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PROJECTIVE LOOPS GENERATE RATIONAL LOOP GROUPS
Journal of the Institute of Mathematics of Jussieu ( IF 0.9 ) Pub Date : 2020-08-17 , DOI: 10.1017/s1474748020000171
Gang Wang 1 , Oliver Goertsches 2 , Erxiao Wang 3
Affiliation  

We generalize Uhlenbeck’s generator theorem of ${\mathcal{L}}^{-}\operatorname{U}_{n}$ to the full rational loop group ${\mathcal{L}}^{-}\operatorname{GL}_{n}\mathbb{C}$ and its subgroups ${\mathcal{L}}^{-}\operatorname{GL}_{n}\mathbb{R}$ , ${\mathcal{L}}^{-}\operatorname{U}_{p,q}$ : they are all generated by just simple projective loops. Recall that Terng–Uhlenbeck studied the dressing actions of such projective loops as generalized Bäcklund transformations for integrable systems. Our result makes a nice supplement: any rational dressing is the composition of these Bäcklund transformations. This conclusion is surprising in the sense that Lie theory suggests the indispensable role of nilpotent loops in the case of noncompact reality conditions, and nilpotent dressings appear quite complicated and mysterious. The sacrifice is to introduce some extra fake singularities. So we also propose a set of generators if fake singularities are forbidden. A very geometric and physical construction of $\operatorname{U}_{p,q}$ is obtained as a by-product, generalizing the classical construction of unitary groups.



中文翻译:

投影循环生成有理循环组

我们将 Uhlenbeck 的${\mathcal{L}}^{-}\operatorname{U}_{n}$生成定理推广到全有理循环群${\mathcal{L}}^{-}\operatorname{GL }_{n}\mathbb{C}$及其子群${\mathcal{L}}^{-}\operatorname{GL}_{n}\mathbb{R}$ , ${\mathcal{L}} ^{-}\operatorname{U}_{p,q}$ :它们都是由简单的投影循环生成的。回想一下,Terng-Uhlenbeck 研究了可积系统的广义 Bäcklund 变换等射影环的修整作用。我们的结果是一个很好的补充:任何合理的敷料都是这些 Bäcklund 变换的组成。这个结论是令人惊讶的,因为李理论表明在非紧实实条件的情况下,幂零环的不可或缺的作用,并且幂零敷料显得相当复杂和神秘。牺牲是引入一些额外的假奇点。因此,如果禁止假奇点,我们还提出了一组生成器。$\operatorname{U}_{p,q}$的一个非常几何和物理的构造 是作为副产品获得的,概括了酉群的经典构造。

更新日期:2020-08-17
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