Global splittings and super Harish-Chandra pairs for affine supergroups

TL;DR

This paper establishes an equivalence between categories of globally split affine supergroups and super Harish-Chandra pairs, using a new functor that extends known results to a broader, characteristic-free geometric setting.

Contribution

It introduces a new functor  that constructs globally strongly split affine supergroups from super Harish-Chandra pairs, extending existing equivalences to a larger, more general context.

Findings

The functor  is a quasi-inverse to  in the category of gs-split affine supergroups.

The equivalence of categories is established in a characteristic-free, geometric framework.

Examples and applications demonstrate the broad applicability of the results.

Abstract

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.