On supergroups and their semisimplified representation categories

The representation category $\mathcal{A} = Rep(G,\epsilon)$ of a supergroup scheme $G$ has a largest proper tensor ideal, the ideal $\mathcal{N}$ of negligible morphisms. If we divide $\mathcal{A}$ by $\mathcal{N}$ we get the semisimple representation category of a pro-reductive supergroup scheme $G^{red}$. We list some of its properties and determine $G^{red}$ in the case $GL(m|1)$.


Introduction
A fundamental fact about finite-dimensional algebraic representations of a reductive group over an algebraically closed field k of characteristic 0 is complete reducibility: Every representation decomposes into a direct sum of irreducible representations. This is no longer true if we consider representations of supergroups. Indeed by a classical result of Djokovic-Hochschild [DH76] the representation category of a Lie superalgebra g is semisimple if and only if g is a semisimple Lie algebra or of the form osp(1|2n) for n ≥ 1. Correspondingly many standard techniques from Lie theory do not work for representations of supergroups. Although a lot of progress has been made on representations of special supergroups such as Gl(m|n) and OSp(m|2n), many classical questions are still open, most notably the tensor product decompositon of two irreducible representations. The category Rep(G), G a supergroup, is a tensor category. Every k-linear tensor category has a largest proper tensor ideal N , the tensor ideal of negligible morphisms. By [AK02] the quotient category ω : Rep(G) → Rep(G)/N is an abelian semisimple k-linear tensor category. 0.1 Theorem. (Theorem 2.2) The quotient Rep(G)/N is a super-tannakian category, i.e. it is of the form Rep(G red , ǫ) where G red is a supergroup scheme with semisimple representation category.
This result follows immediately from a characterization of representation categories due to Deligne [Del02]. A natural question is to understand and possibly determine G red for given G. This is very difficult and not even possible in the general case. We assemble a few general results about these quotients and then focus on the Gl(m|n)-case (m ≥ n).
We show that the classification of the irreducible representations of G red is a wild problem for n ≥ 3 in theorem 3.5. Hence the question should be modified as follows: We should study the subcategory in Rep(G red , ǫ) generated by the images ω(L(λ)) of the irreducible representations of G. To determine this subcategory would amount to determine the tensor product decomposition of irreducible representations up to superdimension 0 and would give a parametrization of the indecomposable summands of non-vanishing superdimension. We study this problem in [HW15] in the case of Gl(n|n). The cases Gl(m|1) and Sl(m|1) are rather special since the blocks are of tame representation type and the indecomposable representations have been classified [Ger98] and we can hope to determine the entire quotient category. From the classification it is easy to determine the irreducible elements in G red in lemma 4.2. We then compute their tensor product decomposition in theorem 4.12.
In this statement we view Gl(m|1) red as a supergroup with trivial odd part and Rep(G red ) as the corresponding category of super representations. In order to determine the tensor product decomposition we use two tools: The theory of mixed tensors [Hei14] gives us the tensor product decomposition between the irreducible Gl(m|1)-representations. We then use cohomological tensor functors DS : Rep(Gl(m|1)) → Rep(Gl(m − 1)) akin to those of [DS05] [HW14] to reduce the tensor product decomposition between indecomposables to the irreducible case. The main point here is that DS(V ) is a Z-graded object for any V , hence DS could be interpreted as a functor to Z × Rep(Gl(m − 1)).
To determine G red is probably in reach for the simple supergroups of maximal atypicality 1. To determine the subgroup of G red corresponding to the irreducible representations is already very difficult for Gl(m|n) and even more so for the other simple supergroups OSp(m|2n), P (n) and Q(n).

Preliminaries
Super Linear Algebra. Throughout the article k is an algebraically closed field of characteristic 0. A super vector space is a finite-dimensional Z 2 -graded vector space V = V 0 ⊕ V 1 over k. Elements in V 0 respectively V 1 are called even respectively odd. An element is homogenuous if it is either even or odd. For a homogenuous element v write p(v) for the parity defined by We denote by Hom(V, W ) the set of k-linear parity-preserving morphism between two super vector spaces V and W . The parity shift functor Π : svec → svec is defined by (ΠV ) 0 = V 1 , (ΠV ) 1 = V 0 and on morphisms f : where v is viewed as an element of ΠV and f (v) as an element of ΠW .
A superring is a superalgebra if it is also a super vector space over k. We denote by salg the category of commutative superalgebras with parity preserving morphisms. A group functor G : salg → sets is called an affine supergroup scheme G if G is a representable group functor. It is called a supergroup if the representing superalgabra is finitely generated. We define the functor Gl(m|n) from the category of commutative superalgebras to the category of groups by sending A = A 0 ⊕ A 1 to the invertible (m + n) × (m + n) matrices of the form where a is an (m× m)-matrix with entries in A 0 , b is an (m× n)-matrix with entries in A 1 , c is an (n × m)-matrix with entries in A 1 and d is an (n × n)-matrix with entries in A 0 . A morphism f : A → B is send to the map that sends the matrix (X ij ) ∈ Gl(m|n)(A) to the matrix (f (X ij )) ∈ Gl(m|n)(B). This group functor is called the General Linear supergroup.
Representations. We define the group functor Gl V : salg → sets to be the functor that assigns to each commutative superalgebra A the even invertible elements of End A (V ⊗A). For finite-dimensional V Gl V is a supergroup. Let G be a group functor and V ∈ svec a finite-dimensional super vector space. A linear representation of G in V is a morphism of group functors G → Gl V . If G has a linear representation on V , we call V a G-module. We denote the category of finite-dimensional representations of G by Rep(G).
Parity automorphisms. Let G be a supergroup scheme and let ǫ be an element of G(k) of order dividing 2 such that the automorphism int(ǫ) of G is the parity automorphism defined by x → (−1) p(x) x for homogenuous x. Then let Rep(G, ǫ) be the category of (finite-dimensional) representations V = (V, ρ) such that ρ(ǫ) is the parity automorphism of V . If G is an affine group scheme, ǫ is central. In this case the category Rep(G, ǫ) identifies itself with Rep(G) with a new commutativity constraint: For every representation (V, ρ) of G the involution ρ(ǫ) defines a Z 2graduation on V and the commutativity isomorphism of the tensor product is given by the Koszul rule. If ǫ is trivial, one recovers Rep(G). For the supergroup Gl(m|n) and ǫ = diag(E m , −E n ) we put Rep(Gl(m|n), ǫ) = R m|n . For the whole category Rep(Gl(m|n)) we also write T m|n . Then T m|n = R m|n ⊕ ΠR m|n [HW14].
The categories T m|n and R m|n are examples of super-Tannakian categories. For background on tensor categories we refer to [DM82]. We denote the unit object of a tensor category by 1. For a partition λ of n let S λ (−) be the associated Schur functor [Del02]. We put Sym n (X) = S (n) (X) (the n-th symmetric power) and Λ n (X) = S (1,...,1) (X) (the n-th alternating power). An object X of A is called Schur-finite if there exists an integer n and a partition λ of n such that S λ (X) = 0.

Theorem. [Del02]
If A is an abelian k-linear rigid tensor category with End(1) ≃ k such that every object is Schur finite, then A is a super tannakian category, i.e. A ≃ Rep(G, ǫ) for some supergroup scheme G.
Following [Ger98] we call a small abelian k-linear category nice if morphism spaces are finite-dimensional, every object has a finite composition series and the category has enough projectives. An example is given by the category R m|n .
1.4 Lemma. [Ger98], lemma 1.1.1. If A is a nice category then (1) The endomorphism ring of any indecomposable object is a local ring.
(2) Every object can be written as a direct sum of indecomposable objects.
(3) Every module has a unique projective cover 2. The universal semisimple quotient 2.1. The universal semisimple quotient. An additive category A is a Krull-Schmidt category if every object has a decomposition in a finite direct sum of elements with local endomorphism rings. An ideal in a k-linear category is for any two objects X, Y the specification of a k-submodule T (X, Y ) of Hom A (X, Y ), such that for all pairs of morphisms f ∈ Hom A (X, Let T be an ideal in A. A/T is the category with the same objects as A and with Hom A/T (X, Y ) = Hom A (X, Y )/T (X, Y ). It is again a Krull-Schmidt category [Liu09], [KZ08]. Suppose that A is abelian and that every object has finite length and let X be an indecomposable element and φ an endomorphism of X. By Fitting's lemma φ is either invertible or nilpotent. An element X is indecomposable if and only if its endomorphism ring is a local ring. We assume in the following that A is a super tannakian category or a pseudoabelian full tensor subcategory. Then all the above conditions hold. An ideal in a tensor category is a tensor ideal if it is stable under id C ⊗ − and − ⊗ id C for all C ∈ A. The ideal is then stable under tensor products from left or right with arbitrary morphisms. Let T r be the trace. The quotient A/N will be called the universal semisimple quotient of A.
2.2 Theorem. The quotient A/N is again a super tannakian category. If A ′ ⊂ A is a pseudoabelian full tensor subcategory, the quotient A ′ /(N ∩ A ′ ) is a super tannakian category.
Proof. The quotient of a k-linear rigid tensor category by a tensor ideal is again a k-linear rigid tensor category. Since N is a tensor ideal the quotient functor ω : A → A/N is a tensor functor. The quotient category is semisimple by construction.
Since Hom-spaces are finite-dimensional one has idempotent lifting, hence A/N is pseudoabelian. A k-linear semisimple pseudoabelian category is abelian by [AK02]. By [Del02] an abelian tensor category is super tannakian if and only if for every object A there exists a Schur functor S µ with S µ (A) = 0. Since ω(S µ (A)) = S µ (ω(A)) any object in A/N is also annulated by a Schur functor.
The category A/N has the following universal property.
2.3 Proposition. Let ω : A → C be a full tensor functor into a semisimple tensor category C. Then ω factorises over the quotient A/N .
Proof. Since C is semisimple there are no negligible morphisms. However the image of a negligible morphism is negligible, since a tensor functor commutes with traces. Hence the image of a negligible morphism under ρ is zero, hence the functor factorizes.
For completeness sake we assemble a few elementary lemmas about this quotient.

2.4
Lemma. An object X of A maps to zero in A/N if and only if id X belongs to N (X, X).
The collection of these elements -called negligible objects -is denoted by N . The dimension of an object X in a tensor category is defined T r Proof. If X ∈ N we have T r(g) = 0 for all g ∈ End(X), in particular for g = id X . Let sdim(X) = 0. We have to show: id X ∈ N (X, X), ie. T r(g) = 0 for all g ∈ End(X). Since X is indecomposable g is either nilpotent or an isomorphism. If g is nilpotent T r(g) = 0 [Bru00], 1.4.3. Let g be an isomorphism. Since X is indecomposable g has a unique eigenvalue λ and T r(g) = λsdim(X), hence T r(g) = 0. Proof. Let X be indecomposable, X / ∈ N . Since A and A/N are abelian and every object has finite lenght, an object X is indecomposable if and only if End(X) is a local ring. We have End A/N (X) = End A (X)/N (X). Since the quotient of a local ring by a (two-sided) ideal is again local, the image of 2.7 Corollary. (i) N is closed under direct sums and direct summands. (ii) If X ∈ N and Y ∈ A, we have X⊗Y in N and each indecomposable summand of X⊗Y has superdimension 0. (iii) Let X / ∈ N and let X = X i be its decomposition into indecomposable elements. Then Hom A/N (X, X) = i, sdim(Xi) =0 k.
2.2. The pro-reductive envelope. Since the quotient A/N is again a supertannakian category, this defines a reductive super group scheme G red with A/N ≃ Rep(G red , ǫ) with ǫ : µ 2 → G such that the operation of µ 2 gives the Z 2 -graduation of the representations. We call G red the pro-reductive envelope of G (following [AK02]). If G is an algebraic group, the pro-reductive envelope has been extensively studied by Andre and Kahn. Their proofs do not apply to the supergroup case. In the tannakian case N = R is equal to the radical ideal. In particular no indecomposable elements maps to zero. Even in the tannakian case the pro-reductive cover will not be of finite type in general.
2.8 Theorem. [AK02], theorem C.5. The proreductive envelope of an affine kgroup G is of finite type over k if and only if G is of finite type over k and the prounipotent radical of G is of dimension ≤ 1.
Consider two examples. If G = G a , then G red = Sl(2). If G = G a ×G a , then G red is no longer of finite type. In fact, the determination of G ֒→ G red is unsolvable since it would include a classification of the indecomposable representations of G which is a wild problem [AK02], 19.7. More generally it seems plausible that Rep(G red , ǫ) is of finite or tame type if and only if Rep(G) is of finite or tame type. It is likely that if Rep(G) is of wild type, the problem of classifying indecomposable modules of non-vanishing superdimension is wild as well. Therefore we should not try to determine G red in this case, but ask the following weaker questions: Given any object V ∈ Rep(G) or Rep(G, ǫ), consider its image in A/N . The tensor category generated by it is a semisimple algebraic tensor category (since A/N is semisimple). The semisimple algebraic tensor categories in characteristic zero were classified in [Wei09]: 2.9 Theorem. Any supergroup G over k such that Rep(G) is semisimple is isomorphic to a semidirect product G ′ ⊳ H of a reductive algebraic k-group H with a product G ′ = r≥1 Spo(1|2r) nr of simple supergroups of BC-type, where the semidirect product is defined by an abstract group homomorphism p : Now consider an irreducible object V ∈ Rep(G, ǫ) and consider the tensor category generated by ω(V ) in Rep(G, ǫ)/N . This tensor subcategory corresponds to an algebraic group G V ֒→ G red . Then this group is of finite type since it has a tensor generator. It is reductive since Rep(G red , ǫ) is semisimple.
2.10 Lemma. Let T be a maximal torus in G V and X * (T ) its character group. Let R be the subgroup generated by the roots of G V . Then the center of G V has cyclic character group X/R and (G V ) 0 der has cyclic center.
Proof. ω(V ) is a tensor generator of Rep(G V ) for the reductive group G V and likewise ω(V ) is a tensor generator of Rep(G V ) 0 der . Now use that a reductive group has a faithful irreducible representation if and only if X/R is cyclic and a semisimple group has a faithful irreducible representation if and only if its center is cyclic [McN].
2.3. The basic classical cases. Let G be basic classical [Ser11b] with underlying basic classical Lie superalgebra g [Kac78]. Duflo and Serganova [DS05] and [Ser11a] constructed for certain elements where g x is a classical Lie algebra or osp(1|2n). These functors are not full, hence need not factorize over the quotient Rep(G)/N . However it should be expected that G red contains groups G x with Lie superalgebra g x . For instance the superdimension of any irreducible representation in Rep(G) equals the superdimension of some representation in Rep(G red ) and in Rep(g x ). Note that this representation in Rep(g x ) might not be irreducible.
3. On the Gl(m|n)-case 3.1. Preliminaries. Let g = gl(m|n) denote the Lie superalgebra of Gl(m|n) with even part g 0 = gl(m) ⊕ gl(n). The Lie superalgebra has a Z-grading gl(m|n) = g −1 ⊕ g0 ⊕ g1 with g 0 = g0 and g 1 = g −1 ⊕ g1 [Kac78]. Let h be the Cartan algebra of diagonal matrices in g. We denote by ǫ i the usual basis elements of h * [Ger98]. For λ ∈ h * let L 0 (λ) be the simple g 0 -module of highest weight λ ∈ h * relative to the Borel subalgebra of upper triangular matrices b 0 . The g 0 -module L 0 (λ) can be extended trivially to g 0 ⊕ g 1 . The Kac-module and the AntiKac-module are by definition [Kac78]  In particular the simple g-modules are up to a parity shift parametrised by the same set of highest weights as the simple g 0 -modules. Hence the (integral dominant) highest weights X + of gl(m|n) are of the form λ = (λ 1 , . . . , λ m |λ m+1 , . . . , λ m+n ).
Here λ 1 ≥ . . . ≥ λ m and λ m+1 ≥ . . . ≥ λ m+n are integers and every λ ∈ Z m+n with these properties parametrises a highest weight of an irreducible g-module. This set of highest weights is called X + , so that the irreducible modules in Rep(g) are given by the {L(λ), ΠL(λ) | λ ∈ X + } where Π denotes the parity shift. In the sl(m|n)case we have to identify two irreducible modules whose highest weights differ only from a weight of the form (k, k, . . . , k| − k, . . . , −k) for k ∈ Z. We say that a module is a Kac-object if it has a filtration whose subquotients are Kac-modules. The full subcategory of these modules is denoted C + . Similarly we have the category C − of objects which have a filtration by AntiKac-modules. By [Ger98] C + ∩ C − = P roj.
Both C + and C − are tensor ideals in R. If K(λ) is irreducible the weight λ is called typical. If not, λ is called atypical. K(λ) is irreducible if and only if K(λ) is projective as a g-module [Kac78]. It is well known that Rep(Gl(m|n)) identifies with the category of integrable supermodules over gl(m|n) [BS12].
3.1.1. Weight diagrams. To each highest weight λ ∈ X + we associate, following [BS12], two subsets of cardinality n of the numberline Z 3.2. Wildness. If A is a nice category we can associate to it its Ext-quiver. The vertex set X + is given by the set of isomorphism classes of simple modules and the number of arrows from λ to µ is given by Ext 1 A (L(λ), L(µ)). A block Γ of X + is a connected component of the Ext-quiver. Let A Γ be the full subcategory of objects of A such that all composition factors are in Γ (also called a block). This gives a decomposition A = Γ A Γ of full abelian subcategories. Every indecomposable module lies in a unique A Γ and all its simple submodules belong to Γ. Two irreducible representations L(λ) and L(µ) are in the same block if and only if the weights λ and µ define labelings with the same position of the labels × and •. The degree of atypicality is a block invariant, and the blocks Γ of atypicality r are in 1-1 correspondence with pairs of disjoint subsets of Z of cardinality m − r resp. n − r.
3.4 Theorem. Assume m, n ≥ 2. Then Gl(m|n) red is not of finite type.
Proof. This follows from the description of the Tannaka group generated by the irreducible elements in [HW15].
The statement also follows from the following lemma. This lemma should of course also hold for m, n ≥ 2, but would require a more difficult argument. Let Q denote the Ext-quiver of R mn . Then there exists a system of relations R on Q such that R mn ≃ kQ/R − mod.
3.5 Lemma. Assume m, n ≥ 3. Then the problem of classifying indecomposable representations of non-vanishing superdimension is wild.
Proof. We show that that the classication is wild for every maximally atypical block for n ≥ 3. Any such block is equivalent to the maximal atypical block Γ of Gl(n|n) Since this subquiver has no path of length > 1, it embeds fully into k(Q)/R. The classifaction of indecomposable representations of the r-subspace quiver is wild for r ≥ 5. The superdimension formula of [Wei10] [HW14] shows that the superdimension is constant of alternating sign away from the diagonal: if [λ] has superdimension d, the [µ i ] have superdimension −d. Hence an indecomposable representation of this subquiver will give an indecomposable representation in Γ of non-vanishing superdimension if and only if We are done when we have shown that the classification of indecomposable representations with ( * ) is wild. Fix the vertex [µ 2n ] and consider an indecomposable representation of the (2n−1)-subspace quiver by specifying a vector space for the vertices . This defines a bijection between the isomorphism classes of indecomposable representations of the (2n − 1)-subspace quiver with a subset of the indecomposable representations of the 2n-subspace quiver satisfying ( * ).
As explained above this means that we should not try to determine G red for general m, n. Instead we should determine the tensor category generated by the irreducible elements in A/N and the corresponding reductive supergroup. In the n = 1-case, where we have tame representation type, we show in this article: 3.6 Theorem. We have: This theorem should be understood in the super sense, i.e. Rep(Gl(m|1))/N ≃ Rep(Gl(m − 1) × Gl(1) × Gl(1)) ⊗ svec k . We prove this by describing explicitely the image of an indecomposable representation I.

4.
Determination of G red for G = Gl(m|1)

Irreducible elements.
Assume from now on that we are in the Gl(m|1)-case and assume that weights are singly atypical. Recall that Kac-modules have a simple socle. The highest weight of the socle is denoted by T − λ. The highest weight of the socle of the AntiKac-module K ′ (λ) is denoted by T + λ. If λ is atypical K(λ) is an extension of L(λ) by L(T − λ): Similarly we have the exact sequence By [Ger98] this sequences are up to equivalence all non-trivial extensions between simple modules: Ext 1 A (L(λ), L(µ)) = C for µ ∈ {T + λ, T − λ} and zero else. An irreducible element is mapped to zero under ω if and only if it is typical or, equivalently, projective.

Indecomposable representations.
We recall the results about the classification of indecomposable modules in the singly atypical case obtained by [Su00] and [Ger98]. We parametrise an atypical block as in [Ger98] by Z and denote the corresponding weight with a ∈ Z. By Germoni the indecomposable modules are either the Kac objects C + or the AntiKac objects C − or V sits in an exact sequence with U ∈ C + and L irreducible, or Q sits in an exact sequence with U ∈ C + and L ′ irreducible. In down to earth terms: Fix an arbitrary a ∈ Z (that is, an arbitrary weight a in the block, or its corresponding simple module L(a)). 4.2. The toy example sl(2|1). The tensor products of the indecomposable modules have been computed by [GQS07] and can be used to compute the formulas in the quotient.
4.3 Lemma. The tensor product of the irreducible modules in A/N is given by the following rules: Proof. This is an inspection using [GQS07]. The tensor products between ZigZagmodules are given by Proposition 4 in loc.cit. as a direct sum of a T -part and a Θ-part. Since ω preserves direct sums we omit any projective module in the formulas. By definiton T (, ) consists of a direct sum of typical modules (formula (44)). For the contributions of the Θ-part see p.836 in loc.cit. Note that Θ maps projective modules to projective modules. The gl(1|1)-formulas yield then the above result.
We use the following reparametrization: We put The irreducible elements in A/N are then parametrized by Z 2 × Z × Z. The Z 2 comes from the fact that we can apply the parity shift Π to an indecomposable representation. The rules for the tensor products read now: Note that this is exactly the tensor product for the group Gl(1) × Gl(1). For the proof of the next statement recall that every representation can be replaced by its parity shift so that its superdimension is positive.
Proof. We assume without loss of generality that the superdimension of all objects (p, q) is positive. We have to define a functor ρ : A/N → Rep(Gl(1) × Gl(1)) which is an equivalence of tensor categories. Use the parametrisation above of the irreducible elements in A/N by Z × Z. Define ρ on objects by mapping the irreducible element corresponding to (p, q) ∈ Z × Z to the irreducible representation t p ⊗ t q of Gl(1) × Gl(1). Note that Hom-spaces are either zero or one-dimensional by Schur's lemma. The results on tensor products show that this is a tensor functor.

Mixed tensors.
We first determine the contribution from the irreducible modules. The subcategory T ⊂ R mn (m ≥ n) of mixed tensors is the pseudoabelian full subcategory of objects, which are direct summands in a tensor product V ⊗r ⊗ (V ∨ ) ⊗s for some r, s. We have the equivalence of tensor categories by [Hei14]. Furthermore by loc.cit every singly atypical irreducible module is a Berezin twist of a mixed tensor. ZigZag or AntiZigZagmodules of length > 1 are never mixed tensors. The indecomposable objects in T are parametrized by certain pairs of partitions (λ L , λ R ) [CW11] and we denote the corresponding indecomposable element by R(λ L , λ R ). By the above result every irreducible representation of non-vanishing superdimension can be written in the form L(λ) ∼ = Ber s(λ) ⊗ R(λ L , λ R ) for some integer s(λ) and a pair of partitions (λ L , λ R ) satisfying l(λ L ) + l(λ R ) ≤ m − 1. It is clear from these results that the reductive supergroup corresponding to the tensor subcategory in A/N generated by the irreducible modules is Gl(1)×Gl(m−1). Indeed the element Ber s(λ) ⊗R(λ L , λ R ) [Hei14] and the determinant representation t.

Indecomposable modules.
It remains to compute the tensor product of two ZigZag-modules up to superdimension 0. As the reparametrisation in the sl(2|1)case shows it is preferable to deviate from the usual Z l (a)-notation. We denote by R(a, . . . , b) the indecomposable module corresponding to the exact sequence 4.5. Cohomological tensor functors. We now define cohomological tensors T m|1 → T m−1|0 . These were first defined in [DS05] and then later refined in [HW14]. We define a similar refined version in the T m|1 -case, but it can be easily extended to the general T m|n -case [Heiar]. For any x ∈ X = {x ∈ g 1 | [x, x] = 0} and any representation (V, ρ), the operator ρ(x) defines a complex since ρ(x) • ρ(x) is zero, and we define V x = ker(ρ(x))/im(ρ(x)). By [DS05] [Ser11a] this defines a tensor functor and V x ∈ T m−1|0 . We fix the following element x ∈ X and denote the corresponding tensor functor V → V x by DS : T m|1 → T m−1|0 .
Parity considerations. If V is in R m|1 , DS(V ) may not be in R m−1|0 . We need to study this more closely. We embed Gl(m − 1|0) as an upper block matrix in Gl(m|1). More precisely we fix the embedding Recall that we defined the categories R m|n in section 1. We now fix the morphism ǫ : Z/2Z → Gl(m)× Gl(n) which maps −1 to diag(E m , −E n ) = ǫ m|n . Then R m|1 is the full subcategory of objects V such that p V = ρ(ǫ m|1 ). If we use the embedding above, we obtain ǫ m|1 = ǫ m−1|0 ǫ 1|1 . If we restrict a representation V ∈ R m|1 to Gl(m − 1), we get the decomposition Since ρ(x) is an odd morphism Hence ρ(x) induces the even morphism ρ(x) : V ± → ΠV ∓ .
We use the notation ∂ for ρ(x).
Z-grading. We equip DS(V ) with a Z-grading as in [HW14]. Although DS(V ) is in T m−1|0 , it still has an action of the torus of diagonal matrices in Gl(m|1). Let V = λ V λ be the weight decomposition and v = λ v λ in V . An easy calculation shows the following lemma. Hence DS(V ) has a weight decompositon with respect to the weight lattice of gl(m|1). The weight decomposition with respect to gl(m − 1|0) is obtained by restriction. The kernel of this restriction map consists of the multiples Zµ. Hence DS(V ) can be endowed with the weight structure coming from the gl(m|1)-module V . This weight decomposition induces then on DS(V ) a decomposition Now let H s denote the torus in the diagonal matrices of elements of the form (1, . . . , 1, t −1 ) and denote by V l the eigenspace of V where H s acts by the eigenvalue t l . Another easy calculation shows the next lemma.
Since ∂V λ ⊂ V λ+µ we obtain a complex which we can write by the last lemma as We denote the cohomology of this complex by H l (V ). Then DS(V ) l = Π l (H l (V )) and we obtain a direct sum decomposition of DS(V ) into Gl(m − 1)-modules This extra structure is very important since it carries a lot more information then just the Z 2 -graded version of DS(L) in T m−1|0 .

Lemma.
A short exact sequence in R m|1 gives rise a long exact sequence for H l .
We denote by σ the automorphism of gl(m|1) defined by σ(x) = −x T where () T denotes the supertranspose. Then σ(x) is still a nilpotent element in g 1 . The corresponding tensor functor is denoted by DS σ : T m|1 → T m−1|0 . We can copy the arguments from above and endow DS σ (V ) with a Z-grading DS σ (V ) = l Π(H l σ (V )). DS and DS σ behave in the same way on irreducible objects, but differ on indecomposable elements, see lemma 4.9.
4.8 Corollary. For any representation V DS(V ) and DS σ (V ) are Z-graded objects.
Another way of looking at this Z-grading is the following. The collection of cohomology functors H i : R m|1 → R m−1|0 for i ∈ Z defines a tensor functor to the category of Z-graded objects in R m−1|0 . Using the parity shift functor Π, this functor can be extended to a tensor functor As in [HW14] we conclude from the support variety calculations in [BKN09], , the following lemma 4.9 Lemma. The kernel of DS is C − and the kernel of DS σ is C + . 4.6. Cohomology computations. We can now calculate DS(M ) for any M of nonvanishing superdimension. By [DS05] DS(L(λ)) = mL core ⊕ m ′ ΠL core for atypical L(λ) where L core is the irreducible Gl(m−1)-representation obtained from L(λ) by replacing the single ∨ in the weight diagram by a ∧. We recall from [Hei14] that we have a commutative diagram where Rep(Gl m−n ) denotes Deligne's category to the parameter δ = m − n. We obtain the same diagram if we replace DS by DS σ since DS σ sends the standard to the standard representation. Since ∆ > 0 these can never be equal. The ∆ = 0 case is easy, in that case both tensor products in the two quotients are equal, and hence neither roofs nor bottoms may appear in the tensor product. The same argument works in the case of the tensor product of two roof modules with a negative ∆. In the case of a tensor product of a roof with a bottom module R(a, . . . , b)⊗B(c, . . . , d) one has roofs in the tensor product for ∆ < 0, Bottoms for ∆ > 0 and an irreducible element for ∆ = 0.
4.8. The pro-reductive envelope. In this section we prove the following theorem.
Proof. For convenience we replace every object by a parity shift so that its superdimension is positive. Then we have to define a functor ρ : T m|1 /N → Rep(Gl(m − 1) × Gl(1) × Gl(1). Any atypical irreducible L(a) can be written as L(a) ∼ = Ber aB ⊗ R(a L , a R ) for some Berezin power and some mixed tensor attached to the bipartition (a L , a R ). The tensor product decomposition above forces the following Ansatz: ρ(L(a)) = 1 × t aB × L(wt(λ)). To each weight a in our block (with irreducible module L(a) = Ber aB ⊗ R(a L ; a R )) are attached the indecomposable modules R(a, . . . , a − s), B(a, . . . , a + r) for some r, s ≥ 0.
Note that the Hom-spaces between the irreducible elements are either zero or onedimensional since Schur's lemma holds in any semisimple tensor category, hence the