Universal tensor categories generated by dual pairs

Let $V_*\otimes V\rightarrow\mathbb{C}$ be a non-degenerate pairing of countable-dimensional complex vector spaces $V$ and $V_*$. The Mackey Lie algebra $\mathfrak{g}=\mathfrak{gl}^M(V,V_*)$ corresponding to this paring consists of all endomorphisms $\varphi$ of $V$ for which the space $V_*$ is stable under the dual endomorphism $\varphi^*: V^*\rightarrow V^*$. We study the tensor Grothendieck category $\mathbb{T}$ generated by the $\mathfrak{g}$-modules $V$, $V_*$ and their algebraic duals $V^*$ and $V^*_*$. This is an analogue of categories considered in prior literature, the main difference being that the trivial module $\mathbb{C}$ is no longer injective in $\mathbb{T}$. We describe the injective hull $I$ of $\mathbb{C}$ in $\mathbb{T}$, and show that the category $\mathbb{T}$ is Koszul. In addition, we prove that $I$ is endowed with a natural structure of commutative algebra. We then define another category $_I\mathbb{T}$ of objects in $\mathbb{T}$ which are free as $I$-modules. Our main result is that the category ${}_I\mathbb{T}$ is also Koszul, and moreover that ${}_I\mathbb{T}$ is universal among abelian $\mathbb{C}$-linear tensor categories generated by two objects $X$, $Y$ with fixed subobjects $X'\hookrightarrow X$, $Y'\hookrightarrow Y$ and a pairing $X\otimes Y\rightarrow \text{\textbf{1}}$ where \textbf{1} is the monoidal unit. We conclude the paper by discussing the orthogonal and symplectic analogues of the categories $\mathbb{T}$ and ${}_I\mathbb{T}$.


Introduction
A tensor category for us is a symmetric, not necessarily rigid, C-linear monoidal abelian category. In this paper we construct and study a tensor category which is universal as a tensor category generated by two objects X, Y with fixed subobjects X ′ ֒→ X, Y ′ ֒→ Y and endowed with a pairing X ⊗ Y → 1, the object 1 being the monoidal unit.
The simpler problem of constructing a universal tensor category generated just by two objects X, Y endowed with pairing X ⊗ Y → 1 was solved several years ago, and explicit constructions of such a category are given in [19] and [6]. The construction in [6] realizes this category as a category T sl(∞) of representations of the Lie algebra sl(∞), choosing X as the natural sl(∞)-module V , and Y as its restricted dual V * . Motivated mostly by a desire to understand better the representation theory of the Lie algebra sl(∞), in [13] a larger category was constructed, denoted T ens sl(∞) , which contains also the algebraic dual modules V * and V * * . It is clear that the category T ens sl(∞) has a completely different flavor as its objects have uncountable length while T sl(∞) is a finite-length category.
However, in [14] the observation was made that the four representations V , V * , V * , V * * generate a finite-length tensor category T 4 gl M (V,V * ) over the larger Lie algebra gl M (V, V * ), see Section 2. We call this latter Lie algebra a Mackey Lie algebra as its introduction has been inspired by G. Mackey's work [12]. The simple objects of T 4 gl M (V,V * ) were determined in [3]. Furthermore, in [5] the tensor category T 3 gl M (V,V * ) , generated by V , V * , and V * , was studied in detail. It was proved that T 3 is Koszul, and it was established that T 3 gl M (V,V * ) is universal as a tensor category generated by two objects X, Y with a pairing X ⊗ Y → 1, such that X has a subobject X ′ ֒→ X. Later, a vast generalization of the results of [5] was given in [4]: here a universal tensor category with two objects X, Y , a paring X ⊗ Y → 1 and an arbitrary (possibly transfinite) fixed filtration of X was realized as category of representations of a certain large Lie algebra.
A main difference of the category T 4 gl M (V,V * ) with previously studied categories is that, as we show in the present paper, the injective hulls of simple objects are not objects of T 4 gl M (V,V * ) but of the ind-completion (T 4 gl M (V,V * ) ) ind which we denote simply by T. In particular, the trivial module has an injective hull I in T of infinite Loewy length, i.e. with an infinite socle filtration. Moreover, remarkably, I has the structure of a commutative algebra.
This leads us to the idea of considering the category I T of I-modules internal to T. The morphisms in this new category are morphisms of gl M (V, V * )-modules as well as of I-modules. The simple objects of I T are of the form I ⊗ M where M is a simple module in T.
A culminating result of the present paper is that the category I T has the universality property stated in the first paragraph of this introduction. The pairs X ′ ֒→ X and Y ′ ֒→ Y are realized respectively as I ⊗ V * ⊂ I ⊗ V * and I ⊗ V ⊂ I ⊗ V * * , I is the unity object, and the tensor product in I T is ⊗ I .
Finally, in Section 4 we study analogues of the tensor categories T o(∞) and T sp(∞) considered in [6] and [19]. Consider a tensor category generated by a single object X with a subobject X ′ ֒→ X and a pairing X ⊗ X → 1. After identifying V * and V , our construction of the category I T yields a universal tensor category also in this setting. However, one can assume in addition that the pairing X ⊗ X → 1 is symmetric or antisymmetric, which leads to new universality problems for tensor categories. With this in mind, we introduce T 2 o(V ) and T 2 sp(V ) where o(V ) and sp(V ) are respective orthogonal and symplectic Lie algebras of a countable-dimensional vector space V . In analogy with our previous constructions, we then produce appropriate categories I ′ T 2 for I ′ = I o(V ) and I ′ = I sp(V ) and prove that these latter categories are universal in the respective new settings. Moreover, the categories I o(V ) T 2 and I sp(V ) T 2 are canonically equivalent as monoidal categories.

Notation
All vector spaces are defined over C (more generally, we could work over an algebraically closed field of characteristic zero); similarly, all abelian categories and all functors between such are assumed C-linear, and we refer to [17] for general background on abelian/additive categories. By S k X and Λ k X we denote respectively the k-th symmetric and exterior powers of a vector space X, and S n stands for the symmetric group on n letters.
Once and for all we fix a non-degenerate pairing V * ⊗ C V → C of countable-dimensional vector spaces V and V * . This pairing defines embeddings For any vector space M we set M * = Hom C (M, C). We abbreviate ⊗ C as ⊗. By ⊗ we denote also tensor product in abstract tensor categories in the hope that this will cause no confusion.
Except in Section 4, g will be the Mackey Lie algebra gl M (V, V * ) of [14] associated to the pairing V * ⊗ V → C. By definition, where ϕ * : V * → V * is the operator dual to ϕ. We will describe g explicitly as Lie algebra of infinite matrices shortly.
We set W * := V * /V * , W := V * * /V and F := W * ⊗ W. There is an extension where Q is defined as the quotient of V * ⊗ V * * by the sum of the kernels of the pairings In Proposition 3.5 below we prove that the extension (2-1) is non-splitting. We model the actions of g on the various modules mentioned above as follows: − The elements of V * ⊂ V * are precisely the finite row vectors.
− g consists of N × N-matrices with finite rows and columns, acting on V * * by left multiplication.
− Similarly, g acts on V * as minus right multiplication.
consists of finite-rank N × N matrices, acted upon by g by commutation.
We will frequently make use of Schur functors S λ attached to Young diagrams λ. Often we write X λ instead of S λ X for a vector space X. Moreover, S k X = S ρ X, Λ k X = S γ X, where ρ, γ are respectively a row and a column with k boxes.
For Young diagrams λ, µ, ν and π we write and similarly, for non-negative integers l, m, n, p we set for appropriate multiplicities m µ,ν . Here |λ| denotes the degree (number of boxes) of a Young diagram λ. Finally, for any subscript s of the form (•, •, •, •) we set where I is the object constructed below in Section 3.
Definition 2.1 We refer to objects involving only the two outside diagrams λ and π as thick (or purely thick for emphasis) and those involving only the two middle diagrams as thin. Everything else is mixed.
It is essential to recall Corollary 4.3 in [5] which claims that L λ,µ,ν,π is a simple g-module, and implies that L l,m,n,p is a semisimple g-module.
The following remark will be used implicitly and repeatedly: given a short exact sequence in a tensor abelian category, the symmetric power S k x has a filtration

Plethysm
Given that F = W ⊗ W * and we have to work with symmetric and exterior powers of F , we will have to understand how such powers decompose as direct sums of objects of the form S λ W ⊗ S µ W * . The result applies to a tensor product W ⊗ W * in any C-linear tensor category, so we work in this generality throughout the present subsection. We call a partition λ special if it satisfies the condition: all hooks of λ with diagonal corner have horizontal and vertical arms of length µ i − 1 and µ i respectively, where µ 1 > µ 2 > . . . > 0 is a partition of k. We now recall the following result.
Proposition 2.2 Let x and y be two objects in a C-linear tensor category. We have the following decompositions: (a) S k (x ⊗ y) is the direct sum of all objects of the form S λ x ⊗ S λ y as λ ranges over all Young diagrams of degree k.
is the direct sum of all objects of the form S λ x ⊗ S λ ⊥ y as λ ranges over all Young diagrams of degree k, where λ ⊥ denotes the conjugate partition.
(c) S k S 2 x is the direct sum of all S λ x for partitions λ of degree k with even parts , i.e. even partitions.

Ordered Grothendieck categories
We recall the following notion from [4, Definition 2.3].
Definition 2.3 Let (P, ) be a poset. An ordered Grothendieck category with underlying order (P, ) is a Grothendieck category C together with objects X s , s ∈ P so that the following conditions hold.
(a) The objects X s are semi-artinian, in the sense that all of their non-zero quotients have non-zero socles.
(b) Every object in C is a subquotient of a direct sum of copies of various X s .
(c) The simple subobjects in S s := {isomorphism classes of simples in socX s } (2)(3)(4)(5) are mutually non-isomorphic for distinct s and they exhaust the simples in C.
(d) Simple subquotients of X s outside the socle socX s are in the socle of some X t , t ≺ s.
(e) Each X s is a direct sum of objects with simple socle.
(f) Let t ≺ s. The maximal subobject X s≻t ⊂ X s whose simple constituents belong to various S r for s r t is the common kernel of a family of morphisms X s → X t . Definition 2.5 For two elements i ≺ j in P the defect d(i, j) is the supremum of the set of non-negative integers q for which we can find a chain We put also d(i, i) := 0. In the context of an ordered Grothendieck category as in Definition 2.3 we adopt the simplified notation d(S, T ) for d(s, t) when S ∈ S s and T ∈ S t .
According to [5,Proposition 2.9] ext functors in an ordered Grothendieck category exhibit the following "upper triangular" behavior. Proposition 2.6 Let S ∈ S s and T ∈ S t be two simple objects in an ordered Grothendieck category.
It is implicit in the statement that, in particular, we have s t (see [5,Lemma 3.8]). One of our goals will be to show that in the ordered Grothendieck category T introduced in Section 3.4 below, we actually have equality, and hence the category T is Koszul in the following sense.
Definition 2.7 An ordered Grothendieck category is Koszul if for every q ≥ 0 and every two simple objects S ∈ S s and T ∈ S t the canonical Yoneda composition map is surjective, where the sum ranges over all isomorphism classes of simples U i . This mimics one of the characterizations of Koszul connected graded algebras, namely the requirement that the graded ext algebra Ext * (k, k) of the ground field k be generated in degree one ([16, §2.1]).
We introduce the following term to capture the desirable situation where defects precisely measure non-vanishing exts.
Definition 2.8 An ordered Grothendieck category is sharp if it satisfies the conclusion of Proposition 2.6 with equality rather than inequality.
The relevance of the concept to the preceding discussion follows from Proof Fix arbitrary simple objects S ∈ S s , T ∈ S t and a positive integer q ≥ 2. It will be enough to show that the Yoneda composition is onto, since we can then proceed by induction on q. Let be the short exact sequence resulting from the embedding of T into its injective hull I T . This sequence constitutes an element of Ext 1 (R T , T ), and Yoneda multiplication by that element induces an isomorphism Ext q−1 (S, R T ) ∼ = Ext q (S, T ).
If Ext q (S, T ) = 0 there is nothing to prove. Otherwise, our sharpness assumption shows that d(S, T ) = q. This together with the finite-length hypothesis then ensures that the socle of Q T is a finite direct sum of simples U with d(S, U ) = q − 1. We furthermore have and hence every non-zero element of Ext q (S, T ) will be contained in the image of the Yoneda map where U ranges over all isomorphism classes of simple constituents of socR T . This finishes the proof.

Comodules
[20, Chapters I and II] will provide sufficient background on coalgebras and comodules. For a coalgebra C over a ground field k we write M C for its category of right comodules and M C f in for its category of finite-dimensional comodules. Since the Grothendieck categories we are interested in will turn out to be of the form M C for coalgebras C we record in this short section a characterization of such categories from [21].
The following is a paraphrase of [21, Definition 4.1], adapted in the context of Grothendieck (as opposed to plain abelian) categories. Moreover, in this case M C f in can be identified with the subcategory of C consisting of finite-length objects.
The following notion (analogous to its dual-ring-theoretic version [1, discussion preceding Theorem 2.1]) will also be relevant below.
Definition 2.12 A coalgebra C is left semiperfect if either of the following conditions, equivalent by [9, Theorem 10], holds: − every indecomposable injective right C-comodule is finite dimensional; − every finite-dimensional left C-comodule has a projective cover.

Tensor categories
The categories we are most interested in are typically monoidal. The latter, in full generality, are covered for instance in [10,Chapter XI]. In the context of abelian categories, we briefly recall the relevant definitions (see also [5, §3.6], where we make the same linguistic conventions).

Definition 2.13
A C-linear abelian category C is monoidal if its monoidal structure has the property that x ⊗ • and • ⊗ x are exact endofunctors for every object x.
If in addition the monoidal structure is symmetric, (C, ⊗) is a tensor category.
A tensor functor between tensor categories is a C-linear symmetric monoidal functor.
Note that this differs from conventions made elsewhere in the literature. In [7, §1.2], for instance, the term 'catégorie tensorielle' implies rigidity.
We occasionally write (C, ⊗, 1) for a monoidal category to specify both the tensor product bifunctor and the monoidal unit object 1.

Definition of the object I
For every nonnegative integer k we have a canonical embedding obtained as the composition where ι : C → Q is the embedding defining Q as an extension of F by C. This gives rise to an exact sequence Taking the colimit (or simply union) we obtain a g-module that has an infinite ascending filtration representable schematically as , (3)(4)(5) where the boxes indicate the layers of the filtration. The morphism ψ : to be defined below will play a central role in the sequel; we will occasionally write ψ for the resulting factorization I/C → F ⊗ I as well, leaving it to context to distinguish between the two possible meanings.
We obtain the morphism ψ as a colimit lim − →k ψ k where The latter map is defined as follows. First, recall that the symmetric algebra has a graded Hopf algebra structure [20, p.228] making the degree-one elements primitive, i.e. such that the comultiplication ∆ : is the unique algebra map defined by The comultiplication (3-8) is a morphism of g-modules. By definition, (3-7) is given by − π is the epimorphism fitting in .
To make sense of lim − →k ψ k we would have to argue that these maps are compatible with the embeddings ι : commute (for arbitrary k). This can be seen by direct examination, fixing a basis (v α ) α for Q with a distinguished element v 0 = 1 ∈ C ⊂ Q and noting that the upper left-hand map in (3-9) is defined on monomials by Proof The kernel of the upper right-hand map in (3-9) is so we are in effect claiming that the preimage of (3-11) through the "partial comultiplication" This is easily seen from the explicit description (3-10) of the comultiplication (3-12).
The morphisms of the first type are such that their joint kernel is L l,m,n,p ⊗ I ⊆ J l,m,n,p ⊗ I = I l,m,n,p .
On the other hand the kernel of ψ 0 is C ⊂ I, and hence the joint kernel of Σ s is L l,m,n,p ∼ = L l,m,n,p ⊗ C ⊂ J l,m,n,p ⊗ I = I l,m,n,p .
This will require some preparation. First, we have the following remark, in the spirit of [5, Lemma 3.1]. Lemma 3.3 Let G be a Lie algebra and I ⊆ G be an ideal. Suppose U ⊆ U ′ is an essential inclusion of G/I-modules and D be a G-module on which I acts densely. Then the inclusion To see this, first choose an arbitrary y ∈ G with image x in G/I. We have On the other hand, by the density assumption there is some a ∈ I such that aw = yw, and we can simply set x = y − a.
Now let w 1 up to w k be linearly independent vectors in D and consider an element Using the fact that U ⊆ U ′ is essential and the claim above, there is an element with q 1 ∈ U . We can now repeat the procedure, acting on the right-hand side of (3-16) with another element x ′ ∈ G so that the resulting sum is of the same form, with q 1 and q 2 in U . By recursion, we will eventually obtain an element of the form in the U (G)-submodule of U ′ ⊗ D generated by f . This finishes the proof.
The following result will require some additional conventions and elaboration. Recall that Q is the quotient . We noted above that we identify the original space V * ⊗ V * * with finite-rank infinite matrices, and hence the quotient consists of equivalence classes of such matrices, where two are declared equivalent whenever they differ along at most finitely many rows or columns (and this is extended minimally to an equivalence relation).
We fix a basis {e α } α∈A of Q as follows: − All other basis elements are classes of rank-1 matrices of the Proof The conclusion will follow from the remark that g acts densely on sets X ∪ X * , i.e. that given x ∈ X and x * ∈ X * , the vectors gx ∈ V and gx * ∈ V * can be prescribed arbitrarily. Keeping this in mind, we can then find g ∈ g such that This choice will meet the requirements of the statement, hence the conclusion.
Clearly then deg(gx) = deg(x) − 1, and we can conclude the argument by using the induction hypothesis.
We can now conclude via Proposition 3.5: an arbitrary non-zero element of the right-hand side of (3-21) is of the form The Lie algebra K F leaves the subspace invariant and its action makes that space isomorphic to (S k Q) ⊕r . The conclusion thus follows from Proposition 3.5.

Simple objects and their endomorphism algebras
The main result of the present subsection is the following (presumably expected) claim.
The arguments, which require some groundwork, will be in the spirit of those used in the proof of the analogous statement [5,Theorem 3.5]. First, recall [5, Lemma 3.1]: Lemma 3.7 Let G be a Lie algebra and J ⊆ G be an ideal. Suppose U , U ′ are two G/J-modules and D a G-module on which J acts densely and irreducibly with End J D = C. Then, the inclusion is an isomorphism.
Proposition 3.8 For any two non-negative integers l, p and Young diagrams µ, ν the endomorphism algebra in T of the object W * l ⊗ V µ,ν ⊗ W p is the group algebra C[S l × S p ], with the two symmetric-group factors acting on the two outer tensorands.
Condition (a). This follows from the fact that all I s have countable filtrations whose subquotients are simple objects of the form L λ,µ,ν,π as in . The latter is clear as the objects J s have finite length and I has the filtration (3)(4).
Condition (b). This holds essentially by construction. Condition (c) is a consequence of [5,Proposition 5.4]. Condition (d). Once more, we filter I s = J s ⊗ I by first refining the socle filtration maximally of J s and then tensor by some maximal refinement of the filtration (3-4).
The successive subquotients of (3-4) can be decomposed as sums of objects of the form W λ ⊗ W * λ by part (a) of Proposition 2.2. Hence, tensoring a simple subquotient S ∈ S s of J s for some s = (l, m, n, p) by such an object has the effect of increasing l and m by the same amount, thus resulting in some t ≺ s according to our ordering (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13).
It thus remains to argue for simple subquotients of instead. In this case though the filtration of J s is obtained either by surjecting one of the tensorands V * onto W * = V * /V * or similarly, one of the tensorands V * * onto W , or by evaluating some V * against some V .
All of these procedures map into J t for t ≺ s, hence the conclusion.
The verification that the joint kernel of these maps is as claimed routine.
We record the following observation.
Remark 3.13 The category T is symmetric with respect to the simultaneous interchange V ↔ V * , V * ↔ V * * . Numerically, this corresponds to l ↔ p and n ↔ m. Lemma 3.12 is compatible with this transformation: according to the last condition in (3-13) we have

Injective resolutions
We will now show that C admits an injective resolution in T We will also see that I j /Im(I j−1 ) admits an ascending filtration with layers where each diagram has j + 1 rows.
To streamline the notation for such Young diagrams we denote by (l, j × 1) the diagram with a row of length l and j single-box rows.
To show that the maps (3-25) fit into a resolution (3-24) we begin with the following simple observation.
Proof That the domain and codomain decompose as Now consider one of the objects Λ j F ⊗ I, j ≥ 0 under discussion. Since I has a filtration (3-5) with subquotients S k F , k ≥ 0, the object Λ j F ⊗ I has a filtration with subquotients Λ j F ⊗ S k F decomposed as Moreover, these decompositions are canonical, i.e. the summands are unique. We write K j := ker ψ j : for the kernel of the map (3-25) and by convention, we set K −1 j = {0}. We now have Lemma 3.15 For each k ≥ 0 the subquotient is the j-row summand of the latter.
Proof The map ψ j respects the filtrations of its domain and codomain, by respectively, and the associated graded map gr ψ j , in degree k, is precisely (3-28) with X = F . By Lemma 3.14 this means that the degree-k kernel of gr ψ j is the j-row summand of Λ j F ⊗ S k F . This verifies the statement at the associated-graded level. To conclude, it will suffice to construct gradings on the domain and codomain of ψ j , compatible with ψ j , that give back the filtrations by . This would then prove that the filtered map ψ j arises from a grading, and hence that its kernel is the direct sum of the kernels of its homogeneous components.
We construct the requisite gradings as in the discussion preceding Lemma 3.1: fix a basis v α for Q with v 0 = 1 ∈ C ⊂ Q, and assign deg v α = 0 for α = 0 1 otherwise.
One checks easily that ψ j preserves degrees, finishing the proof as described above.
We can now finally complete the discussion on the injective resolution (3-24).

Theorem 3.16 The morphisms (3-25) fit into an exact sequence (3-24).
Proof The maps ψ j fit into a sequence −→ · · · (3-33) (not yet known to be exact) of filtered vector spaces. Lemma 3.14 applied to X = F shows that the associated graded sequence is exact, and the conclusion follows from the fact that, as seen in the proof of Lemma 3.15, the filtrations on the terms of (3-33) arise from gradings compatible with the maps ψ j .

Corollary 3.17 For a simple object X of T we have
Proof The statement follows from the existence of the injective resolution (3-24) of the trivial object C, since by Theorem 3.11 which in turn, by Proposition 2.2, (b), decomposes as |λ|=j W * λ ⊗ W λ ⊥ .

Koszulity
We will eventually show that the category T is Koszul. To that end, we first need to strengthen Proposition 2.6 to an equality:

Theorem 3.18 The Grothendieck category T is sharp in the sense of Definition 2.8: for any two simples S, T ∈ T we have
Ext q (S, T ) = 0 ⇒ d(s, t) = q.
We do this in stages, considering the following particular case first.

Proposition 3.19 Theorem 3.18 holds when T is purely thick.
Proof Let T ∈ S t . We have to argue that there is some injective resolution so that the socle of K q is a sum of simple objects S with d(S, T ) = q. This follows from Corollary 3.17 for the trivial object T = C and in general for thick T from Theorem 3.11, which implies that we can obtain an injective resolution for T by simply tensoring (3-24) with T .
In order to push past purely thick objects we need a version of [5, Lemma 3.13], requiring some notation: for a quadruple (λ, µ, ν, π) we write L +ℓ λ,µ,ν,π for the direct sum of all L λ,µ ′ ,ν,π with µ ′ obtained by adding a box to µ. Here ℓ stands for left, and we have a similarly defined object L +r λ,µ,ν,π (for right) obtained by enlarging ν instead.
The case when µ ′ is empty but ν ′ is not proceeds analogously, making use of part (b) of Lemma 3.20 rather than (a).
As a direct consequence of Theorems 2.9 and 3.18 we have  Proof The hypotheses of Theorem 2.11 are met (for the ground field C): T is generated by the finite-length objects in T, since every object is isomorphic to a subquotient of a direct sum of indecomposable injectives I s as defined in (3)(4)(5)(6)(7)(8)(9), and in turn the injectives I s are unions of their finite-length truncations J s ⊗ S k Q for k ∈ Z >0 .
Moreover, according to Theorem 3.6, the endomorphism ring of a simple object L λ,µ,ν,π as in (2-2) is the field C.

An internal commutative algebra and its modules
The object I has a structure of commutative algebra internal to the tensor category g Mod of gmodules. To see this, we observe that I is isomorphic as a g-module to a quotient algebra of the symmetric algebra S • Q. Indeed, denote by a the distinguished element 1 ∈ C ⊂ Q of the degree-one component Q ⊂ S • Q and consider the commutative algebra where 1 ist the unit of the symmetric algebra and (a − 1) is the ideal generated by a − 1. This ideal is clearly g-stable and S • Q/(a − 1) is an algebra in g Mod. Moreover, the definition of I as lim − → S k Q implies that there is an isomorphism of g-modules We will be interested in the category I T of I-modules internal to T. This is clearly a Grothendieck category. Moreover, the forgetful functor forget : I T → T fits into an adjunction We refer to I-modules in the image of I ⊗ • as free. We will see that tensoring with I has the effect of "partially semisimplifying" T, in the following sense.
The morphism σ is the required splitting, concluding the base case n = 1 of the induction.
The argument also shows that we have a decomposition  in I T (and hence also in T), implying that I ⊗ Q ∈ T is injective (by Theorem 3.11 for instance, which shows that both summands in  are injective). We regard Q as a subobject of S 2 Q via the embedding Q ֒→ S 2 Q described in . By the injectivity of I ⊗ Q ∈ T noted above, the embedding Once more, the adjunction (3-38) retrieves a morphism in I T that restricts to the identity on the submodule showing that this embedding splits in I T. This proves the main claim for n = 2 and the fact that there is a splitting We now repeat the procedure recursively to complete the inductive argument.
Since it will be our goal to study the category I T along the same lines as T, we next turn to simple objects therein.

Theorem 3.24
The simple objects in I T are (up to isomorphism) precisely the free I-modules I ⊗S for simples S ∈ T. For each of them, the endomorphism algebra in I T is C.
Proof We first prove that I ⊗ S is simple in I T. The simple objects of T are precisely the various modules L λ,µ,ν,π of (2-2), and according to Theorem 3.11 the injective hull S ⊂ I S contains I ⊗ S (I S exists because I T is a Grothendieck category). Since is essential in I S , it is also essential in I ⊗ S. It follows that any non-zero subobject The claim that this must be C follows from the fact that I⊗S is the injective hull of S (Theorem 3.11) and hence − every morphism S → I ⊗ S in T factors through the socle S ⊂ I ⊗ S, and − End T S ∼ = C.
Since I ⊗ S is an injective hull of S in T, every morphism S → I ⊗ S factors through the socle S ⊂ I ⊗ S, the isomorphism End T S = C implies the existence of an isomorphism .
As for the fact that these are, up to isomorphism, all irreducible objects in I T, consider such an object T and note that it must contain some simple S ∈ T. Hence T must be a quotient of the (irreducible!) free object I ⊗ S ∈ I T. Consequently, we have T ∼ = I ⊗ S as desired.
Since I is a commutative algebra in T, the category I T of internal modules has a natural symmetric monoidal structure with I as the unit object and ⊗ I as the tensor product. Whenever we refer to I T as a tensor category, this will be the structure we consider.
We will similarly consider the (simple, by Theorem 3.24) objects of I T T λ,µ,ν,π := I ⊗ L λ,µ,ν,π  and the semisimple objects T l,m,n,p := I ⊗ L l,m,n,p , that are direct sums of the various T λ,µ,ν,π . We now have the following analogue of Proposition 3.10.

Proposition 3.25 I T is an ordered Grothendieck category in the sense of Definition 2.3.
Proof Taking as above the objects X s to be our I s (this time regarded as objects in I T rather than just T), the argument proceeds much as in the proof of Proposition 3.10 with a small difference in how we define the morphisms I s → I t , t ≺ s in how we define the morphism from Definition 2.3, (f). Once again, said morphisms will be tensor products and compositions of a few building blocks: − projecting one of the tensorands V * of I s = J s ⊗ I onto W * ; − the dual analogue, V * * → W ; − the "pairing" obtained via the adjunction (3-38) from the composition Everything else goes through as sketched in the proof of Proposition 3.10.
The difference to T is that now the free I-modules generated by the full duals V * and V * * admit the pairing (3-42) valued in the unit object I of the category I T under consideration.
We also have an I-module version of Theorem 3.11.
Just as T, the category I T can be realized as comodules over a coalgebra (see Theorem 3.22). As in that previous result, we denote by I T f in ⊂ I T the full subcategory of finite-length objects. Note that the indecomposable injectives I λ,µ,ν,π = I ⊗ J λ,µ,ν,π ∈ I T have finite length: J λ,µ,ν,π have finite filtrations with subquotients simple in T, and according to Theorem 3.24 tensoring these simple objects by I produces simples in I T. Proof The argument is largely parallel to that underpinning Theorem 3.22, via Theorem 2.11 (minus Koszulity, which we have not yet addressed for I-modules).
The additional remark, that D is semiperfect, follows directly from Definition 2.12 and the fact that, as observed above, in I T the indecomposable injectives I λ,µ,ν,π have finite length.
We also need the following remark, which parallels [5, Lemma 2.19] (the proof is virtually identical, so we omit it).

Lemma 3.28
The tensor subcategory I T ′ of I T generated by the morphisms described in the proof of Proposition 3.25 is the full subcategory containing I l,m,n,p .
We next turn to the Koszulity of I T. In keeping with the theme, the argument will be very similar to what we saw in proving Theorems 3.18 and 3.21. We collect all of the statements together as follows.

Theorem 3.29 The Grothendieck category I T is sharp in the sense of Definition 2.8: for any two simples S, T ∈ T we have
Ext q (S, T ) = 0 ⇒ d(s, t) = q.
In particular, the ordered Grothendieck category I T is Koszul in the sense of Definition 2.7.
Proof The last claim follows from sharpness by Theorem 2.9, so we focus on proving the sharpness claim. In turn, the latter follows as in the proof of Theorem 3.18, with the exact sequence (3-34) replaced by its analogue, obtained by simply tensoring it with I.

Remark 3.30
Note that in the present setting the proof of Koszulity is in fact simpler than in §3.6: we do not need a version of Proposition 3.19, since for purely thick simple objects L λ,∅,∅,π the corresponding simple object of I T T λ,∅,∅,π = I ⊗ L λ,∅,∅,π is injective.
As a consequence, we can supplement Theorem 3.27, fully bringing it in line with Theorem 3.22.

Corollary 3.31 The coalgebra D in Theorem 3.27 can be chosen graded and Koszul.
We end the present subsection with description of one possible choice for the graded coalgebra C in Theorem 3.22. This discussion parallels [5, §3.4], which in turn is analogous to [6, §5].
Let T be the tensor algebra in I T of the object Hom I T (I s , I t ).
The algebra A is naturally Z ≥0 -graded by means of the defect introduced in Definition 2.5:

s)=d
Hom I T (I s , I t ).
Finally, the coalgebra C is simply the graded dual of A, with C d = A * d . The fact that C (and hence A) is Koszul then implies

Universality
We can now characterize I T as a universal category in the sense of [5,Theorem 4.23] and [4,Theorem 5.2]. First, note that in I T there is a pairing corresponding to (3-23) through the adjunction (3-38).
(b) if D is additionally a Grothendieck category then F extends uniquely to a coproduct-preserving functor I T → D.
The argument will be analogous to that employed in the proof of [5,Theorem 3.23], revolving around the fact that the algebra A in the preceding discussion is quadratic (Proposition 3.32). For that reason, it will be necessary to understand its components of degree ≤ 2. In degree zero things are simple: the following result is the version of [5,Lemma 3.24] appropriate here.
Lemma 3.34 For (l, m, n, p) ∈ P the endomorphism algebra of the injective object I l,m,n,p ∈ I T is isomorphic to C[S l × S m × S n × S p ], with the symmetric groups acting naturally on the relevant tensorands of I l,m,n,p = I ⊗ J l,m,n,p Proof We have End I T (I ⊗ J l,m,n,p ) ∼ = Hom T (J l,m,n,p , I ⊗ J l,m,n,p ).
It follows that the restricting an arbitrary morphism J l,m,n,p → I ⊗ J l,m,n,p in T to the socle induces an isomorphism Hom T (J l,m,n,p , I ⊗ J l,m,n, We can see that the right-hand-side of (3-44) is naturally identifiable with C[S l × S m × S n × S p ] as in Proposition 3.8.
As for degree 1, we need an analogue of [5,Lemma 3.25]. Stating such an analogue will require some notation. Degree-one morphisms between the objects I l,m,n,p ∈ I T come in three flavors: I l,m,n,p → I l,m−1,n−1,p , I l,m,n,p → I l+1,m−1,n,p , I l,m,n,p → I l,m,n−1,p+1 .
We distinguish families of each flavor, as follows. The morphism φ i,j : I l,m,n,p → I l,m−1,n−1,p for 1 ≤ i ≤ m, 1 ≤ j ≤ n applies the pairing V * ⊗ V * * → I on the i th tensorand V * and the j th tensorand V * * of I l,m,n,p = I ⊗ W ⊗l * ⊗ (V * ) ⊗m ⊗ (V * * ) ⊗n ⊗ W ⊗p and acts as the identity on all other tensorands.
Next, we have the maps i,j π : I l,m,n,p → I l+1,m−1,n,p for 1 ≤ i ≤ m, 1 ≤ j ≤ n which − first permutes cyclically the first i tensorands V * ; − maps the new first (old i th ) tensorand V * onto W * = V * /V * ; − finally permutes the last m − j + 1 tensorands W * cyclically so the newly-created W * becomes the j th .
Finally, we have the left-right mirror image π i,j : I l,m,n,p → I l,m,n−1,p+1 for 1 ≤ i ≤ m, 1 ≤ j ≤ n of i,j π, obtained by substituting V * * for V * , W for W * , reversing the directions of the cyclic permutations, etc.
We write S l,m,n,p := S l × S m × S n × S p for products of symmetric groups and unless specified otherwise, morphism spaces in Lemma 3.35 below are in the category I T.  and finally, the self-dual (I l+1,m−1,n↓,p↑ ⊙ I l↑,m↓,n,p ) ⊕ (I l↑,m↓,n−1,p+1 ⊙ I l,m,n↓,p↑ ) → I l↑,m↓,n↓,p↑ . (3-53) The nine '⊙' symbols above account for the nine possible ways of composing two morphisms, each of one of the three flavors listed in Lemma 3.35.
Remark 3.37 Note that in all cases the product '⊙' conserves the total number of up as well as down arrows.
Proof of Theorem 3.33 (sketch) As in the proof of [5,Theorem 3.23], an appeal to [5,Theorem 2.22] together with Proposition 3.25 proves the statement as soon as we argue that the initial data of x ֒→ x * * , x * ֒→ x * and p : x * ⊗ x * * → 1 in the tensor category D extends to a linear monoidal functor where I T ′ is, as in Lemma 3.28, the full subcategory of I T on the objects I l,m,n,p . Set I T := T ⊗ I for any T ∈ T. Since the objects of I T ′ are precisely the tensor powers (over I ∈ I T) and the morphisms are tensor products and compositions of permutation of tensorands, evaluations (3-43), inclusions I V * ⊂ I V * and I V ⊂ I V * * , etc., there is an obvious candidate for such an extension F , sending etc. What we have to argue is that that extension is in fact well defined.
The fact that, by Proposition 3.32, the algebra A defined in Section 3.8 is quadratic, means that it will be enough to check that the degree-two relations between degree-1 morphisms between the I l,m,n,p (i.e. the kernels of the maps (3-48) to (3-53)) vanish in D upon substituting x for I V , x * for I V * , etc. This would be a somewhat tedious and unenlightening check if done exhaustively, so we exemplify the argument by treating  alone. In that regard, we make the claim: The kernel of the composition  is generated, as an (S l,m−2,n−2,p , S l,m,n,p )-bimodule, by φ m−1,n−1 ⊗ φ m,n − φ m−1,n−1 ⊗ φ m,n • (m, m − 1)(n, n − 1),  where (m, m − 1) is the respective transposition in S m ⊂ S l,m,n,p and similarly, (n, n − 1) ∈ S n ⊂ S l,m,n,p .
Assuming the claim for now, we observe that the relations annihilated by (3-48) hold in any tensor category. It follows that our candidate functor F is indeed compatible with the quadratic relations imposed by composition and hence is well defined. It thus remains to prove the claim; this is the goal we focus on for the duration of the present proof, following the layout of the proof of [5, Lemma 3.27 (a)].
First, note that the morphism (3-48) is surjective by Lemma 3.28. Secondly, the fact that (3-54) belongs to the kernel of (3-48) is immediate: this is because − evaluating the m th tensorand I V * against the n th tensorand I V * * , and then and then repeating the two evaluations above. The proof will thus be complete if we argue that the kernel of (3-48) is not strictly larger than the bimodule generated by . We do this by a dimension count. Tensoring two instances of The desired conclusion that the kernel of the surjection (3-48) cannot contain the bimodule generated by (3-54) strictly will thus follow if we prove that dim I l,m↓↓,n↓↓,p = dim Hom(I l,m,n,p , I l,m−2,n−2,p ) ≥ 1 2 l!m!n!p!.
Since we have an embedding (End I l,0,0,p ) ⊗ I 0,m↓↓,n↓↓,0 → I l,m↓↓,n↓↓,p and the left-hand tensorand is isomorphic to C[S l,p ] by Lemma 3.34, it is enough to assume that l = p = 0 and show that dim or equivalently, via the adjunction (3-38), that This, however, follows by restricting the morphisms on the left to V ⊗m * ⊗ V ⊗n and noting that we already know the analogous inequality from the computation carried out in [6, Lemma 6.3], or from [5, Lemma 3.27 (a)] (which is analogous to the claim being proven here).

Orthogonal and symplectic analogues of the categories T and I T
In this final section we discuss briefly the orthogonal and symplectic versions of the categories T and I T. The orthogonal and symplectic analogues of the Lie algebra gl M (V, V * ) are the Lie algebras o(V ) and sp(V ) where V is now equipped with a nondegenerate symmetric or antisymmetric bilinear form ·, · : V × V → C, yielding a respective linear map S 2 V → C or Λ 2 V → C. The Lie algebras o(V ) and sp(V ) are defined as the respective largest subalgebras of gl M (V, V ) for which the map . Then V is a submodule of V * (via the form ·, · ), and the g-module W := V * /V is irreducible. This can be proved for instance by considering W over the family of Lie subalgebras gl M (V ′ , V ′ * ) ⊂ g for varying decompositions of V as V ′ ⊕ V ′ * for maximal isotropic subspaces V ′ , V ′ * . Over each such subalgebra W is isomorphic to and hence has precisely two proper submodules. Since this submodules vary when V ′ and V ′ * vary, the module W is irreducible over g.
Furthermore, for any Young diagram λ, the irreducible gl M (V, V )-module V λ restricts to g yielding a generally reducible g-module. In all cases the socle of V λ | g is simple, and we denote it by V [λ] for g = o(V ) and by V λ for g = sp(V ). It is clear that the Lie algebras o(∞) and sp(∞) considered in [15] are subalgebras respectively of o(V ) and sp(V ), and by [14,Theorem 7.10] the socle filtrations of V λ | o(∞) and V λ | sp(∞) , described explicitly in [15], coincide with the respective socle filtrations of V λ | o(V ) and V λ | sp(V ) .
Then L λ,µ is a simple g-module. This can be seen by essentially the same argument as in the case of W . Moreover, L λ,µ ∼ = L λ ′ ,µ ′ if and only if λ = λ ′ and µ = µ ′ .
The analogue of the injective object I from Section 3.4 is constructed as follows. One sets F g := S 2 W for g = o(V ), Λ 2 W for g = sp(V ).
Furthermore, the quotient Q g of S 2 V * by the sum of kernels of the pairings V * ⊗ V → C and S 2 V → C admits a non-splitting exact sequence 0 → C → Q g → F g → 0.
The socle filtration of I has the form .

Then the embedding (3-1) induces embeddings
S k Q g ֒→ S k+1 Q g , which allow us to define I g as the colimit Moreover, by the same construction as in Section 3.7 I g is endowed with the structure of a commutative algebra. The category T g is introduced in the same way as in Section 3.4, where now J s = W ⊗l ⊗ V * ⊗m for pairs s = (l, m), l, m ∈ N, and the object I is replaced by I g . In the Introduction we denoted this category by T 2 g to emphasize that is generated as a tensor category by two modules V and V * . In the rest of the paper we use the shorter notation T g . We leave it to the reader to check that Proposition 3.10 holds also for the category T g , and that I g is an injective hull in T g of the object C. The respective partial order (l, m) (l ′ , m ′ ) on N × N is given by l ≥ l ′ , m ≤ m ′ , l + m ′ ≤ l ′ + m. The results of Section 3.3 also hold with obvious modification.
The canonical injective resolution (3-24) stays the same with F replaced by F g , however now the socle of the object (I g ) j = I g ⊗ Λ j F g decomposes as S λ V for g = sp(V ) and S λ ⊥ V for g = o(V ) where λ runs over all special partitions of degree j. Ext j T sp(V ) (X, C) = 0 if X ∼ = L λ ⊥ ,∅ for a special λ with |λ| = j, C if X ∼ = L λ ⊥ ,∅ for a special λ with |λ| = j.
Next, Theorem 3.18 and Proposition 3.19 stay valid with T replaced by T g . We leave it to the reader to modify Lemma 3.20 accordingly. Furthermore, Proposition 3.23, and Theorem 3.24 also hold for I g and Q g (instead of I and Q, respectively). The same applies to Proposition 3.25, Theorem 3.26 (with L λ,µ instead of L λ,µ,ν,π ), Theorem 3.27, Theorem 3.29, and Corollary 3.31.
The universality results from Section 3.9 also carry over to the cases g = o(V ), sp(V ). In particular, the category Ig T is defined in the same way as the category I T: it is the category of internal I g -modules in T g .
Note also that the analogue V * ⊗ V * → Q g ⊂ I g of the map (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14) is well defined and factors through maps S 2 V * → Q g and Λ 2 V * → Q g in the respective cases g = o(V ) and g = sp(V ). This defines pairings I g ⊗ V * ⊗ Ig ⊗I g ⊗ V * → I g ⊗ S 2 V * → I g (4-1) and respectively. Now we have Theorem 4.2 Let (D, ⊗, 1) be a tensor category, and x ֒→ x * be a monomorphism in D. Assume that a morphism in D p : x * ⊗ x * → 1 is given, satisfying p • σ = p for g = o(V ) and p • σ = −p for g = sp(V ), where σ is the flip automorphism of x * ⊗ x * as an object of the tensor category D coming from the assumption that D is a symmetric monoidal category. − the surjection V * → W to x * → x * /x.
(b) if D is additionally a Grothendieck category then F extends uniquely to a coproduct-preserving functor Ig T → D.
The universality of the tensor categories I o(V ) T and I sp(V ) T leads to the fact that they are equivalent as monoidal categories. More precisely, consider the (symmetric) tensor category T − sp(V ) defined in the same way as T sp(V ) but with the flip isomorphism replaced by −σ. One checks that T − sp(V ) is well-defined, i.e. that the new flip isomorphism on V * ⊗ V * induces a well-defined structure of tensor category preserving the monoidal structure on T sp(V ) .
In addition, one checks that there is a well-defined tensor category I sp(V ) T − of internal I-modules in T − sp(V ) which coincides with I sp(V ) T as a monoidal category.