Global splittings and super Harish-Chandra pairs for affine supergroups
Global splittings and super Harish-Chandra pairs for affine supergroups
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仿射超群的全局分裂和超 Harish-Chandra 对
DOI：
10.1090/tran/6456
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发表时间：
2013
期刊：
影响因子：
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通讯作者：
F. Gavarini
中科院分区：
文献类型：
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作者：
F. Gavarini
本文详细讨论了仿射超群理论的两个方面，研究了它们之间的联系。
首先，我讨论了仿射超群的“分裂”性质，即它们可能允许的特殊类型的因子分解-无论是全局的，还是逐点的。
其次，我提出了一个新的贡献，研究仿射超群的超哈里什-钱德拉对（一种方法已经介绍了Koszul，后来扩展了其他作者）。也就是说，我提供了一个显式的函子构造\Psi，它与每个超Harish-Chandra对关联一个始终全局强分裂的仿射超群（简称GS-分裂），从而与论文的第一部分建立了联系。另一方面，存在一个从仿射超群到超Harish-Chandra对的自然函子\Phi：然后我证明了新的函子\Psi -反过来-确实是\Phi的拟逆，只要我们把注意力限制在gs-分裂的仿射超群的子范畴上。因此，\Phi和\Psi（的限制）是gs-分裂仿射超群和超Harish-Chandra对范畴之间的等价。这样的结果在其他背景下是已知的，例如光滑微分或复解析的，或者在某些特殊情况下，通过不同的方法：本文的新奇之处在于，我构造了一个不同的函子\Psi，从而将结果扩展到更大的设置，使用完全不同的，更几何的方法（非常具体，并且特征自由）。
线性代数群的情况下也被视为一个中间，鼓舞人心的一步。
文末给出了一些例子、应用和进一步的推广。
This paper dwells upon two aspects of affine supergroup theory, investigating the links among them.
First, I discuss the "splitting" properties of affine supergroups, i.e. special kinds of factorizations they may admit - either globally, or pointwise.
Second, I present a new contribution to the study of affine supergroups by means of super Harish-Chandra pairs (a method already introduced by Koszul, and later extended by other authors). Namely, I provide an explicit, functorial construction \Psi which, with each super Harish-Chandra pair, associates an affine supergroup that is always globally strongly split (in short, gs-split) - thus setting a link with the first part of the paper. On the other hand, there exists a natural functor \Phi from affine supergroups to super Harish-Chandra pairs: then I show that the new functor \Psi - which goes the other way round - is indeed a quasi-inverse to \Phi, provided we restrict our attention to the subcategory of affine supergroups that are gs-split. Therefore, (the restrictions of) \Phi and \Psi are equivalences between the categories of gs-split affine supergroups and of super Harish-Chandra pairs. Such a result was known in other contexts, such as the smooth differential or the complex analytic one, or in some special cases, via different approaches: the novelty in the present paper lies in that I construct a different functor \Psi and thus extend the result to a much larger setup, with a totally different, more geometrical method (very concrete indeed, and characteristic free).
The case of linear algebraic groups is treated also as an intermediate, inspiring step.
Some examples, applications and further generalizations are presented at the end of the paper.