Algebraic Supergroups Research Articles

Jacobson-Morozov Lemma for algebraic supergroups

Given a quasi-reductive algebraic supergroup G, we use the theory of semisimplifications of symmetric monoidal categories to define a symmetric monoidal functorΦx:Rep(G)→Rep(OSp(1|2)) associated to any given element x∈Lie(G)1¯. For nilpotent elements x, we show that the functor Φx can be defined using the Deligne filtration associated to x.We use this approach to prove an analogue of the Jacobson-Morozov Lemma for algebraic supergroups. Namely, we give a necessary and sufficient condition on odd nilpotent elements x∈Lie(G)1¯ which define an embedding of supergroups OSp(1|2)→G so that x lies in the image of the corresponding Lie algebra homomorphism.

Quotients of complex algebraic supergroups

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.

Affine algebraic super-groups with integral

We generalize to the super context, the known fact that if an affine algebraic group G over a commutative ring k acts freely (in an appropriate sense) on an affine scheme X over k, then the dur sheaf of G-orbits is an affine scheme in the following two cases: (I) G is finite; (II) k is a field, and G is linearly reductive. An emphasize is put on the more difficult generalization in the second case; the replaced assumption then is that an affine algebraic super-group G over an arbitrary field has an integral. Those super-groups which satisfy the assumption are characterized, and are seen to form a large class if Hopf-algebraic techniques including bosonization are applied to prove the results.

Centers of centralizers of nilpotent elements in Lie superalgebras 𝔰𝔩(m|n) or 𝔬𝔰𝔭(m|2n)

Let [Formula: see text] be the simple algebraic supergroup [Formula: see text] or [Formula: see text] over [Formula: see text]. Let [Formula: see text] and let [Formula: see text], where [Formula: see text] is considered as a superalgebra concentrated in even degree. Suppose [Formula: see text] is nilpotent. We describe the centralizer [Formula: see text] of [Formula: see text] in [Formula: see text] and its center [Formula: see text]. In particular, we give bases for [Formula: see text], [Formula: see text] and [Formula: see text]. We also determine the labeled Dynkin diagram [Formula: see text] with respect to [Formula: see text] and subsequently describe the relation between [Formula: see text] and [Formula: see text].

GEOMETRIC CONSTRUCTION OF QUOTIENTS G/H IN SUPERSYMMETRY

It was proved by the first-named author and Zubkov [13] that given an affine algebraic supergroup $\mathbb{G}$ and a closed sub-supergroup $\mathbb{H}$ over an arbitrary field of characteristic $\ne 2$, the faisceau $\mathbb{G} \tilde{/} \mathbb{H}$ (in the fppf topology) is a superscheme, and is, therefore, the quotient superscheme $\mathbb{G}/\mathbb{H}$, which has desirable properties, in fact. We reprove this, by constructing directly the latter superscheme $\mathbb{G}/\mathbb{H}$. Our proof describes explicitly the structure sheaf of $\mathbb{G}/\mathbb{H}$, and reveals some new geometric features of the quotient, that include one which was desired by Brundan [2], and is shown in general, here for the first time.

Borel-Weil theorem for algebraic supergroups

We study the structure of an algebraic supergroup ▪ and establish the Borel-Weil theorem for ▪ to give a systematic construction of all simple supermodules over an arbitrary field. Especially when ▪ has a distinguished parabolic super-subgroup, we show that the set of all simple supermodules of ▪ is parameterized by the set of all dominant weights for the even part of ▪, prove a super-analogue of the Kempf vanishing theorem, and give a description of Euler characteristics.

Tensor ideals, Deligne categories and invariant theory

We derive some tools for classifying tensor ideals in monoidal categories. We use these results to classify tensor ideals in Deligne’s universal categories $${\underline{\mathrm{Rep}}} O_\delta ,{\underline{\mathrm{Rep}}} GL_\delta $$ and $${\underline{\mathrm{Rep}}} P$$ . These results are then used to obtain new insight into the second fundamental theorem of invariant theory for the algebraic supergroups of types A, B, C, D, P. We also find new short proofs for the classification of tensor ideals in $${\underline{\mathrm{Rep}}} S_t$$ and in the category of tilting modules for $${\text {SL}}_2(\Bbbk )$$ with $$\mathrm{char}(\Bbbk )>0$$ and for $$U_q(\mathfrak {sl}_2)$$ with q a root of unity. In general, for a simple Lie algebra $$\mathfrak {g}$$ of type ADE, we show that the lattice of such tensor ideals for $$U_q(\mathfrak {g})$$ corresponds to the lattice of submodules in a parabolic Verma module for the corresponding affine Kac–Moody algebra.

Modular representations of exceptional supergroups

We classify the simple modules of the exceptional algebraic supergroups over an algebraically closed field of prime characteristic.

On functor points of affine supergroups

To construct an affine supergroup from a Harish-Chandra pair, Gavarini [2] invented a natural method, which first constructs a group functor and then proves that it is representable. We give a simpler and more conceptual presentation of his construction in a generalized situation, using Hopf superalgebras over a superalgebra. As an application of the construction, given a closed super-subgroup of an algebraic supergroup, we describe the normalizer and the centralizer, using Harish-Chandra pairs. We also prove a tensor product decomposition theorem for Hopf superalgebras, and describe explicitly by cocycle deformation, the difference which results from the two choices of dualities found in literature.

Algebraic supergroups and Harish-Chandra pairs over a commutative ring

We prove a category equivalence between algebraic supergroups and Harish-Chandra pairs over a commutative ring which is $2$-torsion free. The result is applied to re-construct the Chevalley $\mathbb{Z}$-supergroups constructed by Fioresi and Gavarini [8] and by Gavarini [9, 10]. For a wide class of algebraic supergroups we describe their representations by using their super-hyperalgebras.

Solvability and nilpotency for algebraic supergroups

We study solvability, nilpotency and splitting property for algebraic supergroups over an arbitrary field K of characteristic charK≠2. Our first main theorem tells us that an algebraic supergroup G is solvable if the associated algebraic group Gev is trigonalizable. To prove it we determine the algebraic supergroups G such that dim⁡Lie(G)1=1; their representations are studied when Gev is diagonalizable. The second main theorem characterizes nilpotent connected algebraic supergroups. A super-analogue of the Chevalley Decomposition Theorem is proved, though it must be in a weak form. An appendix is given to characterize smooth Noetherian superalgebras as well as smooth Hopf superalgebras.

Solvable, reductive and quasireductive supergroups

It is well known that if the ground field K has characteristic zero and G is a connected algebraic group, defined over K, then the Lie algebra Lie(G′) of the commutant G′ of G coincides with the commutant Lie(G)′ of Lie(G). We show that this result is no longer true in the category of algebraic supergroups. We also construct a reductive supergroup H=X⋊G, where X and G are connected, reduced and abelian supergroups, such that Xu≠1 and (Hev)u is non-trivial connected (super)group. Quasi-reductive supergroups have been introduced in [10]. We prove that a supergroup H is quasi-reductive if and only if the largest even (super)subgroup of the solvable radical R(H) is a torus, H˜=H/R(H) contains a normal supersubgroup U, which is quasi-isomorphic to a direct product of normal supersubgroups Ui, and H˜/U is a triangulizable supergroup with odd unipotent radical. Moreover, for every i, Lie(Ui)=Ui⊗Sym(ni) are such that either ni=0 and Ui is a classical simple Lie superalgebra, or ni=1 and Ui is a simple Lie algebra.

Corrigendum to: Algebraic supergroups of Cartan type

Abstract In this note I fix a mistake in my previous paper [5]: namely, the result concerning the uniqueness (up to isomorphisms) of such supergroups needs a new formulation and proof. By the same occasion, I explain more in detail the existence result which comes out of the construction of Chevalley supergroups.

Solvable and Unipotent Supergroups

It is proved that an algebraic supergroup G is unipotent iff G ev is unipotent. Here our reasoning involves only induction on dimension and some properties of the adjoint representation. In a similar way, it is shown that over a field of characteristic zero, a connected supergroup G is solvable iff G ev is solvable.

Bernstein–Gelfand–Gelfand Reciprocity and Indecomposable Projective Modules for Classical Algebraic Supergroups

We prove a BGG type reciprocity law for the category of finite dimensional modules over algebraic supergroups satisfying certain conditions. The equivalent of a standard module in this case is a virtual module called Euler characteristic due to its geometric interpretation. In the orthosymplectic case, we also describe indecomposable projective modules in terms of those Euler characteristics.

Algebraic supergroups of Cartan type

Abstract I present a construction of connected affine algebraic supergroups 𝐆 V associated with simple Lie superalgebras 𝔤 of Cartan type and with 𝔤-modules V. Conversely, I prove that every connected affine algebraic supergroup whose tangent Lie superalgebra is of Cartan type is necessarily isomorphic to one of the supergroups 𝐆 V that I introduced. In particular, the supergroup associated in this way with 𝔤 = W ( n ) $ \mathfrak {g}= W(n) $ and its standard representation is described.

Chevalley supergroups

In the framework of algebraic supergeometry, we give a construction of the scheme-theoretic supergeometric analogue of Chevalley groups, namely affine algebraic supergroups associated to simple Lie superalgebras of classical type. In particular both basic (even exceptional) and strange cases are covered. This provides a unified approach to most of the algebraic supergroups considered so far in literature, and an effective method to construct new ones. As an intermediate step, we prove an existence theorem for Chevalley bases of simple classical Lie superalgebras and a PBW-like theorem for their associated Kostant superalgebras.

Modular branching rules for projective representations of symmetric groups and lowering operators for the supergroup 𝑄(𝑛)

There are two approaches to projective representation theory of symmetric and alternating groups, which are powerful enough to work for modular representations. One is based on Sergeev duality, which connects projective representation theory of the symmetric group and representation theory of the algebraic supergroup $Q(n)$ via appropriate Schur (super)algebras and Schur functors. The second approach follows the work of Grojnowski for classical affine and cyclotomic Hecke algebras and connects projective representation theory of symmetric groups in characteristic $p$ to the crystal graph of the basic module of the twisted affine Kac-Moody algebra of type $A_{p-1}^{(2)}$. The goal of this work is to connect the two approaches mentioned above and to obtain new branching results for projective representations of symmetric groups.

Quotient sheaves of algebraic supergroups are superschemes

To generalize some fundamental results on group schemes to the super context, we study the quotient sheaf G / ˜ H of an algebraic supergroup G by its closed supersubgroup H, in arbitrary characteristic ≠ 2. Our main theorem states that G / ˜ H is a Noetherian superscheme. This together with derived results give positive answers to interesting questions posed by J. Brundan.

On Lie Superalgebras of Algebraic Supergroups

Let G be an algebraic supergroup over a field k, whose pure-even group scheme G ev is a reduced algebraic group scheme, i.e., [Formula: see text] is reduced. In this paper, we prove that Lie (G) can be identified with the Lie superalgebra of admissible left-invariant derivations of k[G].