Irregular Riemann-Hilbert correspondence, Alekseev-Meinrenken dynamical r-matrices and Drinfeld twists
Irregular Riemann-Hilbert correspondence, Alekseev-Meinrenken dynamical r-matrices and Drinfeld twists
复制标题
不规则黎曼-希尔伯特对应、Alekseev-Meinrenken 动态 r 矩阵和 Drinfeld 扭曲
DOI：
--
复制
发表时间：
2015
期刊：
影响因子：
--
通讯作者：
Xiaomeng Xu
中科院分区：
文献类型：
--
作者：
Xiaomeng Xu
2004年，Enriquez-Etingof-马歇尔提出了一种新的方法来证明拟三角李双代数$（g，r）$的Ginzburg-Weinstein线性化定理。这种方法是基于解决一个系统的偏微分方程之间的经典r-矩阵r$和阿列克谢耶夫-迈仁肯动态r-矩阵的规范变换。他们证明了半经典极限的容许德林费尔德扭曲引起的解决方案的偏微分方程。
在本文中，我们解释了首选规范变换可以构造为某种非正则Riemann-Hilbert问题的连接映射（假设$r$是标准的经典r-矩阵）。沿着的方式，我们给他们的偏微分方程的辛几何解释，作为一个辛邻域版本的金斯堡-温斯坦线性化定理。我们的建筑是基于Boalch的早期作品。然后，我们证明了半单李代数$g$，规范变换的偏微分方程的任何解决方案是一个容许的德林费尔德扭曲的半经典极限。作为副产品，我们发现了一个令人惊讶的连接映射和德林费尔德扭曲之间的关系，以及一个新的描述卢-温斯坦辛双。
In 2004, Enriquez-Etingof-Marshall suggested a new approach to the Ginzburg-Weinstein linearization theorem for a quasitriangular Lie bialgebra $(g,r)$. This approach is based on solving a system of PDEs for a gauge transformation between the classical r-matrix $r$ and the Alekseev-Meinrenken dynamical r-matrix. They proved that the semiclassical limit of an admissible Drinfeld twist gives rise to a solution of the PDEs.
In this paper, we explain that preferred gauge transformations can be constructed as connection maps for a certain irregular Riemann-Hilbert problem (provided $r$ is the standard classical r-matrix). Along the way, we give a symplectic geometric interpretation of their PDEs, as a symplectic neighborhood version of the Ginzburg-Weinstein linearization theorem. Our construction is based on earlier works by Boalch. We then prove that for semisimple Lie algebra $g$, any solution of the PDEs for the gauge transformation is the semiclassical limit of an admissible Drinfeld twist. As byproducts, we find a surprising relation between the connection maps and Drinfeld twists as well as a new description of the Lu-Weinstein symplectic double.