Abstract
Let \(V_*\otimes V\rightarrow {\mathbb {C}}\) be a nondegenerate pairing of countabledimensional complex vector spaces V and \(V_*\). The Mackey Lie algebra \({\mathfrak {g}}=\mathfrak {gl}^M(V,V_*)\) corresponding to this pairing 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 \(V^*_*\). The category \({{\mathbb {T}}}\) 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 Imodules. 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 \(X\otimes Y\rightarrow {\mathbf{1 }}\) where 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, \({\mathbb {C}}\)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 \(X'\hookrightarrow X\), \(Y'\hookrightarrow Y\) and endowed with a pairing \(X\otimes Y\rightarrow \mathbf{1}\), the object \(\mathbf{1}\) being the monoidal unit.
The simpler problem of constructing a universal tensor category generated just by two objects X, Y endowed with pairing \(X\otimes Y\rightarrow \mathbf{1}\) was solved several years ago, and explicit constructions of such a category are given in [6, 19]. The construction in [6] realizes this category as a category \({{\mathbb {T}}}_{{\mathfrak {sl}}(\infty )}\) of representations of the Lie algebra \({\mathfrak {sl}}(\infty )\), choosing X as the natural \({\mathfrak {sl}}(\infty )\)module V, and Y as its restricted dual \(V_*\). Motivated mostly by a desire to understand better the representation theory of the Lie algebra \({\mathfrak {sl}}(\infty )\), in [13] a larger category was constructed, denoted \({\widetilde{Tens}}_{{\mathfrak {sl}}(\infty )}\), which contains also the algebraic dual modules \(V^*\) and \(V_*^*\). It is clear that the category \({\widetilde{Tens}}_{{\mathfrak {sl}}(\infty )}\) has a completely different flavor as its objects have uncountable length while \({{\mathbb {T}}}_{{\mathfrak {sl}}(\infty )}\) is a finitelength category.
However, in [14] the observation was made that the four representations V, \(V_*\), \(V^*\), \(V_*^*\) generate a finitelength tensor category \({{\mathbb {T}}}^{4}_{\mathfrak {gl}^M(V,V_*)}\) over the larger Lie algebra \(\mathfrak {gl}^M(V,V_*)\), see Sect. 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 \({{\mathbb {T}}}^{4}_{\mathfrak {gl}^M(V,V_*)}\) were determined in [3]. Furthermore, in [5] the tensor category \({{\mathbb {T}}}^{3}_{\mathfrak {gl}^M(V,V_*)}\), generated by V, \(V_*\), and \(V^*\), was studied in detail. It was proved that \({{\mathbb {T}}}^{3}_{\mathfrak {gl}^M(V,V_*)}\) is Koszul, and it was established that \({{\mathbb {T}}}^{3}_{\mathfrak {gl}^M(V,V_*)}\) is universal as a tensor category generated by two objects X, Y with a pairing \(X\otimes Y\rightarrow \mathbf{1}\), such that X has a subobject \(X'\hookrightarrow X\). 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 \(X\otimes Y\rightarrow \mathbf{1}\) 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 \({{\mathbb {T}}}^{4}_{\mathfrak {gl}^M(V,V_*)}\) with previously studied categories is that, as we show in the present paper, the injective hulls of simple objects are not objects of \({{\mathbb {T}}}^{4}_{\mathfrak {gl}^M(V,V_*)}\) but of a colimitcompletion of \({{\mathbb {T}}}^{4}_{\mathfrak {gl}^M(V,V_*)}\) which we denote simply by \({{\mathbb {T}}}\). In particular, the trivial module has an injective hull I in \({{\mathbb {T}}}\) 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 \({}_I{{\mathbb {T}}}\) of Imodules internal to \({{\mathbb {T}}}\). The morphisms in this new category are morphisms of \({\mathfrak {gl}}^M(V,V_*)\)modules as well as of Imodules. The simple objects of \({}_I{{\mathbb {T}}}\) are of the form \(I\otimes M\) where M is a simple module in \({{\mathbb {T}}}\).
A culminating result of the present paper is that the category \({}_I{{\mathbb {T}}}\) has the universality property stated in the first paragraph of this introduction. The pairs \(X'\hookrightarrow X\) and \(Y'\hookrightarrow Y\) are realized respectively as \(I \otimes V_*\subset I \otimes V^* \) and \(I \otimes V\subset I \otimes V_*^*\), I is the unity object in \({}_I{{\mathbb {T}}}\), and the tensor product in \({}_I{{\mathbb {T}}}\) is \(\otimes _I\).
Finally, in Sect. 4 we study analogues of the tensor categories \({{\mathbb {T}}}_{{{\mathfrak {o}}}(\infty )}\) and \({{\mathbb {T}}}_{{{\mathfrak {s}}}{{\mathfrak {p}}}(\infty )}\) considered in [6, 19]. Consider a tensor category generated by a single object X with a subobject \(X'\hookrightarrow X\) and a pairing \(X\otimes X\rightarrow {\mathbf{1 }}\). After identifying \(V_*\) and V, our construction of the category \({}_I{{\mathbb {T}}}\) yields a universal tensor category also in this setting. However, one can assume in addition that the pairing \(X\otimes X\rightarrow {\mathbf{1 }}\) is symmetric or antisymmetric, which leads to new universality problems for tensor categories. With this in mind, we introduce \({{\mathbb {T}}}^2_{{{\mathfrak {o}}}(V)}\) and \({{\mathbb {T}}}^2_{{{\mathfrak {s}}}{{\mathfrak {p}}}(V)}\) where \({{\mathfrak {o}}}(V)\) and \({{\mathfrak {s}}}{{\mathfrak {p}}}(V)\) are respective orthogonal and symplectic Lie algebras of a countabledimensional vector space V. In analogy with our previous constructions, we then produce appropriate categories \(_{I'} {{\mathbb {T}}}^2\) for \(I'=I_{{{\mathfrak {o}}}(V)}\) and \(I'=I_{{{\mathfrak {s}}}{{\mathfrak {p}}}(V)}\) and prove that these latter categories are universal in the respective new settings. Moreover, the categories \({}_{I_{{{\mathfrak {o}}}(V)}}{{\mathbb {T}}}^2\) and \({}_{I_{{{\mathfrak {s}}}{{\mathfrak {p}}}(V)}}{{\mathbb {T}}}^2\) are canonically equivalent as monoidal categories.
Preliminaries
Notation
All vector spaces are defined over \({{\mathbb {C}}}\) (more generally, we could work over an algebraically closed field of characteristic zero); similarly, all abelian categories and all functors between such are assumed \({{\mathbb {C}}}\)linear, and we refer to [17] for general background on abelian/additive categories.
By \(S^k X\) and \(\Lambda ^k X\) we denote respectively the kth 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 nondegenerate pairing \(V_*\otimes _{{{\mathbb {C}}}} V\rightarrow {{\mathbb {C}}}\) of countabledimensional vector spaces V and \(V_*\). This pairing defines embeddings \(V_*\subset V^*\), \(V\subset V^*_*\), where \(V^*={{\,\mathrm{{\mathrm {Hom}}}\,}}_{{{\mathbb {C}}}}(V,{{\mathbb {C}}})\), \(V_*^*={{\,\mathrm{{\mathrm {Hom}}}\,}}_{{{\mathbb {C}}}}(V_*,{{\mathbb {C}}})\). For any vector space M we set \(M^*={{\,\mathrm{{\mathrm {Hom}}}\,}}_{{{\mathbb {C}}}}(M,{{\mathbb {C}}})\). We abbreviate \(\otimes _{{{\mathbb {C}}}}\) as \(\otimes \). By \(\otimes \) we denote also tensor product in abstract tensor categories in the hope that this will cause no confusion.
Except in Sect. 4, \({{\mathfrak {g}}}\) will be the Mackey Lie algebra \({\mathfrak {gl}}^M\) \((V,V_*)\) of [14] associated to the pairing \(V_*\otimes V\rightarrow {{\mathbb {C}}}\). By definition,
where \(\varphi ^*:V^*\rightarrow V^*\) is the operator dual to \(\varphi \). We will describe \({{\mathfrak {g}}}\) explicitly as a Lie algebra of infinite matrices shortly.
We set
There is an extension
where Q is defined as the quotient of \(V^*\otimes V_*^*\) by the sum of the kernels of the pairings
In Proposition 3.5 below we prove that the extension (2.1) is nonsplitting.
We model the actions of \({{\mathfrak {g}}}\) on various modules mentioned above as follows:

\(V_*^*\) consists of infinite column vectors with entries indexed by \({{\mathbb {N}}}=\{0,1,\ldots \}\).

\(V\subset V^*_*\) consists of finite (or finitary) column vectors, i.e. those with at most finitely many nonzero entries.

Dually, \(V^*\) consists of \({{\mathbb {N}}}\)indexed infinite row vectors.

The elements of \(V_*\subset V^*\) are precisely the finite row vectors.

\({{\mathfrak {g}}}\) consists of \({{\mathbb {N}}}\times {{\mathbb {N}}}\)matrices with finite rows and columns, acting on \(V_*^*\) by left multiplication.

Similarly, \({{\mathfrak {g}}}\) acts on \(V^*\) as minus right multiplication.

\(V^*\otimes V^*_*\) consists of finiterank \({{\mathbb {N}}}\times {{\mathbb {N}}}\)matrices with infinite rows and columns, acted upon by \({{\mathfrak {g}}}\) by commutation.
We will frequently make use of Schur functors \({\mathbb {S}}_{\lambda }\) attached to Young diagrams \(\lambda \). Often we write \(X_{\lambda }\) instead of \({{\mathbb {S}}}_{\lambda } X\) for a vector space X. Moreover, \(S^k X={\mathbb {S}}_{\rho } X\), \(\Lambda ^k X={\mathbb {S}}_{\gamma }X\), where \(\rho \), \(\gamma \) are respectively a row and a column with k boxes.
For Young diagrams \(\lambda \), \(\mu \), \(\nu \) and \(\pi \) we write
and similarly, for nonnegative integers l, m, n, p we set
where \(V_{m,n}\) is the socle of \(V^{*\otimes m}\otimes V^{\otimes n}\), i.e.
for appropriate multiplicities \(m_{\mu ,\nu }\). Here \(\lambda \) denotes the degree (number of boxes) of a Young diagram \(\lambda \). Finally, for any subscript s of the form \((\bullet ,\bullet ,\bullet ,\bullet )\) we set
where I is the object constructed below in Sect. 3.
Definition 2.1
We refer to objects involving only the two outside diagrams \(\lambda \) and \(\pi \) as purely thick and those involving only the two middle diagrams as thin. Everything else is mixed.\(\blacklozenge \)
It is essential to recall Corollary 4.3 in [5] which claims that \(L_{\lambda ,\mu ,\nu ,\pi }\) is a simple \({{\mathfrak {g}}}\)module, and implies that \(L_{l,m,n,p}\) is a semisimple \({{\mathfrak {g}}}\)module.
The following remark will be used implicitly and repeatedly: given a short exact sequence
in a tensor abelian category, the symmetric power \(S^kx\) has a filtration
with isomorphisms
Plethysm
Given that \(F=W\otimes W_*\) 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 \({\mathbb {S}}_{\lambda }W\otimes {\mathbb {S}}_{\mu }W_*\). The result applies to a tensor product \(W\otimes W_*\) in any \({{\mathbb {C}}}\)linear tensor category, so we work in this generality throughout the present subsection.
We call a partition \(\lambda \) special if it satisfies the condition: all hooks of \(\lambda \) whose corner lies on the diagonal of \(\lambda \) have horizontal and vertical arms (not counting the corner) of length \(\mu _i1\) and \(\mu _i\), respectively, where \(\mu _1> \mu _2> \ldots > 0\) is a partition. We now recall the following result.
Proposition 2.2
Let x and y be two objects in a \({{\mathbb {C}}}\)linear tensor category. We have the following decompositions:

(a)
\(S^k(x\otimes y)\) is the direct sum of all objects of the form \({\mathbb {S}}_{\lambda }x\otimes {\mathbb {S}}_{\lambda }y\) as \(\lambda \) ranges over all Young diagrams of degree k.

(b)
\(\Lambda ^k(x\otimes y)\) is the direct sum of all objects of the form \({\mathbb {S}}_{\lambda }x\otimes {\mathbb {S}}_{\lambda ^{\perp }}y\) as \(\lambda \) ranges over all Young diagrams of degree k, where \(\lambda ^{\perp }\) denotes the conjugate partition.

(c)
\(S^k S^2 x\) is the direct sum of all \({{\mathbb {S}}}_{\lambda }x\) for partitions \(\lambda \) of degree 2k with even parts , i.e. even partitions.

(d)
\(S^k \Lambda ^2 x\) is \(\bigoplus _{\begin{array}{c} \text {even }\lambda \\ \lambda =2k \end{array}}{{\mathbb {S}}}_{\lambda ^{\perp }}x\).

(e)
\(\Lambda ^k\Lambda ^2 x\) is \(\bigoplus _{\begin{array}{c} \text {special }\lambda \\ \lambda =2k \end{array}}{{\mathbb {S}}}_{\lambda }x\).

(f)
\(\Lambda ^k S^2 x\) is \(\bigoplus _{\begin{array}{c} \text {special }\lambda \\ \lambda =2k \end{array}}{{\mathbb {S}}}_{\lambda ^{\perp }}x\).
Proof
(a) and (b) are reformulations of the Cauchy identities in [18, (6.2.8)]. The other four points paraphrase [11, Example I.8.6]. \(\blacksquare \)
Ordered Grothendieck Categories
We recall the following notion from [4, Definition 2.3].
Definition 2.3
Let \(({{\mathcal {P}}},\preceq )\) be a poset. An ordered Grothendieck category with underlying order \(({{\mathcal {P}}},\preceq )\) is a Grothendieck category \({{\mathcal {C}}}\) together with objects \(X_s\), \(s\in {{\mathcal {P}}}\) so that the following conditions hold.

(a)
The objects \(X_s\) are semiartinian, in the sense that all of their nonzero quotients have nonzero socles.

(b)
Every object in \({{\mathcal {C}}}\) is a subquotient of a direct sum of copies of various \(X_s\).

(c)
The simple subobjects in
$$\begin{aligned} {{\mathcal {S}}}_s:=\{\text {isomorphism classes of simples in }\mathrm {soc}X_s\} \end{aligned}$$(2.5)are mutually nonisomorphic for distinct s and they exhaust the simples in \({{\mathcal {C}}}\).

(d)
Simple subquotients of \(X_s\) outside the socle \(\mathrm {soc}X_s\) are in the socle of some \(X_t\), \(t\prec s\).

(e)
Each \(X_s\) is a direct sum of objects with simple socle.

(f)
Let \(t\prec s\). The maximal subobject \(X_{s\succ t}\subset X_{s}\) whose simple constituents belong to various \({{\mathcal {S}}}_r\) for \(s\succeq r\not \preceq t\) is the common kernel of a family of morphisms \(X_s\rightarrow X_t\).
\(\blacklozenge \)
Ordered Grothendieck categories are well behaved in a number of ways. For instance ([4, Corollary 2.6]):
Proposition 2.4
The indecomposable injective objects in an ordered Grothendieck category \({{\mathcal {C}}}\) are, up to isomorphism, precisely the indecomposable summands of the objects \(X_s\) in Definition 2.3. \(\blacksquare \)
Recall ([5, §3.2] or [4, Definition 2.8]):
Definition 2.5
For two elements \(i\prec j\) in \({{\mathcal {P}}}\) the defect d(i, j) is the supremum of the set of nonnegative 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 \(S\in {{\mathcal {S}}}_s\) and \(T\in {{\mathcal {S}}}_t\).\(\blacklozenge \)
According to [5, Proposition 2.9] ext functors in an ordered Grothendieck category exhibit the following “upper triangular” behavior.
Proposition 2.6
Let \(S\in {{\mathcal {S}}}_s\) and \(T\in {{\mathcal {S}}}_t\) be two simple objects in an ordered Grothendieck category. If \(\mathrm {Ext}^q(S,T)\ne 0\) then \(d(s,t)\ge q\). \(\blacksquare \)
It is implicit in the statement that, in particular, we have \(s\preceq t\) (see [5, Lemma 3.8]). One of our goals will be to show that in the ordered Grothendieck category \({{\mathbb {T}}}\) introduced in Sect. 3.4 below, we actually have equality , and hence the category \({{\mathbb {T}}}\) is Koszul in the following sense.
Definition 2.7
An ordered Grothendieck category is Koszul if for every \(q\ge 0\) and every two simple objects \(S\in {{\mathcal {S}}}_s\) and \(T\in {{\mathcal {S}}}_t\) the canonical Yoneda composition map
is surjective, where the sum ranges over all isomorphism classes of simples \(U_i\).\(\blacklozenge \)
This mimics one of the characterizations of Koszul connected graded algebras, namely the requirement that the graded ext algebra \(\mathrm {Ext}^*(k,k)\) 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 nonvanishing exts.
Definition 2.8
An ordered Grothendieck category is sharp if it satisfies the conclusion of Proposition 2.6 with equality rather than inequality.\(\blacklozenge \)
The relevance of the concept to the preceding discussion follows from
Theorem 2.9
Assume that \({{\mathcal {C}}}\) is an ordered Grothendieck category as in Definition 2.3, such that

the terms of the socle filtration of each indecomposable injective object have finite length;

\({{\mathcal {C}}}\) is sharp in the sense of Definition 2.8.
Then \({{\mathcal {C}}}\) is Koszul.
Proof
Fix arbitrary simple objects \(S\in {{\mathcal {S}}}_s\), \(T\in {{\mathcal {S}}}_t\) and a positive integer \(q\ge 2\). 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 \(\mathrm {Ext}^1(R_T,T)\), and Yoneda multiplication by that element induces an isomorphism
If \(\mathrm {Ext}^q(S,T)=0\) there is nothing to prove. Otherwise, our sharpness assumption shows that \(d(S,T)=q\). Now, the simples in the socle of \(R_T\) are smaller than T (with respect to the ordering), and those that appear as subquotients of \(\overline{R_T}:=R_T/\mathrm {soc}~R_T\) are smaller again. It follows that

if any simple subquotient U of \(\overline{R_T}\) were to satisfy \(d(S,U)=q1\) we would have \(d(S,T)\ge (q1)+2=q+1\) contradicting \(d(S,T)=q\),

and hence no such U can contribute to \(\mathrm {Ext}^{q1}(S,R_T)\).
In conclusion,
By sharpness again, (2.6) can be identified with
with U ranging over those simple summands of \(\mathrm {soc}~R_T\) with \(d(S,U)=q1\). It follows that every nonzero element of \(\mathrm {Ext}^q(S,T)\) will be contained in the image of the Yoneda map
where U ranges over all isomorphism classes of simple constituents of \(\mathrm {soc}R_T\). This finishes the proof. \(\blacksquare \)
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 \({{\mathcal {M}}}^C\) for its category of right comodules and \({{\mathcal {M}}}^C_{fin}\) for its category of finitedimensional comodules. Since the Grothendieck categories we are interested in will turn out to be of the form \({{\mathcal {M}}}^C\) 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.
Definition 2.10
A Grothendieck category is locally finite if it has a set of finitelength generators.\(\blacklozenge \)
We then have the following recognition result for categories of comodules over fields ([21, Theorem 5.1]):
Theorem 2.11
Let k be a field. A klinear Grothendieck category is equivalent to \({{\mathcal {M}}}^C\) for a kcoalgebra C if and only if it is locally finite in the sense of Definition 2.10 and the endomorphism ring of every simple object is finite dimensional over k.
Moreover, in this case \({{\mathcal {M}}}^C_{fin}\) can be identified with the subcategory of \({{\mathcal {C}}}\) consisting of finitelength objects. \(\blacksquare \)
The following notion (analogous to its dual ringtheoretic 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 Ccomodule is finite dimensional;

every finitedimensional left Ccomodule has a projective cover.
\(\blacklozenge \)
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 \({{\mathbb {C}}}\)linear abelian category \({{\mathcal {C}}}\) is monoidal if its monoidal structure has the property that \(x\otimes \bullet \) and \(\bullet \otimes x\) are exact endofunctors for every object x.
If in addition the monoidal structure is symmetric, \(({{\mathcal {C}}},\otimes )\) is a tensor category.
A tensor functor between tensor categories is a \({{\mathbb {C}}}\)linear symmetric monoidal functor.\(\blacklozenge \)
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 \(({{\mathcal {C}}},\otimes , \mathbf{1})\) for a monoidal category to specify both the tensor product bifunctor and the monoidal unit object \(\mathbf{1}\).
The Categories \({{\mathbb {T}}}\) and \({}_I{{\mathbb {T}}}\)
Definition of the Object I
For every nonnegative integer k we have a canonical embedding
obtained as the composition
where \(\iota :{{\mathbb {C}}}\rightarrow Q\) is the embedding defining Q as an extension of F by \({{\mathbb {C}}}\). This gives rise to an exact sequence
Taking the colimit (or simply union)
we obtain a \({{\mathfrak {g}}}\)module that has an infinite ascending filtration representable schematically as
where the boxes indicate the layers (successive quotients) of the filtration.
The morphism
to be defined below will play a central role in the sequel; we will occasionally write \(\psi \) for the resulting factorization \(I/{{\mathbb {C}}}\rightarrow F\otimes I\) as well, leaving it to context to separate the two possible meanings.
We obtain the morphism \(\psi \) as a colimit \(\varinjlim _k \psi ^k\) 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 degreeone elements primitive, i.e. such that the comultiplication
is the unique algebra map defined by
The comultiplication (3.8) is a morphism of \({{\mathfrak {g}}}\)modules. By definition, the map (3.7) is given by
where

the upper lefthand arrow is the \(Q\otimes S^{k1}Q\)component of the comultiplication
$$\begin{aligned} \Delta :S^kQ\rightarrow \bigoplus _{i=0}^k S^iQ\otimes S^{ki}Q \end{aligned}$$described above.

\(\pi \) is the epimorphism fitting in (3.3).
To make sense of \(\varinjlim _k \psi ^k\) we would have to argue that the maps \(\psi ^k\) are compatible with the embeddings
in (3.3), i.e. that the diagrams
commute for arbitrary k. This can be seen by direct examination, fixing a basis \(\{v_{\alpha }\}\) for Q with a distinguished element \(v_0=1\in {{\mathbb {C}}}\subset Q\) and noting that the upper lefthand map in (3.9) is defined on monomials by
Lemma 3.1
The kernel of \(\psi :I\rightarrow F\otimes I\) is precisely \({{\mathbb {C}}}\).
Proof
The kernel of the upper righthand map in (3.9) is
so we are in effect claiming that the preimage of (3.11) through the “partial comultiplication”
is \({{\mathbb {C}}}\subset S^kQ\).
This is easily seen from the explicit description (3.10) of the comultiplication (3.12). \(\blacksquare \)
Order
Following (or rather amplifying) [5], we order the quadruples (l, m, n, p) of nonnegative integers by setting
precisely if
For a quadruple \(s=(l,m,n,p)\) we define a family \(\Sigma _s\) of morphisms
for various \(s'\prec s\) as follows:

first, those of the form \(\theta \otimes {{\,\mathrm{id}\,}}_I\) where \(\theta \in \Theta _s\) as in [5, §3.2];

secondly, \({{\,\mathrm{id}\,}}_{J_s}\otimes \psi _0\) where
$$\begin{aligned} \psi _0:I\rightarrow F\otimes I \end{aligned}$$(3.14)is the morphism in (3.6).
The morphisms of the first type are such that their joint kernel is
On the other hand, the kernel of \(\psi _0\) is \({{\mathbb {C}}}\subset I\) and hence the joint kernel of \(\Sigma _s\) is
We now want to argue that (3.15) is precisely the inclusion of the socle:
Proposition 3.2
For every choice of nonnegative integers l, m, n, and p, the object \(L_{l,m,n,p}\) is the socle of \(I_{l,m,n,p}\) via the inclusion (3.15).
This will require some preparation. First, we have the following remark, in the spirit of [5, Lemma 3.1].
Lemma 3.3
Let \({{\mathfrak {G}}}\) be a Lie algebra and \(I\subseteq {{\mathfrak {G}}}\) an ideal. Suppose \(U\subseteq U'\) is an essential inclusion of \({{\mathfrak {G}}}/I\)modules and D is a \({{\mathfrak {G}}}\)module on which I acts densely. Then the inclusion
is also essential.
Proof
Let \(w_1,~\ldots ,~w_k\) be linearly independent vectors in D (\(k \ge 1\)), and consider an element
with nonzero \(u_i\). We claim that for every \({\overline{x}}\in {{\mathfrak {G}}}/I\), there is \(x\in {{\mathfrak {G}}}\) so that \({\overline{x}}\) is the image of x and
To see this, first choose an arbitrary \(y\in {{\mathfrak {G}}}\) with image \({\overline{x}}\) in \({{\mathfrak {G}}}/\) I, so that
On the other hand, by the density assumption there is some \(a\in I\) satisfying \(aw_i=yw_i\) for all i, and we can simply set \(x=ya\).
Having settled the claim and fixed an element f as above, we can now proceed. The density of \(U\subseteq U'\) means that we can find \({\overline{a}}\) in the universal enveloping algebra \(U({{\mathfrak {G}}}/I)\) such that

all \({\overline{a}}u_i\) belong to U, and

at least one of them (\({\overline{a}}u_j\), say) is nonzero.
Decomposing \({\overline{a}}\) as a polynomial in elements \({\overline{x}}\in {{\mathfrak {G}}}/I\) and lifting each of those to elements \(x\in {{\mathfrak {G}}}\) as in the claim, we obtain an element \(a\in U({{\mathfrak {G}}})\) with
Since \(w_i\) are linearly independent and \({\overline{a}}u_j\ne 0\), this is a nonzero element of \(U\otimes D\), and the proof is complete. \(\blacksquare \)
The following result will require some additional conventions and elaboration. Recall that Q is the quotient
We noted above that we identify the space \( V^*\otimes V^*_*\) with finiterank infinite \(\mathbb {N}\times \mathbb {N}\)matrices, and hence the quotient consists of equivalence classes of such matrices, where two are declared equivalent whenever they differ by a traceless matrix (\(a_{ij}\)) such that \(a_{ij}=0\) for large enough i and j.
We fix a basis \(\{e_{\alpha }\}_{\alpha \in A}\) of Q as follows:

\(e_0=1\in {{\mathbb {C}}}\);

all other basis elements are classes of rank1 matrices of the form \(v^* \otimes v \) for \(v^*\in V^*\) and \(v\in V^*_*\).
Lemma 3.4
Let \(X\subset V_*^*V\) and \(X^*\in V^*V_*\) be finite subsets, linearly independent modulo V and respectively \(V_*\). Fix \(x_0\in X\) and \(x_0^*\in X^*\). There is an element \(g\in {{\mathfrak {g}}}\) such that

\(gx\in V\) for all \(x\in X\),

\(gx^*\in V_*\) for all \(x^*\in X^*\),

\(g( x_0^* \otimes x_0)\) has nonzero trace,

\(g(x^*\otimes x )\) has zero trace for all other choices of \(x\in X\) and \(x^*\in X^*\).
Proof
The conclusion will follow from the remark that \({{\mathfrak {g}}}\) acts densely on sets \(X\cup X^*\), i.e. that given \(x\in X\) and \(x^*\in X^*\), the vectors \(gx\in V \text { and }gx^* \in V_*\) can be prescribed arbitrarily. Keeping this in mind, we can then find \(g\in {{\mathfrak {g}}}\) such that

\(gx=0\) for all \(x\in X\),

\(gx^*=0\) for all \(x\in X^*\setminus \{x_0^*\}\),

the inner product of \(gx_0^*\) with every \(x\in X\setminus \{x_0\}\) vanishes,

the inner product of \(gx_0^*\) with \(x_0\) does not vanish.
This choice will meet the requirements of the statement, hence the conclusion. \(\blacksquare \)
Proposition 3.5
Let \(F\subset W\cup W_*\) be a finite set of vectors and
be the Lie subalgebra that annihilates all elements of F. Then for every positive integer k the inclusion
is essential over \({{\mathfrak {K}}}={{\mathfrak {K}}}_F\).
Proof
We have to show that the \({{\mathfrak {K}}}\)submodule generated by any nonzero element of \(S^kQ\) intersects \({{\mathbb {C}}}\). We fix a basis \(\{e_{\alpha }\}_{\alpha \in A}\) for Q containing \(e_0=1\in {{\mathbb {C}}}\), as in the discussion preceding the statement of the present result. If we put a total order \(\le \) on the index set A, the elements
for tuples
form a basis of \(S^kQ\), where \(\sigma \) denotes the symmetrization operator on \(Q^{\otimes k}\).
We assign \(1=e_0\) degree zero and every other \(e_{\alpha }\) degree 1, thus allowing us to define a degree between 0 and k for each element (3.16) and by extension for each \(x\in S^kQ\), as the largest degree of a basis element (3.16) appearing in a decomposition of x.
We can now prove the claim that
by induction on the degree of x. Since the base case \(\deg (x)=0\) requires no proof, we focus on the induction step.
Decompose
with \(\deg (x)>0\). By Lemma 3.4 we can arrange for an element \(g\in {{\mathfrak {K}}}\) such that

g annihilates all elements of \(F\subset W\cup W_*\),

g sends one of the elements \(1\ne e_{\alpha }\) appearing among the tensorands in (3.17) to a nonzero scalar multiple of \(e_0\),

g annihilates all other \(e_{\alpha }\) appearing in (3.17).
Clearly then
and we can conclude the argument by using the induction hypothesis. \(\blacksquare \)
Proof of Proposition 3.2
We know that \(L_{l,m,n,p}\) is semisimple by [5, Corollary 4.3], so it suffices to show that (3.15) is essential.
Since W, \(W_*\) and I are trivial as \({\mathfrak {sl}}(\infty )\)modules and
is the socle over \({\mathfrak {sl}}(\infty )\), it follows by restricting to the latter subalgebra of \({{\mathfrak {g}}}\) that the inclusion
is essential, reducing the goal to proving that so is the inclusion
We can simplify this further: recall that
Now apply Lemma 3.3 in the following setup:

\({{\mathfrak {G}}}={{\mathfrak {g}}}\) and \(I={\mathfrak {sl}}(\infty )\);

the inclusion \(U\subseteq U'\) is
$$\begin{aligned} W_*^{\otimes l}\otimes W^{\otimes p}\subset W_*^{\otimes l}\otimes W^{\otimes p}\otimes I; \end{aligned}$$(3.19) 
D is any of the simple direct summands \(V_{\mu ,\nu }\) of \(V_{m,n}\).
Lemma 3.3 then shows that the inclusion (3.18) is indeed essential, provided the inclusion (3.19) is. In other words, it is enough to consider \(m=n=0\) in (3.18). Since I is the union of \(S^kQ\) as \(k\rightarrow \infty \), it will furthermore be sufficient to argue that, for every l, p and k, the inclusion
is essential.
We can now conclude via Proposition 3.5: an arbitrary nonzero element of the righthand side of (3.20) is of the form
where \(e_i=\sum _{j} a_{i,j} \otimes b_{i,j}\) are linearly independent elements of \(W_*^{\otimes l}\otimes W^{\otimes p}\) and \(v_i\in S^kQ\). Now let \(F\subset W\cup W_*\) be the finite set of vectors \(\{a_{i,j}, b_{i,j}\}\) and consider the annihilator \({{\mathfrak {K}}}_F\) of F, as in Proposition 3.5.
The Lie algebra \({{\mathfrak {K}}}_F\) leaves the subspace
invariant and its action makes that space isomorphic to \((S^kQ)^{\oplus r}\). 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.
Theorem 3.6
The simple objects \(L_{\lambda ,\mu ,\nu ,\pi }\) are mutually nonisomorphic and have scalar endomorphism algebras.
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 \({{\mathfrak {G}}}\) be a Lie algebra and \(J\subseteq {{\mathfrak {G}}}\) be an ideal. Suppose U, \(U'\) are two \({{\mathfrak {G}}}/J\)modules and D a \({{\mathfrak {G}}}\)module on which J acts densely and irreducibly with \({{\,\mathrm{{\mathrm {End}}}\,}}_J D={{\mathbb {C}}}\). Then, the inclusion
is an isomorphism. \(\blacksquare \)
Proposition 3.8
For any two nonnegative integers l, p and Young diagrams \(\mu \), \(\nu \) the endomorphism algebra in \({{\mathbb {T}}}\) of the object \(W_{*l}\otimes V_{\mu ,\nu }\otimes W_{p}\) is the group algebra \({{\mathbb {C}}}[S_l\times S_p]\), with the two symmetricgroup factors acting on the two outer tensorands.
Proof
We apply Lemma 3.7 to the ideal
with
The density of the action of \({\mathfrak {sl}}(\infty )\) on \(V_{\mu ,\nu }\) and the isomorphism \({{\,\mathrm{{\mathrm {End}}}\,}}_{{\mathfrak {sl}}(\infty )} V_{\mu ,\nu }\cong {{\mathbb {C}}}\) follow by realizing the object \(V_{\mu ,\nu }\) as a colimit of irreducible \({\mathfrak {sl}}_n\)modules, while establishing an isomorphism
is entirely parallel to [5, Proposition 3.2], whose proof we do not reprise here. Lemma 3.7 then implies the desired isomorphism
\(\blacksquare \)
Proof of Theorem 3.6
According to Proposition 3.8, the endomorphism algebra
is semisimple. Since the tensor products \(c_{\lambda }\otimes c_{\pi }\in {{\mathbb {C}}}[S_l]\otimes {{\mathbb {C}}}[S_p ]\) of Young projectors ranging over diagrams with \(\lambda =l,~\pi =p\), form a complete system of equivalence classes of minimal idempotents in (3.21) under inner conjugation, the semisimple object \(W_{*l}\otimes V_{\mu ,\nu }\otimes W_{p}\) has simple constituents isomorphic to
with \(L_{\lambda ,\mu ,\nu ,\pi }\) not isomorphic to \( L_{\lambda ',\mu ,\nu ,\pi '}\) for distinct pairs \((\lambda ,\pi )\ne (\lambda ',\pi ')\) because
Furthermore, using Proposition 3.8, we calculate
Since \((c_{\lambda }\otimes c_{\pi }){{\mathbb {C}}}[S_l\times S_p]=X\) is a simple \({{\mathbb {C}}}[S_l\times S_p] \)module, we have \((c_{\lambda }\otimes c_{\pi }){{\mathbb {C}}}[S_l\times S_p](c_{\lambda }\otimes c_{\pi })={{\,\mathrm{{\mathrm {End}}}\,}}_{{{\mathbb {C}}}[S_l\times S_p]} X={{\mathbb {C}}}\), and the statement is proved.
The Category \(\pmb {{{\mathbb {T}}}}\)
Definition 3.9
The category \({{\mathbb {T}}}\) is the smallest full tensor Grothendieck subcategory of the category \({}_{{{\mathfrak {g}}}}{\mathrm {Mod}}\) of \({{\mathfrak {g}}}\)modules, closed under taking subquotients, and containing

the objects \(V^*\) and \(V_*^*\) (and hence also \(J_s\) of (2.3) for quadruples \(s=(l,m,n,p)\));

the object I of (3.4). \(\blacklozenge \)
The indices \(s=(l,m,n,p)\) form a poset \(({{\mathcal {P}}},\preceq )\) under the ordering introduced in § 3.2. Keeping that in mind, we have
Proposition 3.10
\({{\mathbb {T}}}\) is an ordered Grothendieck category in the sense of Definition 2.3.
Proof
We have to check the conditions listed in Definition 2.3. Here, the objects \(X_s\) will be the objects \(I_s\) from (2.4) for \(s=(l,m,n,p)\).
Condition (a). This follows from the fact that all \(I_s\) have countable filtrations whose subquotients are simple objects of the form \(L_{\lambda ,\mu ,\nu ,\pi }\) as in (2.2). 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 \(I_s=J_s\otimes I\) 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 \(W_{\lambda }\otimes W_{*\lambda }\) by part (a) of Proposition 2.2. Hence, tensoring a simple subquotient \(S\in {{\mathcal {S}}}_s\) 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 \(t\prec s\) according to our ordering (3.13).
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 \(t\prec s\), hence the conclusion.
Condition (e). Indeed, for \(s=(l,m,n,p)\) the object \(I_s\) decomposes as
where the sum ranges over \(\lambda =l\), \(\mu =m\), etc. The summands have simple respective socles \(L_{\lambda ,\mu ,\nu ,\pi }\) by Proposition 3.2.
Condition (f). The morphisms \(I_s\rightarrow I_t\) will be compositions of the obvious ones:

projecting one of the tensorands \(V^*\) of \(I_s=J_s\otimes I\) onto \(W_*\);

the dual analogue, \(V^*_*\rightarrow W\);

the surjection defining Q,
$$\begin{aligned} V^*\otimes V^*_*\rightarrow Q\subset I; \end{aligned}$$(3.22) 
applying the morphism \(I\rightarrow F\otimes I\) in (3.14) to the tensorand I of \(I_s\).
The verification that the joint kernel of these maps is as claimed is routine. \(\blacksquare \)
In particular, [4, Proposition 2.5] and Proposition 3.2 together prove
Theorem 3.11
For every quadruple \((\lambda ,\mu ,\nu ,\pi )\) of Young diagrams, \(I_{\lambda ,\mu ,\nu ,\pi }\) is an injective hull in \({\mathbb {T}}\) of \(L_{\lambda ,\mu ,\nu ,\pi }\).
We record the following observation.
Lemma 3.12
Let \(i=(l,m,n,p)\) and \(i'=(l',m',n',p')\) be two elements of the poset described in (3.13). Then \(i\preceq i'\) implies
Remark 3.13
The category \({{\mathbb {T}}}\) is symmetric with respect to the simultaneous interchange \(V \leftrightarrow V_*\), \(V^*\leftrightarrow V_*^*\). Numerically, this corresponds to \(l\leftrightarrow p\) and \(n\leftrightarrow m\). Lemma 3.12 is compatible with this transformation: according to the last condition in (3.13) we have
so we could have substituted \(pp' + m'm\) for \(ll' + n'n\) in Lemma 3.12.\(\blacklozenge \)
Injective Resolutions
We will now show that \({{\mathbb {C}}}\) admits an injective resolution in \({{\mathbb {T}}}\)
with
We will also see that \(I_j/\mathrm {Im}(I_{j1})\) admits an ascending filtration with layers
where each diagram has \(j+1\) rows.
To streamline the notation, for such Young diagrams we denote by \((l,j\times 1)\) the diagram with a row of length l and j singlebox rows.
To define the maps
we mimic the procedure used in the definition of (3.14). In fact, that notation will be compatible with (3.24), in that we will recover that earlier map by setting \(j=0\) in the latter. As before, (3.24) will be a colimit as \(k\rightarrow \infty \) of maps
The analogue of diagram (3.9) in this context is
where

the upper lefthand map is \({{\,\mathrm{id}\,}}_{\Lambda ^jF}\otimes \Delta \) with
$$\begin{aligned} \Delta :S^kQ\rightarrow Q\otimes S^{k1}Q, \end{aligned}$$the partial comultiplication also appearing in (3.9);

the upper middle map is \({{\,\mathrm{id}\,}}\otimes \pi \otimes {{\,\mathrm{id}\,}}\), with \(\pi :Q\rightarrow F\) the canonical surjection (again as in (3.9));

the upper righthand map is the multiplication
$$\begin{aligned} \Lambda ^j F\otimes F\rightarrow \Lambda ^{j+1}F \end{aligned}$$in the exterior algebra \(\Lambda ^{\bullet } F\), tensored with the identity on \(S^{k1}Q\).
To show that the maps (3.24) fit into a resolution (3.23), we begin with the following simple observation.
Lemma 3.14
Let j, k be two positive integers and X be a vector space of dimension larger than \(j+1\). The map
with \(\Delta :S^k X\rightarrow X\otimes S^{k1}X\) defined as in (3.10) annihilates the direct summand \({{\mathbb {S}}}_{(k+1, (j1)\times 1)} X\) of \(\Lambda ^j X\otimes S^{k}X\) and maps the complementary summand \({{\mathbb {S}}}_{(k, j\times 1)} X\) of \(\Lambda ^j X\otimes S^{k}X\) isomorphically onto the corresponding summand of the codomain \(\Lambda ^{j+1} X\otimes S^{k1}X\).
Proof
That the domain and codomain decompose as
and
respectively, follows from the LittlewoodRichardson rule [8, Appendix A, (A.8)]. The claim can be checked on finitedimensional vector spaces first, where all four direct summands are irreducible representations of the algebraic group GL(X), then passing to arbitrary X by taking a colimit. \(\blacksquare \)
Now consider one of the objects \(\Lambda ^j F\otimes I\), \(j\ge 0\) under discussion. Since I has the filtration (3.5), the object \(\Lambda ^j F\otimes I\) has a filtration by the subobjects \(\Lambda ^{j} F\otimes S^{k}Q\), with consecutive quotients \(\Lambda ^j F\otimes S^k F\). Moreover, these quotients are decomposed as
Moreover, these decompositions are canonical, i.e. the summands are unique.
We write
for the kernel of the map (3.24) and
by convention, we set \(K_j^{1}=\{0\}\). We now have
Lemma 3.15
For each \(k\ge 0\) the quotient
is the jrow summand of \(\Lambda ^{j} F\otimes S^{k}F\).
Proof
The map \(\psi _j\) respects the filtrations of its domain and codomain, by
respectively, and the associated graded map \(\mathrm {gr}~\psi _j\), in degree k, is precisely (3.27) with \(X=F\). By Lemma 3.14 this means that the degreek kernel of \(\mathrm {gr}~\psi _j\) is the jrow summand of \(\Lambda ^j F\otimes S^k F\). This verifies the statement at the associatedgraded level.
To conclude, it will suffice to construct gradings on the domain and codomain of \(\psi _j\), compatible with \(\psi _j\), that give back the filtrations by (3.31). This would then prove that the filtered map \(\psi _j\) 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 follows: as in the discussion preceding Lemma 3.1, we fix a basis \(\{v_{\alpha }\}\) for Q with \(v_0=1\in {{\mathbb {C}}}\subset Q\), and assign
One checks easily that \(\psi _j\) preserves degrees, finishing the proof as described above. \(\blacksquare \)
We can now finally complete the discussion on the injective resolution (3.23).
Theorem 3.16
The morphisms (3.24) fit into an exact sequence (3.23).
Proof
The maps \(\psi _j\) fit into a sequence
(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 (3.32) arise from gradings compatible with the maps \(\psi _j\). \(\blacksquare \)
Corollary 3.17
For a simple object X of \({{\mathbb {T}}}\) we have
Proof
The statement follows from the existence of the injective resolution (3.23) of the trivial object \({{\mathbb {C}}}\), since by Theorem 3.11
which in turn, by Proposition 2.2, (b), decomposes as
\(\blacksquare \)
Koszulity
We will eventually show that the category \({{\mathbb {T}}}\) is Koszul. To that end, we first need to strengthen Proposition 2.6 to an equality:
Theorem 3.18
The Grothendieck category \({{{\mathbb {T}}}}\) is sharp in the sense of Definition 2.8: for any two simples \(S,T\in {{{\mathbb {T}}}}\) we have
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 \(T\in {{\mathcal {S}}}_t\). 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 Definition 3.17 for the trivial object \(T={{\mathbb {C}}}\) and in general for purely thick T from Theorem 3.11, which implies that we can obtain an injective resolution for T by simply tensoring (3.23) with T. \(\blacksquare \)
In order to push past purely thick objects we need a version of [5, Lemma 3.13], requiring some notation: for a quadruple \((\lambda ,\mu ,\nu ,\pi )\) we write \(L^{+\ell }_{\lambda ,\mu ,\nu ,\pi }\) for the direct sum of all \(L_{\lambda ,\mu ',\nu ,\pi }\) with \(\mu '\) obtained by adding a box to \(\mu \). Here \(\ell \) stands for left, and we have a similarly defined object \(L^{+r}_{\lambda ,\mu ,\nu ,\pi }\) (for right) obtained by enlarging \(\nu \) instead.
Lemma 3.20
Consider a simple object \(L_{\lambda ,\mu ,\nu ,\pi }\).

(a)
We have an exact sequence
$$\begin{aligned} 0\rightarrow L^{+\ell }_{\lambda ,\mu ,\nu ,\pi }\rightarrow V^*\otimes L_{\lambda ,\mu ,\nu ,\pi } \rightarrow H \rightarrow 0 \end{aligned}$$(3.33)where H is a sum of simple objects \(L_{\lambda ',\mu ,\nu ,\pi }\) with \(\lambda '=\lambda +1\) and \(L_{\lambda ,\mu ,\nu ',\pi }\) with \(\nu '=\nu 1\).

(b)
Tensoring \(L_{\lambda ,\mu ,\nu ,\pi }\) with \(V_*^*\) produces a similar exact sequence, containing \(L_{\lambda ,\mu ,\nu ,\pi }^{+r}\) rather than \(L_{\lambda ,\mu ,\nu ,\pi }^{+\ell }\).
Proof
We focus on (a), the other half being entirely analogous.
The proof follows the same line of reasoning as that of [5, Lemma 3.13]. We first tensor the extension
with \(L_{\lambda ,\mu ,\nu ,\pi }\) to obtain a sequence (3.33) with an as yet unidentified H, itself fitting into an extension
with \({\widetilde{H}}\) being a direct sum of simples obtained by evaluating one tensorand \(V_*\) against the \(\nu \) component of \(L_{\lambda ,\mu ,\nu ,\pi }\). It follows that \({\widetilde{H}}\) is a direct sum of simple objects \(L_{\lambda ,\mu ,\nu ',\pi }\) with \(\nu '=\nu 1\), and the splitting of (3.34) follows from Proposition 2.6 and the observation that the indices of simple direct summands \(L_{\lambda ',\mu ,\nu ,\pi }\) of \(W_*\otimes L_{\lambda ,\mu ,\nu ,\pi }\) and \(L_{\lambda ,\mu ,\nu ',\pi }\) of \({\widetilde{H}}\) are not comparable with respect to the partial order (3.13). \(\blacksquare \)
Proof of Theorem 3.18
This too presents no substantial difficulties beyond those encountered in [5, Theorem 3.11]. Setting \(s=:(\lambda ,\mu ,\nu ,\pi )\) and \(t=:(\lambda ',\mu ',\nu ',\pi ')\), the argument proceeds by induction on \(q+\mu '+\nu '\). The base case follows from Proposition 3.19 for empty diagrams \(\mu '\) and \(\nu '\), and trivially for \(q=0\).
If \(\mu '\) and \(\nu '\) are empty we can fall back on Proposition 3.19. Otherwise, suppose for instance that \(\mu '\) is nonempty. We can then embed T as a direct summand in \(L^{+\ell }_{\lambda ',\beta ,\nu ',\pi '}\) for \(\beta =\mu '1\). The assumed nonvanishing of \(\mathrm {Ext}^q(S,T)\) and the long exact ext sequence applied to the extension
provided by Lemma 3.20 forces us into one of two cases:
1: \(\mathrm {Ext}^{q1}(S,H')\ne 0.\) By the induction hypothesis we have \(d(S,U)=q1\) for some simple direct summand U of \(H'\), and the conclusion follows from this and the fact that \(d(U,T)=1\) for all such U (by the description of \(H'\) in Lemma 3.20 and the formula for the defect provided by Lemma 3.12).
2: \(\mathrm {Ext}^{q}(S,V^*\otimes L_{\lambda ',\beta ,\nu ',\pi '})\ne 0.\) This means that S is a direct summand of the socle of \(Z_q\) for any injective resolution
By induction we know that the socle of the qth term of an injective resolution
consists of simples U with \(d(U,L_{\lambda ',\beta ,\nu ',\pi '})=q\). Now note that an injective resolution (3.35) can be obtained by tensoring (3.36) with \(V^*\). The simple direct summands of the socle of \(V^*\otimes Y_q\), including S, differ from those of \(Y_q\) in that their \(\mu \) diagrams have one extra box, meaning that indeed
The case when \(\mu '\) is empty but \(\nu '\) 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
Theorem 3.21
The ordered Grothendieck category \({{{\mathbb {T}}}}\) is Koszul in the sense of Definition 2.7. \(\blacksquare \)
Furthermore, we have the following analogue of [5, Corollary 3.19] and [4, Corollary 4.25 (d)]. In the statement, \({{\mathbb {T}}}_{fin}\subset {{\mathbb {T}}}\) denotes the full subcategory consisting of finitelength objects.
Theorem 3.22
The Grothendieck category \({{{\mathbb {T}}}}\) is equivalent to the category \({{\mathcal {M}}}^C\) of comodules over a Koszul graded coalgebra C, with \({{\mathbb {T}}}_{fin}\simeq {{\mathcal {M}}}^C_{fin}\).
Proof
The hypotheses of Theorem 2.11 are met (for the ground field \({{\mathbb {C}}}\)): \({{{\mathbb {T}}}}\) is generated by the finitelength objects in \({{\mathbb {T}}}\), since every object is isomorphic to a subquotient of a direct sum of indecomposable injectives \(I_s\) as defined in (3.9), and in turn the injectives \(I_s\) are unions of their finitelength truncations
Moreover, according to Theorem 3.6, the endomorphism ring of a simple object \(L_{\lambda ,\mu ,\nu ,\pi }\) as in (2.2) is the field \({{\mathbb {C}}}\). \(\mathrel {\Box }\)
An Internal Commutative Algebra and Its Modules
The object I has a structure of commutative algebra internal to the tensor category \({}_{{{\mathfrak {g}}}}{\mathrm {Mod}}\) of \({{\mathfrak {g}}}\)modules. To see this, we observe that I is isomorphic as a \({{\mathfrak {g}}}\)module to a quotient algebra of the symmetric algebra \(S^{\bullet }Q\). Indeed, denote by a the distinguished element \(1\in {{\mathbb {C}}}\subset Q\) of the degreeone component \(Q\subset S^{\bullet }Q\) and consider the commutative algebra
where 1 ist the unit of the symmetric algebra and \((a1)\) is the ideal generated by \(a1\). This ideal is clearly \({{\mathfrak {g}}}\)stable and \(S^{\bullet }Q/(a1)\) is an algebra in \({}_{{\mathfrak {g}}}{\mathrm {Mod}}\). Moreover, the definition of I as \(\varinjlim S^k Q\) implies that there is an isomorphism of \({{\mathfrak {g}}}\)modules
We will be interested in the category \({}_I{{\mathbb {T}}}\) of Imodules internal to \({{\mathbb {T}}}\). This is clearly a Grothendieck category. Moreover, the forgetful functor
fits into an adjunction
We refer to Imodules in the image of the functor \(I\otimes \bullet \) as free. We will see that tensoring with I has the effect of “partially semisimplifying” \({{\mathbb {T}}}\), in the following sense.
Proposition 3.23
For every positive integer n, the filtration
splits in \({}_I{{\mathbb {T}}}\) upon tensoring it with I. Consequently, for every n the object \(I\otimes S^nQ\) is injective in \({{\mathbb {T}}}\).
Proof
We prove this inductively on n. For \(n=1\) the claim is that the embedding
splits in \({}_I{{\mathbb {T}}}\). To see this, consider the embedding
in \({{\mathbb {T}}}\). It corresponds, via the adjunction (3.37), to a morphism in \(_I{{\mathbb {T}}}\)
that is clearly the identity on the Isubmodule
The morphism \(\sigma \) is the required splitting, concluding the base case \(n=1\) of the induction.
The argument also shows that we have a decomposition
in \({}_I{{\mathbb {T}}}\) (and hence also in \({{\mathbb {T}}}\)), implying that \(I\otimes Q\) is injective in \({{\mathbb {T}}}\) (by Theorem 3.11 for instance, which shows that both summands in (3.38) are injective).
We regard Q as a subobject of \(S^2Q\) via the embedding \(Q\hookrightarrow S^2Q\) described in (3.2). By the injectivity of \(I\otimes Q\in {{\mathbb {T}}}\) noted above, the embedding
extends to a morphism in \({{\mathbb {T}}}\)
Once more, the adjunction (3.37) retrieves a morphism in \(_I{{\mathbb {T}}}\)
that restricts to the identity on the submodule
showing that the embedding Definition 3.39 splits in \({}_I{{\mathbb {T}}}\). This proves the main claim for \(n=2\) and the fact that there is a splitting
meaning that \(I\otimes S^2Q\) is injective in \({{\mathbb {T}}}\). We now repeat the procedure recursively to complete the inductive argument. \(\mathrel {\Box }\)
Since it will be our goal to study the category \({}_I{{\mathbb {T}}}\) along the same lines as \({{\mathbb {T}}}\), we next turn to simple objects therein.
Theorem 3.24
The simple objects in \({}_I{{\mathbb {T}}}\) are (up to isomorphism) precisely the free Imodules \(I\otimes S\) for simples \(S\in {{\mathbb {T}}}\). For each of them, the endomorphism algebra in \({}_I{{\mathbb {T}}}\) is \({{\mathbb {C}}}\).
Proof
We first prove that \(I\otimes S\) is simple in \({}_I{{\mathbb {T}}}\). The simple objects of \({{\mathbb {T}}}\) are precisely the various modules \(L_{\lambda ,\mu ,\nu ,\pi }\) of (2.2), and according to Theorem 3.11 the injective hull \(S\subset I_S\) contains \(I\otimes S\) (\(I_S\) exists because \({}_I {{\mathbb {T}}}\) is a Grothendieck category). Since the embedding
is essential in \(I_S\), it is also essential in \(I\otimes S\). It follows that any nonzero subobject
in \({}_I{{\mathbb {T}}}\) contains S and hence the Imodule \(I\otimes S\) it generates, so \(T=I\otimes S\). This concludes the proof of the claim that all \(I\otimes S\) are simple.
The existence of an isomorphism
follows from the observation that by the adjunction (3.37) there is an isomorphism
Indeed, since \(I\otimes S\) is an injective hull of S in \({{\mathbb {T}}}\), every morphism \(S\rightarrow I\otimes S\) factors through the socle \(S\subset I\otimes S\), and therefore the isomorphism \({{\,\mathrm{{\mathrm {End}}}\,}}_{{{\mathbb {T}}}} S={{\mathbb {C}}}\) implies (3.40).
As for the fact that \(I\otimes S\) are, up to isomorphism, all irreducible objects in \({}_I{{\mathbb {T}}}\), consider an arbitrary object T and note that it must have a simple subobject \(S\in {{\mathbb {T}}}\). Hence T must be a quotient of the (irreducible!) free object \(I\otimes S\in {}_I{{\mathbb {T}}}\). Consequently, we have \(T\cong I\otimes S\) as desired. \(\blacksquare \)
Since I is a commutative algebra in \({{\mathbb {T}}}\), the category \({}_I{{\mathbb {T}}}\) of internal modules has a natural symmetric monoidal structure for which I is the unit object and \(\otimes _I\) is the tensor product. Whenever we refer to \({}_I{{\mathbb {T}}}\) as a tensor category, this will be the structure we consider.
The Category \({}_I{{\mathbb {T}}}\)
We are now ready to apply to \({}_I{{\mathbb {T}}}\) the same treatment we subjected \({{\mathbb {T}}}\) to. We work with precisely the same poset \(({{\mathcal {P}}},\preceq )\) of quadruples (l, m, n, p) of nonnegative integers with the ordering described in (3.13), and the corresponding objects \(I_s=I\otimes J_s\in {}_I{{\mathbb {T}}}\) for \(s\in {{\mathcal {P}}}\), as in Eqs. (2.2)–(2.4).
We will similarly consider the simple (by Theorem 3.24) objects of \({}_I{{\mathbb {T}}}\)
and the semisimple objects \(T_{l,m,n,p}:=I\otimes L_{l,m,n,p}\), that are direct sums of the various \(T_{\lambda ,\mu ,\nu ,\pi }\).
We now have the following analogue of Proposition 3.10.
Proposition 3.25
\({}_I{{\mathbb {T}}}\) 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 \({}_I{{\mathbb {T}}}\) rather than just \({{\mathbb {T}}}\)), the argument proceeds much as in the proof of Proposition 3.10 with a small difference in how we define the morphisms \(I_s\rightarrow I_t\) for \(t\prec s\) 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 \(I_s=J_s\otimes I\) onto \(W_*\);

the dual analogue, \(V^*_*\rightarrow W\);

the “pairing”
$$\begin{aligned} I_{0,1,0,0}\otimes _I I_{0,0,1,0} = (I\otimes V^*)\otimes _{I} (I\otimes V_*^*)\cong I\otimes V^*\otimes V_*^*\rightarrow I \end{aligned}$$(3.42)obtained via the adjunction (3.37) from the composition
$$\begin{aligned} V^*\otimes V_*^*\rightarrow Q\subset I \end{aligned}$$in (3.22).
Everything else goes through as sketched in the proof of Proposition 3.10. \(\blacksquare \)
The difference from \({{\mathbb {T}}}\) is that now the free Imodules generated by the full duals \(V^*\) and \(V_*^*\) admit the pairing (3.42) valued in the unit object I of the category \({}_I{{\mathbb {T}}}\) under consideration.
We also have an Imodule version of Theorem 3.11.
Theorem 3.26
For every quadruple \((\lambda ,\mu ,\nu ,\pi )\) of Young diagrams, the inclusion
obtained by applying the functor \(I\otimes \bullet \) to the inclusion
is an injective hull in \({}_I{{\mathbb {T}}}\). \(\blacksquare \)
Just as \({{\mathbb {T}}}\), the category \({}_I{{\mathbb {T}}}\) can be realized as comodules over a coalgebra (see Theorem 3.22). As in that previous result, we denote by \({}_I{{\mathbb {T}}}_{fin}\subset {}_I{{\mathbb {T}}}\) the full subcategory of finitelength objects. Note that the indecomposable injectives
have finite length: \(J_{\lambda ,\mu ,\nu ,\pi }\) have finite filtrations with subquotients simple in \({{\mathbb {T}}}\), and according to Theorem 3.24 tensoring these simple objects by I produces simples in \({}_I{{\mathbb {T}}}\).
Theorem 3.27
The Grothendieck category \({}_I{{{\mathbb {T}}}}\) is equivalent to the category \({{\mathcal {M}}}^D\) of comodules over a coalgebra D, with \({}_I{{\mathbb {T}}}_{fin}\simeq {{\mathcal {M}}}^D_{fin}\). Furthermore, the coalgebra D is left semiperfect in the sense of Definition 2.12.
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 Imodules).
The additional remark, that D is semiperfect, follows directly from Definition 2.12 and the fact that, as observed above, in \({}_I{{\mathbb {T}}}\) the indecomposable injectives \(I_{\lambda ,\mu ,\nu ,\pi }\) have finite length. \(\blacksquare \)
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 \({}_I{{\mathbb {T}}}'\) of \({}_I{{\mathbb {T}}}\) generated by the morphisms described in the proof of Proposition 3.25 is the full subcategory containing \(I_{l,m,n,p}\). \(\blacksquare \)
We next turn to the Koszulity of \({}_I{{\mathbb {T}}}\). In keeping with the theme, the argument will be very similar to what we saw in proving Theorems 3.18 and 3.21.
Theorem 3.29
The Grothendieck category \({}_I{{\mathbb {T}}}\) is sharp in the sense of Definition 2.8: for any two simples \(S,T\in {{{\mathbb {T}}}}\) we have
In particular, the ordered Grothendieck category \({}_I{{\mathbb {T}}}\) 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 (3.33) replaced by its analogue, obtained by simply tensoring it with I. \(\blacksquare \)
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 \(L_{\lambda ,\emptyset ,\emptyset ,\pi }\) the corresponding simple object of \({}_{I}{{\mathbb {T}}}\)
is injective.\(\blacklozenge \)
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. \(\blacksquare \)
We end the present subsection with description of one possible choice for the graded coalgebra C from Theorem 3.22. This discussion parallels [5, §3.4], which in turn is analogous to [6, §5].
Let T be the tensor algebra in \({}_I{{\mathbb {T}}}\) of the object
with \(T_d\) denoting its degreed component, and the nonunital algebra of endomorphisms
The algebra \({\mathcal {A}}\) is naturally \({{\mathbb {Z}}}_{\ge 0}\)graded by means of the defect introduced in Definition 2.5:
Finally, the coalgebra C is simply the graded dual of \({\mathcal {A}}\), with \(C_d={\mathcal {A}}_d^*\).
The fact that C (and hence \({\mathcal {A}}\)) is Koszul then implies
Proposition 3.32
The algebra \({\mathcal {A}}\) is quadratic.
Proof
Koszul algebras are well known to be quadratic; see e.g. [2, §2.3]. \(\blacksquare \)
Universality
We can now characterize \({}_I{{\mathbb {T}}}\) as a universal category in the sense of [5, Theorem 4.23] and [4, Theorem 5.2]. First, note that in \({}_I{{\mathbb {T}}}\) there is a pairing
corresponding to (3.22) through the adjunction (3.37). We will occasionally indicate tensoring with I by a lefthand ‘I’ subscript, as in \({}_IX:= I\otimes X\).
Theorem 3.33
Let \(({{\mathcal {D}}},\ \otimes ,\ \mathbf{1})\) be a (\({{\mathbb {C}}}\)linear abelian) tensor category, \(x\hookrightarrow x_*^*\) and \(x_*\hookrightarrow x^*\) be monomorphisms in \({{\mathcal {D}}}\), and
be a morphism in \({{\mathcal {D}}}\).

(a)
There is a unique (up to monoidal natural isomorphism) left exact symmetric monoidal functor \(F:{}_I{{\mathbb {T}}}_{fin}\rightarrow {{\mathcal {D}}}\) sending

the pairing (3.43) to p;

the surjection \({}_IV_*^*\rightarrow {}_IW\) to \(x_*^*\rightarrow x_*^*/x\);

the surjection \({}_IV_*^*\rightarrow {}_IW_*\) to \(x^*\rightarrow x^*/x_*\).


(b)
if \({{\mathcal {D}}}\) is additionally a Grothendieck category then F extends uniquely to a coproductpreserving functor \({}_I{{\mathbb {T}}}\rightarrow {{\mathcal {D}}}\).
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 \(\le 2\). In degree zero things are simple: the following result is the version of [5, Lemma 3.24] appropriate here.
Lemma 3.34
For \((l,m,n,p)\in {{\mathcal {P}}}\) the endomorphism algebra of the injective object \(I_{l,m,n,p}\in {}_I{{\mathbb {T}}}\) is isomorphic to \({{\mathbb {C}}}[S_l\times S_m\times S_n\times S_p]\), with the symmetric groups acting naturally on the relevant tensorands of
Proof
We have
The quotient of \(J_{l,m,n,p}\) by its socle in \({{\mathbb {T}}}\) has a filtration by subquotients \(L_{\lambda ',\mu ',\nu ',\pi '}\) with
which thus admit no nonzero morphisms to
It follows that restricting an arbitrary morphism
in \({{\mathbb {T}}}\) to the socle induces an isomorphism
We can see that the righthand side of (3.44) is naturally identifiable with \({{\mathbb {C}}}[S_l\times S_m\times S_n\times S_p]\) as in Proposition 3.8. \(\blacksquare \)
As for degree 1, we need an analogue of [5, Lemma 3.25]. Stating such an analogue will require some notation. Degreeone morphisms between the objects \(I_{l,m,n,p}\in {}_I{{\mathbb {T}}}\) come in three flavors:
We distinguish families of each flavor, as follows. The morphism
executes the pairing
of the ith tensorand \(V^*\) and the jth tensorand \(V_*^*\) in
and acts as the identity on all other tensorands.
Next, we have the map
which

first permutes cyclically the first i tensorands \(V^*\);

maps the new first (old ith) tensorand \(V^*\) onto \(W_*=V^*/V_*\);

finally permutes the last \(mj+1\) tensorands \(W^*\) cyclically so the newlycreated \(W_*\) becomes the jth.
Finally, we have the leftright mirror image
of \({}_{i,j}\pi \), obtained by substituting \(V_*^*\) for \(V^*\), W for \(W_*\), reversing the directions of the cyclic permutations, etc.
We write
for products of symmetric groups and, unless specified otherwise, morphism spaces in Lemma 3.35 below are in the category \({}_I{{\mathbb {T}}}\).
Lemma 3.35
Let \((l,m,n,p)\in {{\mathcal {P}}}\).

(a)
\({{\,\mathrm{{\mathrm {Hom}}}\,}}(I_{l,m,n,p},I_{l,m1,n1,p})\) is isomorphic to \({{\mathbb {C}}}[S_{l,m,n,p}]\) as a bimodule over
$$\begin{aligned} {{\,\mathrm{{\mathrm {End}}}\,}}I_{l,m1,n1,p} \cong {{\mathbb {C}}}[S_{l,m1,n1,p} ]\text { and }{{\,\mathrm{{\mathrm {End}}}\,}}I_{l,m,n,p}\cong {{\mathbb {C}}}[S_{l,m,n,p}], \end{aligned}$$with any of the morphisms \(\phi _{i,j}\) as a generator for the right \({{\mathbb {C}}}[S_{l,m,n,p} ]\)module structure while identifing the subgroups
$$\begin{aligned} S_{m1}\subset S_{m} \text { and } S_{n1} \subset S_{n} \end{aligned}$$with the isotropy groups of i and j respectively.

(b)
\({{\,\mathrm{{\mathrm {Hom}}}\,}}(I_{l,m,n,p},I_{l+1,m1,n,p})\) is isomorphic to the induced module
$$\begin{aligned} {{\mathbb {C}}}[S_{l+1,m,n,p}]\cong \mathrm {Ind}_{S_l}^{S_{l+1}} {{\mathbb {C}}}[S_{l,m,n,p}] = {{\mathbb {C}}}[S_{l+1}]\otimes _{{{\mathbb {C}}}[S_l]}{{\mathbb {C}}}[S_{l,m,n,p}] \end{aligned}$$as a bimodule over
$$\begin{aligned} {{\,\mathrm{{\mathrm {End}}}\,}}I_{l+1,m1,n,p}\cong {{\mathbb {C}}}[S_{l+1,m1,n,p}]\text { and }{{\,\mathrm{{\mathrm {End}}}\,}}I_{l,m,n,p} \cong {{\mathbb {C}}}[S_{l,m,n,p}], \end{aligned}$$with \({}_{i,j}\pi \) as a generator for the right \({{\mathbb {C}}}[S_{l,m,n,p}]\)module structure while identifying the subgroups
$$\begin{aligned} S_l\subset S_{l+1}\text { and } S_{m1}\subset S_m \end{aligned}$$with the isotropy groups of j and i respectively.

(c)
The leftright mirror image of (b): \({{\,\mathrm{{\mathrm {Hom}}}\,}}(I_{l,m,n,p},I_{l,m,n1,p+1})\) is isomorphic to the induced module
$$\begin{aligned} {{\mathbb {C}}}[S_{l,m,n,p+1}]\cong \mathrm {Ind}_{S_p}^{S_{p+1}} {{\mathbb {C}}}[S_{l,m,n,p}] = {{\mathbb {C}}}[S_{p+1}]\otimes _{{{\mathbb {C}}}[S_p]}{{\mathbb {C}}}[S_{l,m,n,p}] \end{aligned}$$as a bimodule over
$$\begin{aligned} {{\,\mathrm{{\mathrm {End}}}\,}}I_{l,m,n1,p+1}\cong {{\mathbb {C}}}[S_{l,m,n1,p+1}]\text { and }{{\,\mathrm{{\mathrm {End}}}\,}}I_{l,m,n,p}\cong {{\mathbb {C}}}[S_{l,m,n,p}], \end{aligned}$$with \(\pi _{i,j}\) as a generator for the right \({{\mathbb {C}}}[S_{l,m,n,p}]\)module structure while identifying the subgroups
$$\begin{aligned} S_p\subset S_{p+1}\text { and } S_{n1}\subset S_n \end{aligned}$$with the isotropy groups of j and i respectively.
Proof
(a) Having fixed i and j as in the statement, we have a morphism
of \(({{\mathbb {C}}}[S_{l,m1,n1,p}], {{\mathbb {C}}}[S_{l,m,n,p}])\)bimodules, sending 1 to \(\phi _{i,j}\). Lemma 3.28 implies that this morphism is surjective, so it is the injectivity that we focus on.
Note that the morphism (3.45) factors as
where the downward arrow is the tensor product (over the unit object \(I\in {}_I{{\mathbb {T}}}\)) of morphisms in \({}_I{{\mathbb {T}}}\). Since the vertical maps are injective, the bottom morphism (which is in our focus) will be onetoone if and only if the top arrow is. The lefthand tensorand
of the top map in (3.46) is an isomorphism by Lemma 3.34, so it is enough to consider the right hand tensorand
of that map; equivalently, it suffices to resolve the present discussion in the case \(l=p=0\). But this follows from [5, Lemma 3.25 (a)] (which is analogous to the result being proven here), by noting that the restrictions of the compositions
to
are precisely the morphisms proven linearly independent there.
(b) We again have an \(({{\mathbb {C}}}[S_{l+1,m1,n,p}],{{\mathbb {C}}}[S_{l,m,n,p}])\)bimodule map
sending 1 to \({}_{i,j}\pi \), and its surjectivity is a consequence of Lemma 3.28. The injectivity follows as in part (a), by first decomposing (3.47) as a tensor product of maps
and
The latter is an isomorphism by Lemma 3.34, and the former is an injection as in the proof of (a), by appealing to [5, Lemma 3.25 (b)].
(c) As noted in the statement, this is entirely parallel to part (b), interchanging the roles of \(V^*\) and \(V_*^*\), l and p, m and n, etc. \(\blacksquare \)
The composition map from the degreeone to the degreetwo component of A comes in several varieties, depending on the domain. Before listing the various options, it will be convenient to introduce
Notation 3.36
For a quadruple \((l,m,n,p)\in {{\mathcal {P}}}\) we write
and similarly for other morphism spaces, with arrows indicating whether the respective index increases or decreases, and multiple arrows to indicate the amount. Other examples are
and so on.
For composable morphism spaces in \({}_I{{\mathbb {T}}}\) we denote by ‘\(\odot \)’ the tensor product over the endomorphism algebra of the intermediate object. For example:
With this in place, the possibilities for composition of degree1 morphisms are:
which is mirrorselfdual,
and its mirror image
and its mirror image
and finally, the selfdual morphism
The nine ‘\(\odot \)’ symbols above account for the nine possible ways of composing two morphisms, each being of one of the three flavors listed in Lemma 3.35.
Remark 3.37
Note that in all cases the product ‘\(\odot \)’ conserves the total number of up as well as down arrows.\(\blacklozenge \)
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
in the tensor category \({{\mathcal {D}}}\) extends to a linear monoidal functor
where \({}_I{{\mathbb {T}}}'\) is, as in Lemma 3.28, the full subcategory of \({}_I{{\mathbb {T}}}\) on the objects \(I_{l,m,n,p}\). Set \({}_I T:=T\otimes I\) for any \(T\in {{\mathbb {T}}}\). Since the objects of \({}_I{{\mathbb {T}}}'\) are precisely the tensor powers (over \(I\in {}_I{{\mathbb {T}}}\)) and the morphisms are tensor products and compositions of permutation of tensorands, evaluations (3.43), inclusions \({}_IV_*\subset {}_IV^*\) and \({}_IV\subset {}_IV_*^*\), 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 \({\mathcal {A}}\) defined in Sect. is quadratic, means that it will be enough to check that the degreetwo relations between degreeone morphisms between the \(I_{l,m,n,p}\) (i.e. the kernels of the maps (3.48) to (3.53)) vanish in \({{\mathcal {D}}}\) upon substituting x for \({}_IV\), \(x_*\) for \({}_IV_*\), etc. This would be a somewhat tedious and unenlightening check if done exhaustively, so we exemplify the argument by treating (3.48) alone. In that regard, we make the claim:
The kernel of the composition (3.48) is generated, as an \((S_{l,m2,n2,p},S_{l,m,n,p})\)bimodule, by
where \((m,m1)\) is the respective transposition in \(S_m\subset S_{l,m,n,p}\) and similarly,
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 mth tensorand \({}_IV^*\) against the nth tensorand \({}_IV_*^*\), and then

evaluating the \((m1)\)st tensorand \({}_IV^*\) against the \((n1)\)st tensorand \({}_IV_*^*\)
has the same effect as

permuting the mth and \((m1)\)st tensorands \({}_IV^*\),

permuting the nth and \((n1)\)st tensorands \({}_IV_*^*\)
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 (3.54). We do this by a dimension count. Tensoring two instances of Lemma 3.35, (a) over \({{\mathbb {C}}}[S_{l,m1,n1,p}]\), we conclude that the domain
of (3.48) is isomorphic to \({{\mathbb {C}}}[S_{l,m,n,p}]\) as an \(({{\mathbb {C}}}[S_{l,m2,n2,p}],{{\mathbb {C}}}[S_{l,m,n,p}])\)bimodule, with

\(\phi _{m,n}\) identified with the generator \(1\in {{\mathbb {C}}}[S_{l,m,n,p}]\), and

\(S_{l,m2,n2,p}\subset S_{l,m,n,p}\) being the subgroup fixing m, \(m1\), n and \(n1\).
This identification turns the putative generator (3.54) of the kernel of (3.48) into
The \(({{\mathbb {C}}}[S_{l,m2,n2,p}],{{\mathbb {C}}}[S_{l,m,n,p}])\)bimodule generated by (3.55) coincides with the right \(S_{l,m,n,p}\)module generated by the same element. The dimension of that module is half that of \({{\mathbb {C}}}[S_{l,m,n,p}]\), and hence
The desired conclusion that the kernel of the surjection (3.48) cannot strictly contain the bimodule generated by (3.54) will thus follow if we prove that
Since we have an embedding
and the lefthand tensorand is isomorphic to \({{\mathbb {C}}}[S_{l,p}]\) by Lemma 3.34, it is enough to assume that \(l=p=0\) and show that
or equivalently, via the adjunction (3.37), that
This, however, follows by restricting the morphisms on the left to \(V_*^{\otimes m}\otimes V^{\otimes n}\) 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 \(\pmb {{{\mathbb {T}}}}\) and \(\pmb {{}_I{{\mathbb {T}}}}\)
In this final section we discuss briefly the orthogonal and symplectic versions of the categories \({{\mathbb {T}}}\) and \(_I {{\mathbb {T}}}\). The orthogonal and symplectic analogues of the Lie algebra \({\mathfrak {gl}}^M(V,V_*)\) are the Lie algebras \({{\mathfrak {o}}}(V)\) and \({{\mathfrak {s}}}{{\mathfrak {p}}}(V)\) where V is now equipped with a nondegenerate symmetric or antisymmetric bilinear form \(\langle \cdot , \cdot \rangle :V\times V \rightarrow {\mathbb {C}}\), yielding a respective linear map \(S^2 V\rightarrow {{\mathbb {C}}}\) or \(\Lambda ^2 V \rightarrow {{\mathbb {C}}}\). The Lie algebras \({{\mathfrak {o}}}(V)\) and \({{\mathfrak {s}}}{{\mathfrak {p}}}(V)\) are defined as the respective largest subalgebras of \({\mathfrak {gl}}^M(V,V)\) for which the map \(S^2 V\rightarrow {{\mathbb {C}}}\) or \(\Lambda ^2 V \rightarrow {{\mathbb {C}}}\) is a morphism of representations.
Let \({{\mathfrak {g}}}={{\mathfrak {o}}}(V)\), \({{\mathfrak {s}}}{{\mathfrak {p}}}(V)\). Then V is a submodule of \(V^*\) (via the form \(\langle \cdot ,\cdot \rangle \)), and the \({{\mathfrak {g}}}\)module \(W:=V^*/V\) is irreducible. This can be proved for instance by considering W over the family of Lie subalgebras \({\mathfrak {gl}}^M(V',V'_*)\subset {{\mathfrak {g}}}\) arising from varying decompositions of V as \(V'\oplus V'_{*}\) for maximal isotropic subspaces \(V'\), \(V'_{*}\). Over each such subalgebra W is isomorphic to
and hence has precisely two proper submodules. Since these submodules vary when \(V'\) and \(V'_*\) vary, the module W is irreducible over \({{\mathfrak {g}}}\).
Furthermore, for any Young diagram \(\lambda \), the irreducible \({\mathfrak {gl}}^M(V,V)\)module \(V_{\lambda }\) restricts to \({{\mathfrak {g}}}\) yielding a generally reducible \({{\mathfrak {g}}}\)module. In all cases the socle of \(V_{\lambda }_{{{\mathfrak {g}}}}\) is simple, and we denote it by \(V_{[\lambda ]}\) for \({{\mathfrak {g}}}={{\mathfrak {o}}}(V)\) and by \(V_{\langle \lambda \rangle }\) for \({{\mathfrak {g}}}={{\mathfrak {s}}}{{\mathfrak {p}}}(V)\). It is clear that the Lie algebras \({{\mathfrak {o}}}(\infty )\) and \({{\mathfrak {s}}}{{\mathfrak {p}}}(\infty )\) considered in [15] are subalgebras respectively of \({{\mathfrak {o}}}(V)\) and \({{\mathfrak {s}}}{{\mathfrak {p}}}(V)\), and by [14, Theorem 7.10] the socle filtrations of \(V_{\lambda }_{{{\mathfrak {o}}}(\infty )}\) and \(V_{\lambda }_{{{\mathfrak {s}}}{{\mathfrak {p}}}(\infty )}\), described explicitly in [15], coincide with the respective socle filtrations of \(V_{\lambda }_{{{\mathfrak {o}}}(V)}\) and \(V_{\lambda }_{{{\mathfrak {s}}}{{\mathfrak {p}}}(V)}\).
If \(\lambda \), \(\mu \) is a pair of Young diagrams, we set
Then \(L_{\lambda ,\mu }\) is a simple \({{\mathfrak {g}}}\)module. This can be seen by essentially the same argument as in the case of W. Moreover,
The analogue of the injective object I from Sect. 3.4 is constructed as follows. One sets
Furthermore, the quotient \(Q_{{{\mathfrak {g}}}}\) of \(S^2V^*\) by the sum of kernels of the pairings \(V^*\otimes V\rightarrow {{\mathbb {C}}}\) and \(S^2 V\rightarrow {{\mathbb {C}}}\) admits a nonsplitting exact sequence
The socle filtration of I has the form
Then the embedding (3.1) induces embeddings
which allow us to define \(I_{{{\mathfrak {g}}}}\) as the colimit
Moreover, by the same construction as in Sect. 3.7, \(I_{{{\mathfrak {g}}}}\) is endowed with the structure of a commutative algebra.
The category \({{\mathbb {T}}}_{{{\mathfrak {g}}}}\) is introduced in the same way as in Sect. 3.4, where now \(J_s=W^{\otimes l} \otimes V^{*\otimes m}\) for pairs \(s=(l,m)\), l, \(m\in {{\mathbb {N}}}\), and the object I is replaced by \(I_{{{\mathfrak {g}}}}\). In the Introduction we denoted this category by \({{\mathbb {T}}}^2_{{{\mathfrak {g}}}}\) 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 \({{\mathbb {T}}}_{{{\mathfrak {g}}}}\). We leave it to the reader to check that Proposition 3.10 holds also for the category \({{\mathbb {T}}}_{{{\mathfrak {g}}}},\) and that \(I_{{{\mathfrak {g}}}}\) is an injective hull in \({{\mathbb {T}}}_{{{\mathfrak {g}}}}\) of the object \({{\mathbb {C}}}\). The respective partial order \((l,m)\preceq (l',m')\) on \({{\mathbb {N}}}\times {{\mathbb {N}}}\) is given by \(l\ge l'\), \(m\le m'\), \(l+m'\le l'+m\). The results of Sect. 3.3 also hold with obvious modification.
The canonical injective resolution (3.23) stays the same with F replaced by \(F_{{{\mathfrak {g}}}}\), however now the socle of the object \(\left( I_{{{\mathfrak {g}}}}\right) _j=I_{{{\mathfrak {g}}}}\otimes \Lambda ^j F_{{{\mathfrak {g}}}} \) decomposes as \(\bigoplus {{\mathbb {S}}}_{\lambda } V\) for \({{\mathfrak {g}}}={{\mathfrak {s}}}{{\mathfrak {p}}}(V)\) and \(\bigoplus {{\mathbb {S}}}_{\lambda ^{\perp }} V\) for \({{\mathfrak {g}}}={{\mathfrak {o}}}(V)\) where \(\lambda \) runs over all special partitions of degree 2j.
Corollary 4.1
For any simple object X of \({{\mathbb {T}}}_{{{\mathfrak {o}}}(V)}\) we have
and for any simple object X of \({{\mathbb {T}}}_{{{\mathfrak {s}}}{{\mathfrak {p}}}(V)}\) we have
Next, Theorem 3.18 and Proposition 3.19 stay valid with \({{\mathbb {T}}}\) replaced by \({{\mathbb {T}}}_{{{\mathfrak {g}}}}\). We leave it to the reader to modify Lemma 3.20 accordingly. Furthermore, Proposition 3.23, and Theorem 3.24 also hold for \(I_{{{\mathfrak {g}}}}\) and \(Q_{{{\mathfrak {g}}}}\) (instead of I and Q, respectively). The same applies to Proposition 3.25, Theorem 3.26 (with \(L_{\lambda ,\mu }\) instead of \(L_{\lambda ,\mu ,\nu ,\pi }\)), Theorem 3.27, Theorem 3.29, and Corollary 3.31.
The universality results from Sect. 3.9 also carry over to the cases \({{\mathfrak {g}}}={{\mathfrak {o}}}(V)\), \({{\mathfrak {s}}}{{\mathfrak {p}}}(V)\). In particular, the category \({}_{I_{{{\mathfrak {g}}}}}{{\mathbb {T}}}\) is defined in the same way as the category \({}_{I}{{\mathbb {T}}}\): it is the category of internal \(I_{{{\mathfrak {g}}}}\)modules in \({{\mathbb {T}}}_{{{\mathfrak {g}}}}\).
Note also that the analogue
of the map (3.22) is well defined and factors through maps
in the respective cases \({{\mathfrak {g}}}={{\mathfrak {o}}}(V)\) and \({{\mathfrak {g}}}={{\mathfrak {s}}}{{\mathfrak {p}}}(V)\). This defines pairings
and
respectively.
Now we have
Theorem 4.2
Let \(({{\mathcal {D}}},\ \otimes ,\ \mathbf{1})\) be a tensor category, and \(x\hookrightarrow x^*\) be a monomorphism in \({{\mathcal {D}}}\). Assume that a morphism in \({{\mathcal {D}}}\)
is given, satisfying \(p\circ \sigma =p\) for \({{\mathfrak {g}}}={{\mathfrak {o}}}(V)\) and \(p\circ \sigma =p\) for \({{\mathfrak {g}}}={{\mathfrak {s}}}{{\mathfrak {p}}}(V)\), where \(\sigma \) is the flip automorphism of \(x^*\otimes x^*\) as an object of the tensor category \({{\mathcal {D}}}\) coming from the assumption that \({{\mathcal {D}}}\) is a symmetric monoidal category.

(a)
There is a unique (up to monoidal natural isomorphism) left exact symmetric monoidal functor \(F:{}_{I_{{{\mathfrak {g}}}}}{{\mathbb {T}}}_{fin}\rightarrow {{\mathcal {D}}}\), where \({{\mathfrak {g}}}={{\mathfrak {o}}}(V)\) if \(p\circ \sigma =p\) and \({{\mathfrak {g}}}={{\mathfrak {s}}}{{\mathfrak {p}}}(V)\) if \(p\circ \sigma =p\), sending

(b)
if \({{\mathcal {D}}}\) is additionally a Grothendieck category then F extends uniquely to a coproductpreserving functor \({}_{I_{{{\mathfrak {g}}}}}{{\mathbb {T}}}\rightarrow {{\mathcal {D}}}\).
The universality of the tensor categories \({}_{I_{{{\mathfrak {o}}}(V)}}{{\mathbb {T}}}\) and \({}_{I_{{{\mathfrak {s}}}{{\mathfrak {p}}}(V)}}{{\mathbb {T}}}\) leads to the fact that they are equivalent as monoidal categories. More precisely, consider the (symmetric) tensor category \({{\mathbb {T}}}^_{{{\mathfrak {s}}}{{\mathfrak {p}}}(V)}\) defined in the same way as \({{\mathbb {T}}}_{{{\mathfrak {s}}}{{\mathfrak {p}}}(V)}\) but with the flip isomorphism
replaced by \(\sigma \). One checks that \({{\mathbb {T}}}^_{{{\mathfrak {s}}}{{\mathfrak {p}}}(V)}\) is welldefined, i.e. that the new flip isomorphism on \(V^*\otimes V^*\) induces a welldefined structure of tensor category preserving the monoidal structure on \({{\mathbb {T}}}_{{{\mathfrak {s}}}{{\mathfrak {p}}}(V)}\).
In addition, one checks that there is a welldefined tensor category \({}_{I_{{{\mathfrak {s}}}{{\mathfrak {p}}}(V)}}{{\mathbb {T}}}^\) of internal Imodules in \({{\mathbb {T}}}^_{{{\mathfrak {s}}}{{\mathfrak {p}}}(V)}\) which coincides with \({}_{I_{{{\mathfrak {s}}}{{\mathfrak {p}}}(V)}}{{\mathbb {T}}}\) as a monoidal category.
Corollary 4.3
The tensor categories \({}_{I_{{{\mathfrak {o}}}(V)}}{{\mathbb {T}}}\) and \({}_{I_{{{\mathfrak {s}}}{{\mathfrak {p}}}(V)}}{{\mathbb {T}}}^\) are canonically equivalent.
Proof
By Theorem 4.2, there are distinguished functors
sending \(V^* \otimes I_{{{\mathfrak {o}}}(V)}\) to \(V^*\otimes I_{{{\mathfrak {s}}}{{\mathfrak {p}}}(V)}\), \(V_* \otimes I_{{{\mathfrak {o}}}(V)}\) to \(V_* \otimes I_{{{\mathfrak {s}}}{{\mathfrak {p}}}(V)}\) and \(W \otimes I_{{{\mathfrak {o}}}(V)}\) to \(W \otimes I_{{{\mathfrak {s}}}{{\mathfrak {p}}}(V)}\), and vice versa. Again, by Theorem 4.2 the functors F and \(F^\) must be mutually inverse. \(\mathrel {\Box }\)
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Acknowledgements
We thank a referee for a thorough reading of an earlier draft of the paper and for excellent suggestions. A.C. was partially supported by NSF Grants DMS1801011 and DMS2001128. I.P. was partially supported by DFG Grants PE 9807/1 and PE 9808/1.
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Chirvasitu, A., Penkov, I. Universal Tensor Categories Generated by Dual Pairs. Appl Categor Struct 29, 915–950 (2021). https://doi.org/10.1007/s10485021096402
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Keywords
 Mackey Lie algebra
 Tensor module
 Monoidal category
 Koszulity
 Grothendieck category
 Semiartinian
Mathematics Subject Classification
 17B65
 17B10
 18M05
 18E10
 16T15
 16S37