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, I discuss the "splitting" properties of affine supergroups, i.e. special kinds of factorizations they may admit -either globally, or point-wise. Almost everything should be more or less known, but seems to be not as clear in literature (to the author's knowledge) as it ought to.Second, I present a new contribution to the study of affine supergroups by means of super HarishChandra pairs (a method already introduced by Koszul, and later extended by other authors). Namely, Iprovide a new functorial construction Ψ 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 from affine supergroups to super Harish-Chandra pairs: then I show that the new functor Ψ -which goes the other way round -is indeed a quasi-inverse to Φ , provided we restrict our attention to the subcategory of affine supergroups that are gs-split. Therefore, (the restrictions of) Φ and Ψ 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 (see [16], [19], [7]): nevertheless, the novelty in the present paper lies in that I construct a different functor Ψ and thus extend 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: I present it 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. 1

Solvability and nilpotency for algebraic supergroups

“…Remark 3.4. Theorem 4.23 of [18] generalizes the equivalence ASG → HCP above, replacing K with a commutative ring, say R, which is 2-torsion free in the sense that 2 : R → R is injective; see also [8,19]. Working over such a ring one poses in [18] some additional assumptions to the objects, to define ASG and HCP; it would, however, deserve to remark that the splitting property (3.4), which is included in the added assumptions, was proved to hold, very recently by [19], and so it is needless to assume.…”

“…The definition below is reproduced from [17]. Its presentation is slightly different from the one in fashion found in [2,3,8,28], and is indeed a translation of [16,Definition 7] given in Hopf-algebraic terms.…”

Algebraic supergroups and Harish-Chandra pairs over a commutative ring

“…After an earlier version of this paper was submitted, the article [11] by Gavarini was in circulation. Theorem 4.3.14 of [11] essentially proves our category equivalence theorem in the generalized situation that k is an arbitrary commutative ring.…”

“…After an earlier version of this paper was submitted, the article [11] by Gavarini was in circulation. Theorem 4.3.14 of [11] essentially proves our category equivalence theorem in the generalized situation that k is an arbitrary commutative ring. A point is to use the additional structure, called 2-operations, on Lie superalgebras g, which generalizes the map v → 1 2 [v, v], g 1 → g 0 given on an admissible Lie superalgebra in our situation.…”

“…In the appendix of this paper we will refine his category equivalence, using our construction and giving detailed arguments on 2-operations, in particular. This would not be meaningless because such detailed arguments are not be given in [11]; see Remark A.11.…”