_M$ could be a quasi-Poisson bivector, we say that we work with a generalized Hamiltonian torus action. The reduced systems are in general superintegrable owing to the large set of invariants of the proper Hamiltonian action of $\mathrm{U}(1)^{\ell_1} \times \mathbb{R}^{\ell_2}$. We present several examples and apply our construction for solving open problems regarding the integrability of systems obtained previously by reductions of master systems on doubles of compact Lie groups: the cotangent bundle, the Heisenberg double and the quasi-Poisson double. Furthermore, we offer numerous applications to integrable systems living on moduli spaces of flat connections, using the quasi-Poisson approach.