有理顶及其经典R-矩阵
Rational Top and its Classical R-matrix
摘要：
在${\rm SL}_N$李群的余伴随轨道上构造了一个有理可积系统（有理顶）.它由带谱参数的Lax算子和经典的非动力学反对称$r$ -矩阵描述。在最小维轨道的情况下，该模型与有理Calogero-Moser（CM）系统是规范等价的。为了获得结果，我们将CM模型的Lax算子表示为两种不同的因式分解形式--不带谱参数（与无自旋情况相关）和另一种带谱参数。后者给出了合理的顶部，而第一个是有关广义Cremmer-Gervais $r$ -矩阵。与有理顶和CM模型相关的规范变换提供了IRF-Vertex对应的经典有理版本。从几何的角度描述了退化椭圆曲线上${\rm SL}(N,\mathbb C)$ -丛的变形。从辛Hecke对应的观点出发，将有理陀螺与有理自旋CM模型联系起来.建议的建设可能的应用和推广进行了讨论。特别地，所获得的$r$ -矩阵定义了一类KZB方程。
英文摘要：
We construct a rational integrable system (the rational top) on a coadjoint orbit of ${\rm SL}_N$ Lie group. It is described by the Lax operator with spectral parameter and classical non-dynamical skew-symmetric $r$-matrix. In the case of the orbit of minimal dimension the model is gauge equivalent to the rational Calogero-Moser (CM) system. To obtain the results we represent the Lax operator of the CM model in two different factorized forms -- without spectral parameter (related to spinless case) and another one with the spectral parameter. The latter gives rise to the rational top while the first one is related to generalized Cremmer-Gervais $r$-matrices. The gauge transformation relating the rational top and CM model provides a classical rational version of the IRF-Vertex correspondence. From a geometrical point of view it describes the modification of ${\rm SL}(N,\mathbb C)$-bundles over degenerated elliptic curve. In view of Symplectic Hecke Correspondence the rational top is related to the rational spin CM model. Possible applications and generalizations of the suggested construction are discussed. In particular, the obtained $r$-matrix defines a class of KZB equations.
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