Quotients of complex algebraic supergroups

To the memory of V.S. Varadarajan
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Abstract

In this paper we prove that the etale sheafification of the functor arising from the quotient of an algebraic supergroup by a closed subsupergroup is representable by a smooth superscheme.

Introduction

The purpose of this paper is to provide a construction of the quotient of a complex algebraic supergroup by a closed subsupergroup. This construction is already available in a more general setting in the literature (see [17]), however here we present a different and more geometric proof, that is closer to the original approach by Chevalley (see [2] Ch. II).

We start by reviewing the ordinary construction. Suppose G is a complex algebraic group and H a closed subgroup. Then, G/H admits a unique algebraic variety structure, compatible with the group multiplication. In fact, there exists a rational representation of G in a finite dimensional vector space V and a line L in V whose stabilizer is H. Hence, we have an action of G on the projective space P(V) and H is the stabilizer subgroup of the point [L] in P(V). We can thus identify set-theoretically the quotient G/H with the orbit Y of the point [L]; Y being an orbit is also an algebraic variety, because of Chevalley's theorem. The uniqueness of this structure is obtained by the universal property of the quotient (see [2] Ch. II).

We want to replicate this geometric construction in the super setting. There are two major obstructions: the C-points of a supervariety do not carry enough information on its geometry, as it happens for the ordinary counterpart. Also, quotients of supergroups may not admit a projective embedding. We overcome the first difficulty by making use of the functor of points of superschemes and introducing etale coverings and etale sections, which mimic in some sense the differential approach to the construction of quotients (see [13], [1]). As for the latter problem, we replace the projective superspace with Grassmannian superschemes. In supergeometry the projective superspace appears somehow too rigid and it is necessary to allow for more general structures, as the Grassmannians. In this way we can realize an embedding of an orbit of a supergroup action into a suitable Grassmannian, hence identifying it with a smooth superscheme. In this sense, our proof will also provide a variation of the ordinary construction of quotients of complex algebraic groups and goes beyond a mere translation of the known recipe into the super context.

Our main result is the following.

Theorem 1.1

Let G be a complex algebraic supergroup, H a closed subsupergroup. Then, the sheafification in the etale topology of the functor TG(T)/H(T), T a superscheme, is representable in the category of superschemes, by a smooth superscheme.

We shall prove this result in several steps. In Sec. 2 we give some preliminaries and notation on algebraic supervarieties and superschemes, while in Sec. 3 we establish some results on smoothness. In Sec. 4 we prove the representability of the etale sheafification of the functor TG(T)/H(T), when H is the stabilizer of a point for an action of G on a superscheme. Finally, in Sec. 5 we give our main result, Theorem 5.5 and a comparison with [17] and the definition by Brundan in [3]. In the end we provide a section with some examples.

Acknowledgements. We are indebted with prof. V.S. Varadarajan for many illuminating discussions, for all the help and encouragement given to us through the many years of mutual interactions. We also thank Prof. D. Gieseker, Prof. A. Maffei, Prof. V. Serganova and Prof. T. Graber for helpful comments. Finally, we thank our Referee for suggestions that helped us to improve our paper. R.F. and S.D.K. wish to thank the UCLA Department of Mathematics for the kind hospitality while this work was done.

Section snippets

Supervarieties and superschemes

In this section we collect some facts of supergeometry. For more details see [6], [16], [4], [20], [15].

Let C be our ground field. Let (salg) be the category of commutative superalgebras and let A=A0A1(salg). Let us consider a non-zero fA0 and Af the localization of the A0-module A at f. The assignment:Uf:={x|Spec(A0)||f(x)0}Af defines a B-sheaf on |Spec(A0)|. Hence, there exists a unique sheaf of superalgebras OA on |Spec(A0)| such that OA|Uf=Af.

Definition 2.1

We define affine superscheme X associated

Smooth morphisms

We now introduce the notion of smooth morphism of relative dimension. For the ordinary setting see [19] Ch. 5.

Definition 3.1

A morphism of affine superschemesf:Spec_A[x1xm+r,ξ1,,ξn+s]/(f1,,fr,ϕ1,,ϕs)Spec_A is smooth of relative dimension m|n at a topological point x, if the rank of the Jacobian is maximal, i.e.rk(fi,ϕj)(xk,ξl)(x)=r|s (1ir, 1js, 1km+r, 1ln+s).

We say that a morphism of superschemes f:XY is smooth at x|X| of relative dimension m|n, if there exists two affine neighbourhoods UX

Etale sections and quotients

In this section we examine supergroup actions and homogeneous superspaces.

Definition 4.1

An affine supergroup is an affine supervariety whose functor of points is group valued, that is to say, it associates a group to each superalgebra.

If G is an affine supergroup, then G is a closed subgroup of GL(m|n) and the superalgebra O(G) has a natural Hopf superalgebra structure (see [4] Ch. 11). Furthermore, G is smooth (see [9]).

Definition 4.2

Let V be a super vector space. We define linear representation of G in V a group

Quotients

In this section we prove our main result.

Proposition 5.1

Let the G be an affine algebraic supergroup and H a closed subsupergroup. Then, there exists a finite dimensional representation ρ of G in V and a subspace WV, such that:H(T)={gG(T)|ρ(g)W=W},

Proof

See [4], 11.7.11. 

Once we fix suitable coordinates, the subsuperspace WV corresponds to a point p|Gr|, where Gr is the Grassmannian of r|s subsuperspaces of Cm|n, where r|s=dimW and m|n=dimV (see [4] Ch. 10 for the definition of Gr as superscheme and our Appendix

Examples

In this section we present some examples of the theory developed so far.

Example 6.1

Quotients of Chevalley Supergroups and flag superschemes. Let g be a complex contragredient Lie superalgebra (see [14]), with Cartan subalgebra h and root space decomposition g=h+αΔgα, Δ=Δ0Δ1. Let B={Hi,Xα}αΔ,1i be a Chevalley basis (see [10], Ch. 3) and let G:(sschemes)(sets) be the Chevalley supergroup obtained as in [10], Ch. 5, Def. 5.9, with ground field k=C (so g=gC in 5.2 in [10]):TG(T):=g0g1,g0G0(T0),g1=α

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