Global splittings and super Harish-Chandra pairs for affine supergroups

Abstract

This paper dwells upon two aspects of affine supergroup theory, investigating the links among them. First, the “splitting” properties of affine supergroups are discussed, i.e., special kinds of factorizations they may admit — either globally, or pointwise. Almost everything should be more or less known, but seems to be not as clear in the literature (to the author’s knowledge) as it ought to. Second, a new contribution to the study of affine supergroups by means of super Harish-Chandra pairs is presented (a method already introduced by Koszul, and later extended by other authors). Namely, a new functorial construction Ψ \Psi is provided 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. One knows that there exists a natural functor Φ \Phi from affine supergroups to super Harish-Chandra pairs. Then we show that the new functor Ψ \Psi — which goes the other way round — is indeed a quasi-inverse to Φ \Phi \, , provided we restrict our attention to the subcategory of affine supergroups that are gs-split. Therefore, (the restrictions of) Φ \Phi and Ψ \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, via different approaches. Nevertheless, the novelty in the present paper lies in the construction of a different functor Ψ \Psi and thus extends the result to a much larger setup, with a totally different, more geometrical method. In fact, this method (very concrete, indeed) is universal and characteristic-free and is presented here for the algebro-geometric setting, but actually it can be easily adapted to the frameworks of differential or complex analytic supergeometry. The case of linear supergroups is treated also as an intermediate, inspiring step. Some examples, applications and further generalizations are presented at the end of the paper.