Algebraic Supergroups Research Articles

Affine Quotients of Supergroups

In this paper we consider sheaf quotients of affine superschemes by affine supergroups that act on them freely. The necessary and sufficient conditions for such quotients to be affine are given. If G is an affine supergroup and H is its normal supersubgroup, then we prove that a dur K-sheaf \( \mathop {G/H}\limits^ \approx \) is again an affine supergroup. Additionally, if G is algebraic, then a K-sheaf \( \mathop {G/H}\limits^ \sim \) is also an algebraic supergroup and it coincides with \( \mathop {G/H}\limits^ \approx \). In particular, any normal supersubgroup of an affine supergroup is faithfully exact.

Smoothness of algebraic supervarieties and supergroups

In this paper we discuss the notion of smoothness in complex algebraic supergeometry and we prove that all affine complex algebraic supergroups are smooth. We then prove the stabilizer theorem in the algebraic context, providing some useful applications.

Real forms of complex Lie superalgebras and complex algebraic supergroups

The paper concerns two versions of the notion of real forms of Lie superalgebras. One is the standard approach, where a real form of a complex Lie superalgebra is a real Lie superalgebra whose complexification is the original complex Lie superalgebra. The second arises from considering A-points of a Lie superalgebra over a commutative complex superalgebra A equipped with superconjugation. The first kind of real form can be obtained as the set of fixed points of an antilinear involutive automorphism; the second is related to an automorphism f such that f2 is the identity on the even part and the negative identity on the odd part. The generalized notion of real forms is then introduced for complex algebraic supergroups.

On Algebraic Supergroups, Coadjoint Orbits and Their Deformations

In this paper we study algebraic supergroups and their coadjoint orbits as affine algebraic supervarieties. We find an algebraic deformation quantization of them that can be related to the fuzzy spaces of non commutative geometry.

ON ALGEBRAIC SUPERGROUPS AND QUANTUM DEFORMATIONS

We give the definitions of affine algebraic supervariety and affine algebraic supergroup through the functor of points and we relate them to the other definitions present in the literature. We study in detail the algebraic supergroups GL(m|n) and SL(m|n) and give explicitly the Hopf algebra structure of the algebra representing the functors of points. In the end we also give the quantization of GL(m|n) together with its coaction on suitable quantum spaces according to Manin's philosophy.

Modular representations of the supergroup Q(n), I

The representation theory of the algebraic supergroup Q(n) has been studied quite intensively over the complex numbers in recent years, especially by Penkov and Serganova [18, 19, 20] culminating in their solution [21, 22] of the problem of computing the characters of all irreducible finite dimensional representations ofQ(n). The characters of one important family of irreducible representations, the so-called polynomial representations, had been determined earlier by Sergeev [24], exploiting an analogue of Schur-Weyl duality connecting polynomial representations of Q(n) to the representation theory of the double covers Ŝn of the symmetric groups. In [2], we used Sergeev’s ideas to classify for the first time the irreducible representations of Ŝn over fields of positive characteristic p > 2. In the present article and its sequel, we begin a systematic study of the representation theory of Q(n) in positive characteristic, motivated by its close relationship to Ŝn. Let us briefly summarize the main facts proved in this article by purely algebraic techniques. Let G = Q(n) defined over an algebraically closed field k of characteristic p 6= 2, see §§2-3 for the precise definition. In §4 we construct the superalgebra Dist(G) of distributions on G by reduction modulo p from a Kostant Z-form for the enveloping superalgebra of the Lie superalgebra q(n,C). This provides one of the main tools in the remainder of the paper: there is an explicit equivalence between the category of representations of G and the category of “integrable” Dist(G)supermodules (see Corollary 5.7). In §6, we classify the irreducible representations of G by highest weight theory. They turn out to be parametrized by the set X p (n) = {(λ1, . . . , λn) ∈ Z | λ1 ≥ · · · ≥ λn with λi = λi+1 only if p|λi}. For λ ∈ X+ p (n), the corresponding irreducible representation is denoted L(λ), and is constructed naturally as the simple socle of an induced representation H(λ) := indBu(λ) where B is a Borel subgroup of G and u(λ) is a certain irreducible representation of B of dimension a power of 2. The main difficulty here is to show that H0(λ) 6= 0 for λ ∈ X+ p (n), which we prove by exploiting the main result of [2] classifying the irreducible polynomial representations of G: so ultimately the proof that H0(λ) 6= 0 depends on a counting argument involving p-regular conjugacy classes in Ŝn.

SUPERGROUPS, QUANTUM SUPERGROUPS AND THEIR HOMOGENEOUS SPACES

We give a more algebraic definition of algebraic supergroup via its Hopf algebra. The Hopf algebra of the supergroup SL (m|n) is given together with its quantization. The example of supercoadjoint orbits is examined at the end.