论文标题
托有泊松在紧凑型谎言组上的双重托管系统减少
Poisson reductions of master integrable systems on doubles of compact Lie groups
论文作者
论文摘要
我们考虑了任何半圣经的三个“古典双打”，连接并简单地连接的紧凑型谎言组$ g $：cotangent Bundle，Heisenberg double和内部融合的Quasi-poisson double。在每个双倍上，我们都会确定一对“主体可集成系统”，并研究其泊松减少。在最简单的cotangent捆绑案例中，减少的定义是通过$ g $本身的共轭动作的cotangengent提升来定义的，这自然而然地概括为其他两个双打。在每种情况下，我们都会为减少的泊松结构和运动方程提供明确的公式，并发现它们与众所周知的经典动力学$ r $ - amatrices相关。我们的主要结果是，我们提供了一个统一的统一处理，其中包含新模型以及旋转Sutherland和Ruijsenaars的示例 - 先前已经研究过的Schneider模型。我们认为，在泊松商的通用符号叶子上，减少的系统在堕落的意义上是可以集成的，尽管需要进一步的工作来严格证明这一点。
We consider three 'classical doubles' of any semisimple, connected and simply connected compact Lie group $G$: the cotangent bundle, the Heisenberg double and the internally fused quasi-Poisson double. On each double we identify a pair of 'master integrable systems' and investigate their Poisson reductions. In the simplest cotangent bundle case, the reduction is defined by taking quotient by the cotangent lift of the conjugation action of $G$ on itself, and this naturally generalizes to the other two doubles. In each case, we derive explicit formulas for the reduced Poisson structure and equations of motion and find that they are associated with well known classical dynamical $r$-matrices. Our principal result is that we provide a unified treatment of a large family of reduced systems, which contains new models as well as examples of spin Sutherland and Ruijsenaars--Schneider models that were studied previously. We argue that on generic symplectic leaves of the Poisson quotients the reduced systems are integrable in the degenerate sense, although further work is required to prove this rigorously.
