Symplectic extensions of the Kirillov-Kostant and Goldman Poisson structures and Fuchsian systems.
Symplectic extensions of the Kirillov-Kostant and Goldman Poisson structures and Fuchsian systems.
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基里洛夫-科斯坦特和戈德曼泊松结构以及福克斯系统的辛扩张。
DOI：
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发表时间：
2019
期刊：
影响因子：
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通讯作者：
D. Korotkin
中科院分区：
文献类型：
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作者：
M. Bertola;D. Korotkin
我们重新讨论了黎曼球面上Fuchsian系统的单调映射的辛性质。推广了Hitchin，Aleksev-Malkin和Korotkin-Samtleben的结果，证明了Mondromy映射是介于系数空间上的Kirillov-Kostant Poisson结构和Mondromy特征标集上的Goldman括号之间的Poisson态射。这种延伸是通过在两侧定义更大的空间来提供的，这些空间配备了自然地投影到正则结构的辛结构。在系数方面，我们的辛结构对应于由有理动力$r-矩阵表示的非退化二次泊松结构；当投影到标准空间时，它退化为Kirillov-Kostant括号。在单边，我们得到了一个辛结构，它诱导了Goldman Poisson括号的叶上的Aleksev-Malkin的辛结构。我们利用Malgrange和其中一位作者的形式证明了单向映射提供了一个辛同构。作为推论，我们证明了A.Its，O.Liosovyy和A.Prokhorov最近的猜想在其“强”版本中，而原始的“弱”版本是从先前已知的结果得到的。我们还证明了等单形的Jimbo-Miwa tau函数与这种变换的母函数密切相关。
We revisit symplectic properties of the monodromy map for Fuchsian systems on the Riemann sphere. We extend previous results of Hitchin, Alekseev-Malkin and Korotkin-Samtleben where it was shown that the monodromy map is a Poisson morphism between the Kirillov-Kostant Poisson structure on the space of coefficients, on one side, and the Goldman bracket on the monodromy character variety on the other. The extension is provided by defining larger spaces on both sides which are equipped with symplectic structures naturally projecting to the canonical ones. On the coefficient side our symplectic structure corresponds to a non-degenerate quadratic Poisson structure expressed via the rational dynamical $r$-matrix; it reduces to the Kirillov-Kostant bracket when projected to the standard space. On the monodromy side we get a symplectic structure which induces the symplectic structure of Alekseev-Malkin on the leaves of the Goldman Poisson bracket. We prove that the monodromy map provides a symplectomorphism using the formalism of Malgrange and one of the authors. As a corollary we prove the recent conjecture by A.Its, O.Lisovyy and A.Prokhorov in its "strong" version while the original "weak" version is derived from previously known results. We show also that the isomonodromic Jimbo-Miwa tau-function is intimately related to a generating function of such transformation.