Classical Lie superalgebras at infinity

Abstract

We study certain categories of modules over direct limits of classical Lie superalgebras. In many cases these categories are equivalent to similar categories for classical Lie algebras. The functors establishing this equivalence can be used to obtain a new result for representation theory of direct limits of Lie algebras.

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1 Introduction

There are several generalizations of simple Lie algebras and superalgebras in the infinite-dimensional case. In this paper, we discuss representations of locally simple Lie algebras, i.e. Lie algebras and superalgebras we consider are the direct limits $g = \underrightarrow {lim}\,{g_i}$ of finite-dimensional simple Lie algebras (or superalgebras) g _i . In particular,we are interested in the cases when g = sl(∞),so(∞) or sp(∞).

In [5] we tried to define a nice analogue of the category of finite-dimensional modules for g. The most obvious analogue, the category of integrable modules, is rather difficult to study. Even the problem of classifying simple modules involves infinitely many continuous parameters.

On the other hand, g has a very natural class of representations in the tensor powers of the standard and costandard modules. In [2] we give an intrinsic definition of a category T _g that contains all such representations. It turns out that T _g has many remarkable properties. Although it is not semi-simple, it is a Koszul category in the sense of [1]. That allows one to calculate extensions between simple modules and their injective resolutions. We also prove that the categories T _g for g = so(∞) and sp(∞) are equivalent. In the recent preprint [8] the same categories are studied from slightly different point of view.

The goal of the present paper is to define and study analogues of T _g for direct limits of classical Lie superalgebras. As follows from the Kac classification, [4], there are four such superalgebras sl(∞,∞), osp(∞,∞), P(∞) and Q(∞). We will see that in the first three cases we do not obtain new categories. Namely, T _sl(∞,∞) is equivalent to T _sl(∞), and T _osp(∞,∞) and T _P(∞) are equivalent to T _o(∞). The latter fact can be used to construct a direct equivalence functor T _o(∞) → T _sp(∞). This result is somewhat surprising, since it appears that these categories are easier than the corresponding categories for finite-dimensional superalgebras. The rather complicated matter of atypical representations disappears after going to direct limits.

In the case of Q(∞) we obtain a completely new category. It is interesting to study it in detail.

2 Direct limits of classical Lie algebras

2.1 General and special Lie algebras at infinity

Let V and V _* be countable-dimensional vector spaces with non-degenerate pairing tr : V ⊗ V _* → ℂ.

Definition 1 gl(∞) = V ⊗ V _* has a natural Lie algebra structure given by

$$[{v_1} \otimes {u_1},{v_2} \otimes {u_2}] = tr({v_2} \otimes {u_1}){v_1} \otimes {u_2} - tr({v_1} \otimes {u_2}){v_2} \otimes {u_1}.$$

Ker (tr) = sl(∞) is a simple Lie subalgebra of gl(∞).

One can also realize g as a direct limit

$$sl(\infty ) = \underrightarrow {\lim }\,sl(n),gl(\infty ) = \underrightarrow {\lim }\,gl(n),$$

and identify gl (∞) with the space of infinite matrices (a _ij) _{i, j∈ℕ} with finitely many non-zero entries and sl(∞) with the subalgebra of traceless matrices in gl(∞).

It is clear that V and V _* are simple g-modules. Furthermore, the classical Schur-Weyl duality works in the infinite-dimensional case.

Theorem 1 (Schur-Weyl duality) Let g = gl (∞) or sl (∞). Then

$${V^{ \otimes n}} = \mathop \oplus \limits_{|\lambda | = n} \;{V^\lambda } \otimes {Y_\lambda },V_*^{ \otimes n} = \mathop \oplus \limits_{|\lambda | = n} \;V_*^\lambda \otimes {Y_\lambda },$$

where λ runs the set of partitions of size n and Y _λ denotes the corresponding irreducible representation ofS_n and V^λ = π _λ (V ^⊗n), where π _λ is a Young projector with the Young diagram λ.

However, the representation of g in the space of mixed tensors V ^⊗n ⊗ V _* ^⊗m is not completely reducible in contrast with finite-dimensional case. Indeed, for instance, the exact sequence of sl(∞)-modules

$$0 \to sl(\infty ) \to V \otimes {V_*} \to \mathbb{C} \to 0$$

does not split. The following result gives a description of the g-module structure on V ^⊗n ⊗ V _* ^⊗m.

Theorem 2 ([6]) Let g = gl(∞) or sl(∞). Then

$${V^{ \otimes n}} \otimes V_*^{ \otimes m} = \mathop \oplus \limits_{|\lambda | = n,|\mu | = m} {\tilde V^{\lambda ,\mu }} \otimes ({Y_\lambda } \boxtimes {Y_\mu }),$$

where each ${\tilde V^{\lambda ,\mu }} = {V^\lambda } \otimes V_*^\mu$ is an indecomposable g-module with irreducible socle V ^λ,η and Y _λ ⊠ Y µ is the exterior tensor product of irreducible S_n and S_m-modules.

The socle filtration of ${\tilde V^{\lambda ,\mu }}$ is given by

$$so{c^k}({\tilde V^{\lambda ,\mu }})/so{c^{k - 1}}({\tilde V^{\lambda ,\mu }}) = \mathop \oplus \limits_{|\lambda | = k} N_{\gamma ,\lambda '}^\lambda N_{\gamma ,\mu '}^\mu {V^{\lambda ',\mu '}}.$$

Here $N_{\gamma ,\lambda '}^\lambda$, stand for Littlewood-Richardson coefficients.

The proof is based on the results of Howe, Tan and Willenbring [3] who calculated asymptotic decomposition of mixed tensor products in the finite-dimensional case.

2.2 Orthogonal and symplectic Lie algebras

Assume now that a countable-dimensional vector space V has a non-degenerate symmetric (resp. skew-symmetric) form ω : V ⊗ V → ℂ.

Definition 2 so (∞) (resp. sp(∞))is the Lie subalgebra of finite rank linear operators in V preserving ω.

One can use identification

$$so(\infty ) = {\Lambda ^2}(V),sp\,(\infty ) = {S^2}(V)$$

given by

$${X_{v \wedge w}}(u) = \omega (v,u)w - \omega (u,w)v,\forall v,w,u \in V.$$

((1))

Another way to define g is via direct limits

$$so(\infty ) = \underrightarrow {\lim }\,so\,(n),sp\,(\infty ) = \underrightarrow {\lim }\,sp\,(n).$$

The representation of g in the tensor algebra T(V) were also described by Penkov and Styrkas.

Theorem 3 ( [6]) Let g = so(∞) or sp(∞). Then

$${V^{ \otimes n}} = \mathop \oplus \limits_{|\lambda | = n} \;{\tilde V^\lambda } \otimes {Y_\lambda },$$

where each ${\tilde V^\lambda }$ is an indecomposable g -module with irreducible socle V ^λ.

The socle filtration of ${\tilde V^\lambda }$ is given by

$$so{c^k}({\tilde V^\lambda })/so{c^{k - 1}}({\tilde V^\lambda }) = \mathop \oplus \limits_{|\lambda | = k} \,N_{2\gamma ,\lambda '}^\lambda {V^{\lambda '}},$$

for g = so(∞), and

$$so{c^k}({\tilde V^\lambda })/so{c^{k - 1}}({\tilde V^\lambda }) = \mathop \oplus \limits_{|\lambda | = k} \,N_{2{\gamma ^ \bot },\lambda '}^\lambda \,{V^{\lambda '}},$$

for g = sp(∞).

3 The category T _g

Definition 3 Let g = gl(∞) or sl(∞). A subalgebra k ⊂ g is a finite corank sub-algebra if there exist finite dimensional subspaces W ⊂ V and W _* ⊂ V _* such that the restriction of the canonical pairing to W ⊗ W _* → ℂ is non-degenerate and $t \supset g \cap (W_*^ \bot \otimes {W^ \bot })$.

If g = so(∞) or sp(∞), then k ⊂ g is a finite corank subalgebra if there exists a finite dimensional subspace W ⊂ V such that the restriction of ω on W is non-degenerate and k ⊃ Λ²(W⊥) or S ²(W⊥) respectively.

We define T _g as a full subcategory of g-modules whose objects M satisfy the following conditions

• • M is integrable.

• • For every m∈M the annihilator of m in g is a finite corank subalgebra.

• • M has finite length.

It is not difficult to see that T _g is closed under tensor product, hence it is a mono-idal category. However, it is not rigid since there is no a reasonable duality functor on T _g . The following results relate tensor representations of g with T _g .

Theorem 4 ([2])

• • For g = gl(∞) or sl(∞) all (up to isomorphism) simple objects of T _g are V ^λ,µ.

• • For g = so(∞) or sp(∞) all (up to isomorphism) simple objects of T _g are V ^λ .

• • ${\tilde V^{\lambda ,\mu }}$ (respectively ${\tilde V^\lambda }$) are all up to isomorphism indecomposable injective in T _g .

To prove injectivity of ${\tilde V^{\lambda ,\mu }}$ we use the fact (proven in [5]) that for any integrable g-module M, the integrable part of M* is injective in the category of integrable modules. From this it is easy to see that the functor Γ from the category of all integrable g-modules to T _g defined by

$$\Gamma (M) = \cup {M^t}$$

(where the union is taken over all finite corank k ⊂ g) maps an injective module in the category of integrable modules to an injective module in T _g . On the other hand, by a direct calculation done in [2]

$$\Gamma ({V^{ \otimes m}} \otimes V_*^{ \otimes n}) = \mathop \oplus \limits_{k \leqslant m,l \leqslant n} \,{({V^{ \otimes k}} \otimes V_*^{ \otimes l})^{ \oplus c\,(k,l)}}.$$

Note that Γ can be considered as a certain version of the Zuckerman functor [9].

There exists a Borel subalgebra b ⊂ g such that all simple modules of T _g are highest weight modules. For instance, if g = sl(∞), considered as the algebra of matrices (a _ij) _i,j ∈ℕ, we define a nonstandard total order on ℕ by1 <3 <5 <⋯ <6 <4 <2 and positive roots ε _i — ε _j for all i < j. The corresponding infinite “Dynkin diagram” is

$$ \circ - \circ - \cdots - \circ - \circ .$$

Lemma 1 If S is a non-zero quotient of the Verma module and S ∈ T _g , then S is simple.

To show that the the category T _g is Koszul we use the following

Lemma 2 ([2])

• • If g = gl (∞)or sl (∞),then

  $$Ex{t^k}({V^{\lambda ,\mu }},{V^{v,\kappa }}) \ne 0.$$

  implies |λ| — |v| = |µ| — |K| = k.

• • If g =so (∞) or sp(∞), then

  $$Ex{t^k}({V^\lambda },{V^v}) \ne 0$$

  implies |λ|—|v| = 2k.

Let T = T(V) for g =so(∞) or sp(∞) and T = ⨁ _{m n≤0} V^⊗m ⊗ V _* ^⊗n for g = sl(∞) or gl(∞). For g = gl (∞) or sl(∞) we set

$$A_g^k = \mathop \oplus \limits_{m,n \geqslant 0} \,Ho{m_g}({V^{ \otimes m}} \otimes V_*^{ \otimes n},{V^{ \otimes m - k}} \otimes V_*^{ \otimes n - k}),{A_g} = \mathop \oplus \limits_{k \geqslant 0} \,A_g^k.$$

For g=so (∞) or sp (∞)set

$$A_g^k = \mathop \oplus \limits_{n \geqslant 0} \,Ho{m_g}({V^{ \otimes n}},{V^{ \otimes n - 2k}}),\;{A_g} = \mathop \oplus \limits_{k \geqslant 0} \,A_g^k.$$

Note that A _g is by definition a graded algebra. It does not have the identity but it is a direct limit of unital algebras.

Theorem 5 ([2])

• • The category T _g is antiequivalent to the category A _g -fmod of locally unitary finite-dimensional A _g -modules.

• • A _g is a direct limit of Koszul rings.

For g = gl(∞) and sl(∞) the corresponding algebras A _g are the same. It is shown in [2] that $A_g^0 = { \oplus _{m,n \geqslant 0}}\,\mathbb{C}[{S_n} \times {S_m}]$ and $A_g^1$ is generated by contractions.

In this case one can prove that A _g is Koszul self-dual, i.e. ${(A_g^!)^{op}} \simeq {A_g}.$ That implies the following formulas for extensions between simple modules

$$\dim Ex{t^k}({V^{\lambda ',\mu '}},{V^{\lambda ,\mu }}) = \mathop \sum \limits_{|\gamma | = k} \,N_{\gamma ,\lambda '}^\lambda N_{{\lambda ^ \bot },\mu '}^\mu .$$

It is also shown in [2] that for g = so(∞) and sp(∞) $sp(\infty )A_g^0 = { \oplus _{n \geqslant 0}}\mathbb{C}[{S_n}]$ and $A_g^1$ is generated by contractions. Knowing this it is easy to obtain an isomorphism

$${A_{so(\infty )}} \simeq {A_{sp(\infty )}}.$$

The latter implies an equivalence of abelian categories T _sp(∞) and T _so(∞) by Theorem 5. Under this equivalence V ^λ goes to V ^λ⊥. It is proven in [8] that this is an equivalence of monoidal categories. A different proof of this fact is given in the next section.

4 Superalgebras

4.1 Direct limits of classical Lie superalgebras

Let V = V ₀ ⨁ V ₁ be a superspace, both V ₀ and V ₁ are countable-dimensional. Below we consider the following possibilities.

• • There is a countable-dimensional V _* and a non-degenerate pairing str :V ⊗V _*→ ℂ. Then we set gl(∞,∞) = V ⊗ V _* and sl(∞,∞) = Ker (str) with the commutator defined in the same way as in the purely even case (with the usual sign convention). Note that g = gl(∞,∞) has a ℤ-grading (compatible with ℤ₂-grading) g = g _-1 ⨁ g ₀ ⨁ g ₁, where

  $$\begin{gathered} {g_0} = {V_0} \otimes {({V_0})_*} \oplus {({V_1})_*} \otimes {V_1} \simeq (gl(\infty )) \oplus (gl(\infty )), \hfill \\ {g_1} = {V_0} \otimes {({V_1})_*},\quad {g_{ - 1}} = {({V_0})_*} \otimes {V_1}. \hfill \\ \end{gathered} $$

  This grading naturally can be restricted to sl(∞,∞). The Lie superalgebra sl (∞,∞) is simple since it can be obtained as a direct limit of simple Lie superalgebras.

• • We fix an even non-degenerate symmetric form ω : S ²(V) → ℂ and define osp(∞,∞) as the subalgebra of operators in V of finite rank preserving ω. One can identify osp(∞,∞) with ∧² (V) using (1). In this case

  $${g_0} = so(\infty ) \oplus sp\,(\infty ),{g_1} = {V_0} \otimes {V_1}.$$

  The Lie superalgebra osp(∞,∞) is simple because it is isomorphic to a direct limit of finite-dimensional simple Lie superalgebras.

• • We fix an odd non-degenerate form ω : S ² (V) →ℂ and define the Lie superalge-bra P(∞) as the subalgebra of linear operators of finite rank in V preserving ω. The superalgebra P(∞) has a ℤ-grading g = g _-1 ⨁ g ₀ ⨁ g ₁, with

  $${g_0} = {V_0} \otimes {V_1}\,{g_1} = {S^2}({V_0}),\,g - 1 = {\Lambda ^2}({V_1}).$$

  Note that g ₀ is isomorphic to gl(∞), V ₀ and V ₁ are standard and costandard representation of g ₀. Furthermore, g ₁ (resp. g _-1) is identified with S ² (V ₀) (resp. ∧² (V₁)) by setting

  $${X_{u,w}}(v) = (u,v)w + (w,v)u,$$

  for any u,w ∈ V ₀, v ∈V and

  $${X_{u,w}}(v) = (u,v)w - (w,v)u,$$

  for any u,w∈ V ₁, v ∈ V. Observe that $P(\infty ) = \mathop {\lim }\limits_ \to P(n)$ is not a simple Lie algebra.

  Its commutator is a simple ideal SP(∞) of all traceless matrices in P(∞).

• • Let J : V → V be an odd operator such that J ² = - 1. The Lie superalgebra Q (∞) is the centralizer of J in gl (∞, ∞). As in the finite-dimensional case g ₀ = gl(∞) and g ₁ is the adjoint representation of g ₀. Note that $Q(\infty ) = \mathop {\lim }\limits_ \to Q(n)$ is not simple. It contains a simple ideal SQ (∞) of odd codimension 1 consisting of operators X such that str(XJ) = 0. Note that in contrast with all other cases, SQ(∞) is the direct limit of SQ (n), but SQ (n) are not simple.

  We leave to the reader the definition of finite corank subalgebras in this case.

4.2 T _g for Lie superalgebras

Now let g denote one of the Lie superalgebras defined in the previous section. Let T _g be a full subcategory of g g-modules M satisfying the following three conditions:

• (1) M is integrable over g ₀, and therefore over g,

• (2) the annihilator of every vector in M is a finite corank subalgebra in g,

• (3) M has finite length over g ₀.

It is clear that T _g is an abelian monoidal category. If g = Q(∞), in order to avoid the annoying but not essential parity chasing we allow morphisms which change parity, i. e. the standard functor П changing the parity is an isomorphism in our category. In fact, it is not difficult to show that if g = Q(∞), then

$${T_g} = T_g^ + \oplus T_g^ - $$

with $\Pi :T_g^ + \to T_g^ -$ defining an equivalence of categories. Threfore, admitting odd isomorphisms in the category T _g is the same as studying $T_g^ +$ instead of T _g .

4.3 Orthosymplectic superalgebra

Let g = osp(∞,∞). The goal of this subsection is to prove the following

Theorem 6 The monoidal categories T _g , T_sp(∞) and T _so(∞) are equivalent.

We start with studying tensor powers of the standard representation V. If M is a g-module and k ⊂ g is a subalgebra, then M _k denotes the space of k-invariants in M.

Lemma 3 (a) ${({V^{ \otimes n}})^{so(\infty )}} = V_1^{ \otimes n}$ and ${({V^{ \otimes n}})^{sp(\infty )}} = V_0^{ \otimes n}.$.

(b)$V_0^{ \otimes n}$ or $V_1^{ \otimes n}$ generates V^⊗n over g.

Proof. We have the obvious isomorphism of g⁰-modules

$${V^{ \otimes n}} \simeq \mathop \oplus \limits_{p - q = n} (V_0^{ \otimes p} \otimes )V_1^{ \otimes q}{)^{ \oplus C(n,p)}}.$$

The identity

$${(V_0^{ \otimes p} \otimes V_1^{ \otimes q})^{so\,(\infty )}} = {(V_0^{ \otimes p})^{so(\infty )}} \otimes V_1^{ \otimes q}$$

together with the fact that ${({V_0}^{ \otimes p})^{so(\infty )}} = 0$ for p ≠ 0 imply ${({V^{ \otimes n}})^{so(\infty )}} = V_1^{ \otimes n}.$ The second statement of (a) is similar.

Now we prove that ${V_0}^{ \otimes n}$ generates V ^⊗n. Assume that the statement is not true. Define the grading on V ^⊗n by setting the degree of a homogeneous indecomposable tensor u = u ₁⊗⋯⊗u _n equal the number of u _i ∈ V ₁ . Pick up an indecomposable u of minimal degree that does not belong to $U\,(g)\,{V_0}^{ \otimes n}.$ Then k = deg (u) > 0 and ui ∈ V ₁ for some i. Pick up e, e′ G V ₀ such that (e, e′) = 1, (e, u ₁) = ⋯ = (e, un) = 0. Then

$$u = \pm {X_{e \wedge {u_i}}}({u_1} \otimes \cdots \otimes {u_{i - 1}} \otimes e' \otimes {u_{i + 1}} \otimes \cdots {u_n}) + v,$$

for some v of degree k — 2. Note that deg(u ₁ ⊗ ⋯ ⊗u _i-1 ⊗e′ ⊗u _{i + 1} ⊗… u _n) = k - 1. Hence both v and (u ₁ ⊗ ⋯ ⊗ u _{i- 1} ⊗ e′ ⊗ u _{i +1} ⊗ …u _n) belong to $U\,(g)\,{V_0}^{ \otimes n}.$ Therefore $u \in U\,(g)\,{V_0}^{ \otimes n}.$ Contradiction. In the same way one can prove that ${V_1}^{ \otimes n}$ generates V ^⊗n over g.

Let

$$T(V) = \mathop \oplus \limits_{n \geqslant 0} \,{V^{ \otimes n}},{T^{ \geqslant m}}(V) = \mathop \oplus \limits_{n \geqslant m} \,{V^{ \otimes n}},{T^{ \leqslant m}}(v)\mathop \oplus \limits_{n \leqslant m} \,{V^{ \otimes n}}.$$

A linear operator X ∈ End _k (V) is called bounded if the there are n and m such that T ^≥n (V) ⊂ KerX and ImX ⊂ T ^≥m (V).

We denote by A _g the subalgebra of all bounded operators in End _g (T(V)). Note that A _g = ⨁ _m,n≥0Hom _g (V ^⊗m,V^⊗n). By A _{s o(∞)} (resp. A _sp(∞) ) we denote the algebras of bounded operators in End _so(∞) (T(V ₀)) (resp. End _sp(∞) (T(V ₀))).

Lemma 3 (a) implies that there are natural homomorphisms

$${\rho _{so}}:{A_g} \to {A_{so(\infty )}},:{A_g} \to {A_{sp(\infty )}}$$

given by the restriction to T (V)^sp(∞) and T (V)^so(∞) respectively.

Lemma 4 Both ρ _so and ρ _sp are isomorphisms.

Proof. Injectivity of ρ _so and ρ _sp follows from Lemma 3 (b). To prove surjectiv-ity recall from [2] that A _so(∞) (resp. A _sp(∞) ) are generated by permutation groups acting on ${V_0}^{ \otimes n}$ (resp. ${V_1}^{ \otimes n}$) and contractions. Both permutations and contraction are defined on T (V) and hence lie in the image of ρ _so (resp. ρ _sp ).

Let λ be a partition and π _λ be the corresponding Young projector in ℂ[S _n]. We define

$${\tilde V^\lambda } = {\pi _\lambda }({V^{ \otimes n}}),\tilde V_0^\lambda = {\pi _\lambda }(\tilde V_0^{ \otimes n}),\tilde V_1^\lambda = {\pi _\lambda }(V_1^{ \otimes n}).$$

Recall that the socles of ${\tilde V_0}^\lambda$ and ${\tilde V_1}^\lambda$ are simple g₀-modules. We denote them ${V_0}^\lambda$ and ${V_1}^\lambda$ respectively.

Lemma 5 (a) ${V_0}^\lambda \otimes {V_1}^\mu$ is a simple g ₀-module.

(b) Every simple module in T _g0 is isomorphic to ${V_0}^\lambda \otimes {V_1}^\mu$ for some partitions λ and µ.

(c) ${\tilde V_0}^\lambda \otimes {\tilde V_1}^\mu$ is indecomposable injective in T_g0 with socle equal to ${V_0}^\lambda \otimes {V_1}^\mu .$

Proof. (a) can be proven by the standard argument using the Jacobson density theorem. Let $v = \sum _{j = 1}^me{\,_j} \otimes {f_j} \ne 0$ for some linearly independent ${e_j} \in V_0^\lambda$ and ${f_j} \in V_1^\lambda .$ Let u = e ⊗f. Since $V_0^\lambda$ and $V_1^\mu$ are simple there exist X ∈ U(so(∞)) and Y ∈ U(sp(∞)) such thatX(e1) = e,Y(f ₁) = f and X (ej) = Y (fj) =0 for j > 1. Hence u = X ⊗ Y (v). Thus any non-zero v generates the whole $V_0^\lambda \otimes V_1^\mu .$

(b) Let M ∈ T _g0 be simple. Then M as an so(∞)-module lies in a slightly bigger category ${\hat T_{so(\infty )}}$ of modules satisfying (1) and (2). Therefore M contains a subquotient isomorphic to $V_0^\lambda$ for some λ. Since M is simple $M = V_0^\lambda \otimes W,$ where W is some simple module in ${\hat T_{sp(\infty )}}.$ Hence $W = V_1^\mu$ for some partition µ.

(c) Injectivity of $\tilde V_0^\lambda$ in T_so(∞) implies injectivity of $\tilde V_0^\lambda \otimes \tilde V_1^\mu$ in T _so(∞). By the same reason $\tilde V_0^\lambda \otimes \tilde V_1^\mu$ is injective in T _sp(∞). For any simple $V_0^{\lambda '} \otimes V_1^{\mu '}$ we have that an exact sequence

$$0 \to \tilde V_0^\lambda \otimes V_1^\mu \to M \to V_0^{\lambda '} \otimes V_1^{\mu '} \to 0$$

splits over so(∞) and sp(∞). Hence it splits over g ₀. That proves injectivity of $\tilde V_0^\lambda \otimes \tilde V_1^\mu$. Irreducibility of the socle follows from the identity

$$Ho{m_{go}}(V_0^{\lambda '} \otimes V_1^{\mu '},\tilde V_0^\lambda \otimes \tilde V_1^\mu ) \simeq Ho{m_{so(\infty )}}(V_0^{\lambda '},\tilde V_0^\lambda ) \otimes Ho{m_{sp(\infty )}}(V_1^{\mu '},\tilde V_1^\mu ).$$

We define functors R _so : T _g → T _so(∞) and R _sp : T _g → T _sp(∞) by

$${R_{so}}(M) = {M^{sp(\infty )}},{R_{sp}}(M) = {M^{so(\infty )}}.$$

Lemma 6 If M ∈ T _g and M ≠ 0, then R _so (M) ≠ 0 and R _sp (M) ≠ 0.

Proof. Let $L \simeq V_0^\lambda \otimes V_1^\mu$ be a simple submodule in soc _g0(M) with maximal |λ|. Consider the natural morphism θ : L ⊗g ₁ → M of g ₀-modules given by θ(u ⊗X _w) = X _w (u). Then soc _g0(Imθ) has only constituents $V_0^\kappa \otimes V_1^v$ with |κ| < |λ|. Therefore if u ⊗w ∈ L is such that u = π _λ (u ₁ ⋯ ••• ⊗ u _n), w ∈ V ^⊗|µ| and e ⊗ f ∈ V ₀ ⊗ V ₁ is such that (e, u _i) = 0 for all i = 1, … , n, then

$$\theta (u \otimes w \otimes e \otimes f) = {X_{e \wedge f}}(u \otimes w) = 0.$$

((2))

Pick up e, e′ ∈ V ₀ , f, f′ ∈ V ₁ such that (e, u _i) = (e′, u _i) = 0 for all i = 1,…,n, (e, e′) = 1 and (f,f′) = 0. ThenX _{f Λ f′} = [X _{e Λf} ,X_{e′ Λ f′} ]. By (2) X _{f Λ f′} (u ⊗ w) = 0. It is easy to see that X _{f Λf′} for all orthogonal f, f′ generate sp (∞) we obtain sp (∞) w = 0. Hence µ = 0, i.e. L ⊂ R _so (M).

The proof that R _sp (M ) ≠ 0 is similar.

Now we are ready to describe simple objects in T _g . Let λ be a partition with |λ| = n. Choose a Cartan subalgebra h such that the roots of g are as in D(∞, ∞). The even roots of g are {±ε _i ± ε_j|i, j > 0, i ≠ j} ⋃ {±δ _i ± δ_j|i, j > 0} and the odd roots are {± (ε _i ± δ_j|i, j > 0}. Let b be the Borel subalgebra defined by the set of positive roots

$$\{ {\varepsilon _i} \pm {\varepsilon _j}|i < j\} \cup \{ {\delta _i} \pm {\delta _j}|i < l\} \cup \{ 2{\delta _i}|i > 0\} \cup \{ {\varepsilon _i} \pm {\delta _j}|i,j > 0\} .$$

Let V ^λ denote the simple highest weight module with highest weight λ = Σ λ _iε_i. We introduce the standard order on weights by setting λ ≤ µ if µ— λ is a non-negative integral linear combination of positive roots.

Lemma 7 V ^λ ∈ T _g . Any simple object in T _g is isomorphic to V ^λ for some partition λ.

Proof. Recall that $V_0^\lambda$ is a highest weight module over so(∞). Hence there exists a unique up to proportionality $v \in V_0^\lambda \subset {\tilde V^\lambda }$ of weight λ = Σ λ _iε_i. An easy calculation shows that n(v ) = 0. By Frobenius reciprocity there exists a non-zero homomor-phism ψ from the Verma module M ^λ to ${\tilde V^\lambda }.$ The image of ψ has a unique simple quotient isomorphic to V ^λ . Thus V ^λ ∈ T _g .

Now let M ∈ T _g be a simple module. Pick up a g ₀-submodule $L \simeq {V_0}^\lambda$ in R _so (M) with maximal |λ|. Then a non-zero v ∈ L of weight λ is annihilated by n. Therefore M ≃ V ^λ.

Lemma 8 Let M ∈ T _g be a non-zero quotient of the Verma module M^λ for some partition λ. Then M ≃ V ^λ.

Proof. Suppose that the statement is false. Then there exists V ^µ ⊂ M with µ < λ. Let v _µ ∈ V ^µ be a highest vector of weight µ and vλ be a non-zero vector of weight λ. Then v _µ ∈ U(n-)v _λ and therefore v _µ ∈ U (n^- ∩ g′)v _λ for some finite-dimensional g′ = osp(2p,2q) ⊂ g. Without loss of generality we may assume p > 2q + |µ|. Therefore the quadratic Casimir element in U (g′) has the same eigenvalue on v _µ and v _λ. This is impossible since (λ+ρ,λ+ρ) ≠ (µ+ρ,µ+ρ) if p > 2q+ |µ|, here ρ is the half sum of even positive roots minus the half sum of odd positive roots of g′.

Lemma 9 Let

$$0 \to {V^\lambda } \to M \to {V^\mu } \to 0$$

be a non-split exact sequence in T _g , then µ < λ in the standard order. Furthermore, |µ| < |λ|.

Proof. If µ and λ are not comparable then a vector of weight ¡i spans in M a sub-module isomorphic to V ^µ and the sequence splits. Since M ∈ T_g0, M is semisimple over h, the sequence also splits in the case µ = λ.If µ > λ, then M is a quotient of the Verma module M ^µ. By Lemma 8M ≃ V ^µ. Therefore the only possibility is µ < λ. Note that this implies |µ| ≤ |λ|. We claim that |µ| < |λ|. Indeed, let |µ| = |λ|. For any module N set N ⁺ be the span of all weight spaces of weights Σ a _iε_i + Σ b _jδ_j with Σ a _i = |λ|. If k ≃ gl(∞) be the subalgebra in g generated by the roots of the form ε _i — ε _j, then N ⁺ is obviously k-stable. It is easy to see that (V ^λ)⁺ and (V ^µ)⁺ are simple k-submodules in the tensor algebra of the standard k-module. Therefore

$$0 \to {({V^\lambda })^ + } \to {M^ + } \to {({V^\mu })^ + } \to 0$$

splits over k. But then the original sequence must split as well by Lemma 8.

Lemma 10 ${\tilde V^\lambda }$ is injective in T _g .

Proof. It is sufficient to prove that for any µ an exact sequence

$$0 \to {\tilde V^\lambda } \to M \to {V^\mu } \to 0$$

of modules in T _g splits. If |µ| ≥ |λ|, then the sequence splits by Lemma 9, as |µ| ≥ |v| for any simple subquotient V ^v in ${\tilde V^\lambda }$. Hence we may assume |μ| < |λ|.

By Lemma 5 ${\tilde V^\lambda }$ is injective in T _g0. Therefore the sequence splits over g ₀. Thus, we can write $M = {V^\mu } \oplus {\tilde V^\lambda }$ as a g ₀-module. The action of g ₁ is given by X(u, w) = (Xu,c(u ⊗X) + Xw) for any u ∈ V ^µ,$u \in {V^\mu },w \in {V^\lambda },X \in {g_1}$ g ₁ and some $c \in Ho{m_{go}}({V^\mu } \otimes {g_1},{\tilde V^\lambda })$. By Lemma 5

$$so{c_{go}}\,{\tilde V^\lambda } = \mathop \oplus \limits_{(v,v')} \,V_0^v \otimes V_1^{v'}$$

for some set of pairs (v, v′) such that |v| + |v′| = |λ|. Therefore we have

$$c(V_0^\mu \otimes {g_1}) \cap so{c_{go}}{\tilde V^\lambda } = \mathop \oplus \limits_v \,V_0^v \otimes {V_1},$$

where the summation is taken over some set of partitions v such that |v| = |λ| — 1. If $u \in V_0^\mu$, then

$$c(u \otimes {X_{e \wedge f}}) = \sum {a_v}{\pi _v}(u \otimes e) \otimes f,$$

for some a _v ∈ ℂ. We claim that in fact all a _v = 0. Indeed, assume a _v ≠ 0. Let $u = {\pi _\lambda }({u_1} \otimes \cdots \otimes {u_n}) \in V_0^\lambda$ for some linearly independent isotropic mutually orthogonal u1,…,u _n. For any e ∈ V ₀ linearly independent of u1,…,u _n and any non-zero f ∈ V ₁, we have c(u ⊗X _{e Λ f} ) ≠ 0. But then the annihilator of u ∈ M is not a finite corank subalgebra, hence M is not in T _g . Contradiction.

Thus, $c(V_0^\mu \otimes {g_1}) = 0$. Let ${v_\mu } \in V_0^\mu$ be the highest vector. Then (v _µ,0) ∈ M is n-invariant. Consider the submodule N ⊂ M generated by (v _µ,0). By Lemma 8 N ≃ V ^µ Therefore the exact sequence splits.

Corollary 1 The socle of ${\tilde V^\lambda }$ coincides with V^λ.

Proof. Follows from Lemma 6 and 7 since ${R_{so}}({\tilde V^\lambda }) = \tilde V_0^\lambda$ .

Corollary 2

$${R_{so}}({V^\lambda }) = V_0^\lambda ,{R_{sp}}({V^\lambda }) = V_1^{{\lambda ^ \bot }}.$$

((3))

Proof. By Corollary 1 and Lemma 10

$${V^\lambda } = \quad \quad \mathop \cap \limits_{\varphi \in Ho{m_g}({{\tilde V}^\lambda },{T^{ \leqslant |\lambda | - 1}}(V))} \;Ker\varphi .$$

Therefore Lemma 4 implies the statement.

One can prove as in [2] that T _g is antiequvivalent to the category of locally unitary finite-dimensional A _g - modules. Therefore Lemma 4 implies that T _osp (∞,∞) and T _so (∞) are equivalent abelian categories.

Define now the functors S _so : T _so(∞) → T_g and S _sp : T_sp(∞) → T_g as follows. Let M ∈ T _so(∞) (resp. T _{s p(∞)}). By I(M) we denote the induced module U(g) ⊗ _U(g0) M, where we define the action of sp(∞) (resp. so(∞)) on M to be trivial. We set

$${S_{so}}(M) = I(M)/(\mathop \cap \limits_{\varphi \in Ho{m_g}(I(M),T(V))} \;Ker\varphi ),$$

respectively

$${S_{sp}}(M) = I(M)/(\mathop \cap \limits_{\varphi \in Ho{m_g}(I(M),T(V))} \;Ker\varphi ).$$

Observe that Hom _g (I(M), T ^≤n(V)) = Hom _g0(M, T ^≤n(V)) = 0 for sufficiently large n. Thus, S _so (M) (resp. S _sp (M)) have finite length over g ₀ and hence lie in T _g . It is not hard to see that S _so(M) (resp. S _sp (M)) is the maximal quotient of I(M) belonging to T _g . Hence by Frobenius reciprocity

$$\begin{gathered} Ho{m_g}({S_{so}}(M),N) \simeq Ho{m_{so(\infty )}}(M,{R_{so}}(N)), \hfill \\ Ho{m_g}({S_{sp}}(M),N) \simeq Ho{m_{sp(\infty )}}(M,{R_{sp}}(N)). \hfill \\ \end{gathered} $$

((4))

Proposition 1 The functors S _so (resp. S _sp ) and R _so (resp. R _sp ) are mutually inverse equivalences between T _so(∞) (resp. T _sp(∞)) and T_g .

Proof. A functor F : T_g → T_so(∞) establishing an equivalence can be taken as a composition of F1 : A _g-fmod → T_so(∞) and F ₂ : T_g → A _g-fmod, where F ₁ = HomA _g(•, T(V ₀)) and F ₂ = Hom _g (·, T(V)). Lemma 4 implies R _so = F ₁ ◯ F ₂. Since S _so is left adjoint to R _so by (4), S _so must be inverse of R _so by general nonsense. The case of sp(∞) is similar.

To prove Theorem 6 we claim the following.

Proposition 2 The functors S _so (resp. S _sp ) and R _so (resp. R _sp ) are equivalences of monoidal categories.

The proof of the above Proposition amounts to showing that R _so andR _sp preserve tensor products. It is an easy consequence of the following curious fact. We leave its proof to the reader.

Lemma 11 Let k = gl(∞),sl(∞),so(∞),sp(∞). In the category Tk the functor of invariants preserves tensor product. In other words, (M ⊗ N)^k = M^k ⊗ N ^k.

4.4 The case of gl (∞,∞)

The case g = gl(∞, ∞) is similar to the case of osp(∞,∞). Therefore we only state the results omitting the proofs. Recall that the even part g ₀ is a direct sum of two copies of gl(∞).Let k = V ₀⊗(V ₀)_* and l = V ₁ ⊗(V ₁)_*. We define the functors R _k : T_g → T_k, R _l : T_g → T _l by R _k(M)=M ^l,R_l(M) =M ^k and S: T_gl(∞) → T_gl(∞|∞) by

$$S(M) = I(M)/(\mathop \cap \limits_{\varphi \in Hom(I(M),T(V \oplus {V_*}))} \;Ker\varphi ,$$

where I(M) = U(g) ⊗ _U (g₀) M.

It is not hard to see that S is left adjoint to R _k and R _l .

Theorem 7 The functors R _k and S establish an equivalence of monoidal categories T_{gl (∞,∞)} and T g _{l (∞)} .

Note that the composition S ◯ R _l : T _k → T _l defines an autoequivalence of the category T _{gl (∞)} that sends V ^λ,µ → V ^λ⊥ ,µ^⊥.

4.5 The case of P (∞)

It turns out that for g = P(∞) the category T_g is also equivalent to T_so(∞) and hence to T_sp(∞) and T _osp(∞|∞). However, the proof is different in this case.

We claim that any module M in T_g can be equipped with ℤ-grading M = ⨁ M ^k such that g _i M^k ⊂ M ^k+2i. Indeed, note that any simple g ₀-subquotient of M is isomorphic $V_0^{\lambda ,\mu }$, and we assign to it degree |λ| — |µ|.

Define a functor R : T_g → T_g0 by

$$R(M) = {M^{{g_1}}}.$$

Lemma 12 (a) For any M ∈ T_g , R(M) ≠ 0.

(b) If M is simple then R(M) is simple.

(c) If R(M) ≃ R(L) for two simple M,L ∈ T_g , then M ≃ L.

Proof. Since M has finite length over g0 there exists a maximal k such that M ^k ≠ 0. Then g ₁ (M ^k) ⊂ M ^{k +2} = 0. That proves (a).

(b) Let k be maximal such that M ^k ≠ 0. If N is a proper submodule in M ^k, then U(g)N = U(g _-1)N is a proper submodule in M since (U(g)N)^k =N. Therefore M ^k is a simple g ₀-module. If R(M)i ≠ 0 for some i < k, then U(g)(R(M)ⁱ) is a proper non-zero submodule in M. Hence if M is simple, then R(M) = M ^k is simple.

(c) It is clear that both M and L are simple quotients of the parabolically induced module U(g) ⊗ _U(g0⨁g1) R (M). But the latter has a unique simple quotient. Hence (c).

Lemma 13 Let M ∈ T_g .

(a) If $V_0^\lambda \subset$ M is an embedding of g ₀ -modules, then $V_0^\lambda$ ⊂ R (M).

(b) Any simple submodule in R(M) is isomorphic to $V_0^\lambda$ for some partition λ.

Proof. (a) Consider the map $\psi :{g_1} \otimes V_0^\lambda \to M$ defined by ψ (X ⊗ v) = Xv. Suppose |λ| = k. Consider linearly independent u ₁ ,…, u _k ∈ V ₀ . Then $u = {\pi _\lambda }({u_1} \otimes \cdots \otimes {u_k}) \in V_0^\lambda$ is not zero. Moreover, for any w, v ∈ V ₀ we have

$$\psi ({X_{w,v}} \otimes u) = \mathop \sum \limits_v {a_v}{\pi _v}(w \otimes v \otimes u)$$

for some v obtained from λ by adding two boxes not in the same column and some a _v ∈ ℂ. Note that if some a _v ≠ 0, then for any w, v such that u ₁ ,…,u _k,v,w are linearly independent, we have ψ (X _w,v ⊗u) ≠ 0. But then the annihilator of $v \in V_0^\lambda$ is not of finite corank. Hence ψ = 0, i.e. $V_0^\lambda \subset R(M)$.

(b) Let L be a simple g ₀-submodule in R(M). Then $L \simeq so{c_{g0}}(V_0^\lambda \otimes V_1^\mu )$ We want to show that µ = 0. Assume the opposite. Consider the g ₀-homorphism ψ : L ⊗ g _-1 → M given by ψ ( w⊗ X) = X(w) for all X ∈ g _-1 ,w ∈ L. Since the annihilator of any vector in M has finite corank we have

$$\psi (so{c_{{g_0}}}(L{ \otimes _{g - 1}})) = 0.$$

((5))

Let w G L be of the form

$$w = {\pi _\lambda }({v_1} \otimes \cdots \otimes {v_n}) \otimes {\pi _\mu }({\mu _1} \otimes \cdots \otimes {u_m}),$$

where v _i ∈ V ₀ , u _j ∈ V ₁ are linearly independent, n = |λ|,m = |µ|, (v _i,u_j) = 0 for all i ≤n, j ≤ m. Let f ₁ , f ₂ ∈ V ₁ be orthogonal to v ₁ ,…, v _n and linearly independent with u ₁ ,…, u _m.By (5)

$$\psi (w \otimes {X_{{f_1}}}_{{f_2}}) = {X_{{f_1},}}_{{f_2}}(w) = 0.$$

((6))

Let e ∈ V ₀. Then for any v ∈ V ₀ , u ∈ V ₁ we have

$$\begin{array}{*{20}{l}} {[{X_{e,e}},{X_{{f_1},}}_{{f_2}}](v) = 2(({f_1},v)({f_2},e) - ({f_2},v)({f_1},e))e,} \\ {[{X_{e,e}},{X_{{f_1},}}_{{f_2}}](u) = 2(e,u)(({f_2},e){f_2} - ({f_2},e){f_1}).} \end{array}$$

((7))

Pickup e ∈ V ₀ , f ₁ , f ₂ ∈ V ₁ such that (e,f ₁) = 1, (e, f ₂) = 0, (e,u ₁) = 1 and (e,u _i) = 0 for i > 1. Then by (6) [X _e,e,X_{f 1, f2}] (w) = 0 and by (7)

$$[{X_{e,e}},{X_{{f_1},}}_{{f_2}}](w) = 2{\pi _\lambda }({v_1} \otimes \cdots \otimes {v_n}) \otimes {\pi _\mu }({f_2}{u_2} \otimes \cdots \otimes {u_m}) \ne 0.$$

Contradiction.

Corollary 3 For any simple $M \in {T_g},R(M) \simeq V_0^\lambda$ for some λ.

We use the notation V ^λ for the simple M ∈ T_g with $R(M) = V_0^\lambda$ We will prove now that V ^⊗n is injective in T_g. Consider the action of S _n on V ^⊗n such that an adjacent transposition σ _i,i+1 acts by

$${\sigma _{i,i + 1}}({v_1} \otimes \cdots \otimes {v_n}) = {( - 1)^{p({v_i})p({v_{i + 1}})}}{v_1} \otimes \cdots \otimes {v_{i + 1}} \otimes {v_i} \otimes \cdots \otimes {v_n}.$$

This action commutes with the action of g.

Proposition 3 (a) R(V ^⊗n) = (V ₀)^⊗n;

(b) $soc{V^{ \otimes n}} = U(g0)(V_0^{ \otimes n})$ ;

(c) $En{d_g}({V^{ \otimes n}}) = En{d_{g0}}(V_0^{ \otimes n}) = \mathbb{C}[{S_n}]$ ;

(d) ${V^{ \otimes n}} = { \oplus _{|\lambda | = n}}{\tilde V^\lambda } \otimes {Y_\lambda }$, where Y_λ is the irreducible S _n-module associated to λ and ${\tilde V^\lambda }$ is an indecomposable module with socle V^λ.

Proof. (a) Consider V ^⊗n as a g ₀-module. It splits into indecomposable sumands V ₀ ^µ ⊗ V ₁ ^V with |µ| + |v| = n. Hence the statement follows from Lemma 13(b)

(b) By Lemma 12(a) and (a) each $V_0^\mu \subset V_0^{ \otimes n}$ generates a simple submodule. Therefore (b) follows from (a).

(c) The restriction map: $En{d_g}({V^{ \otimes n}}) \to En{d_{g0}}(V_0^{ \otimes n}) = \mathbb{C}[{S_n}$; is obviously surjec-tive. We claim that the quotient of V ^ℂn by the socle can have only simple subquotients V ^µ for |µ| < n. Indeed, if V ^µ is a simple subquotient of V ^⊗n, then $V_0^\mu$ is a simple g ₀-subquotient of V ^ℂn. Hence |µ| ≤ n. On the other hand, if |µ| = n, then $V_0^\mu \subset V_0^{ \otimes n} \subset soc{V^{ \otimes n}}$. Therefore any φ ∈ End _g (V ^⊗n) that kills the socle must be zero. That implies injectivity of the restriction map.

(d) is a consequence of (c) and (b).

Note that (b) also implies Corollary 4 If Hom _g (V ^⊗n, V^⊗m) ≠ 0, then n — mis non-negative even.

Lemma 14 Let $M \in {T_g},V_0^\lambda \subset R(M)$ generates M, then M ≃ V ^λ.

Proof. Let |λ| =n. Consider the parabolically induced module $K = U(g){ \otimes _{U(g0 \oplus g1)}}V_0^\lambda$ . We have an isomorphism $K \simeq \Lambda (g - 1) \otimes V_0^\lambda$ of g ₀-modules. Let ${N_0} = so{c_{g0}}(g - 1 \otimes V_0^\lambda )$. First, we show that g ₁(N ₀) = 0. For this we fix the Borel sub-algebra of g ₀ such that all tensor modules are highest weight modules (see [2] and Sect. 3). If $v \in V_0^\lambda$ is a highest vector, Y ∈ g _-1 is a highest vector with respect to the adjoint action of g ₀, then Y ⊗ v generates N ₀ (as a g ₀-module). Let X ∈ g ₁, then X(Y ⊗v) = 1⊗ [X,Y]v. By a straightforward check [X,Y]v = 0. Thus, g ₁(Y ⊗v) =0 and hence the whole N ₀ is annihilated by g ₁ .

Now let N ⊂ K be the submodule generated by N ₀ and Q = K/N. Let π : K → Q denote the natural projection. Note that the annihilator of π ( 1 ⊗ v) is of finite corank. Since U (g) π (1 ⊗ v) = Q,Q satisfies (1) and (2). Note that M is a quotient of K. Consider the natural projection σ : K → M. If σ (N ₀) ≠ 0, then Zσ (1 ⊗ v) ≠ 0 for any Z ∈ g _-1, that contradicts our assumption M ∈ T_g, as σ (1 ⊗ v) does not have the annihilator of finite corank. Hence σ (N ₀) = 0 and therefore M is a quotient of Q.

Note that although K ∉ T_g, it is still equipped with the ℤ-grading such that ${K^{n - 2k}} = {\Lambda ^k}(g - 1) \otimes V_0^\lambda$ . Hence both N and Q are also graded. We claim that for all µ and k > 0

$$Ho{m_{go}}(V_0^\mu ,{Q^{n - 2k}}) = 0.$$

((8))

Indeed, N ^n-2k is generated by Λ^k-¹(g _-1)(Y ⊗ v) over g ₀. Any weight vector in Λ^k-1(g _-1)(Y ⊗ v) has weight Σ a _iε_i with at least two negative a _i and hence belongs to $soc_{g0}^{2k - 1}{K^{n - 2k}}$. But then

$${N^{n - 2k}} \subset soc_{{g_0}}^{2k - 1}{K^{n - 2k}}.$$

Therefore we have an embedding of g ₀-modules

$$so{c_{{g_0}}}{Q^{n - 2k}} \subset (soc_{{g_0}}^{2k}{K^{n - 2k}})/{N^{n - 2k}}.$$

All g ₀-simple subquotients of $soc_{g0}^{2k}{K^{n - 2k}}$ are of the form $so{c_{g0}}(V_0^\mu \otimes V_1^v)$ with |v| ≥ 1. Hence (8).

Now we can prove that Q is simple and hence isomorphic to V ^λ. Indeed, it is equivalent to proving that $R(Q) = V_0^\lambda$. Suppose the latter is false, i.e. R(Q)ⁿ-^2k ≠ 0 for some k > 0. By Lemma 13(a) there exists µ such that $V_0^\lambda \subset R{(Q)^{n - 2k}}$. But this contradicts (8).

Theorem 8 ${\tilde V^\lambda }$ is the injective hull of V ^λ in T _g .

Proof. It suffices to prove that any exact sequence in T_g of the form

$$0 \to {\tilde V^\lambda } \to M \to {V^\mu } \to 0$$

splits. Since ${\tilde V^\lambda }$ is injective in T _g0 the sequence splits over g₀. In particular, we have an embedding $V_0^\mu \subset M$ of g ₀-modules. By Lemma 13(a) $V_0^\mu \subset R(M)$. By Lemma 14 $V_0^\mu$ generates V ^µ ⊂ M. Hence the statement.

The above theorem and Proposition 3(d) imply

Corollary 5 V ^⊗n is injective in T_g .

Recall that A _g denotes the subalgebra of all bounded operators in End _g (T(V)). Theorem 8 implies

Corollary 6 T_g is antiequivalent to the category of locally unitary finite-dimensional modules over A _g .

By Corollary 4 we have a ℤ-grading

$${A_g} = \mathop \oplus \limits_{i \geqslant 0} A_{g,}^iA_g^i = \mathop \oplus \limits_{n \geqslant 0} Ho{m_g}({V^{ \otimes n}},{V^{ \otimes n - 2i}}).$$

By Proposition 3

$$A_g^0 = \mathop \oplus \limits_{n \geqslant 0} {\kern 1pt} {\kern 1pt} {\kern 1pt} \mathbb{C}[{S_n}].$$

For any 1 ≤ i < j ≤ n define $\tau _{ij}^n$ Hom_g ( V ^⊗n, V^⊗n-²) by the formula

$$\tau _{ij}^n({v_1} \otimes \cdots \otimes {v_n}) = {( - 1)^s}({v_{i,}}{v_j}){v_1} \otimes \cdots \otimes {\hat v_i} \otimes \cdots \otimes {\hat v_j} \otimes \cdots \otimes {v_n},$$

where s = (p (v _i) + p (v _j)) (p (v ₁) + ••• + p(v _{i- 1})) + p (v _j) (p(v _{i +1}) +••• + p ( v _j-1 )) .

Lemma 15 $\{ \tau _{ij}^n\}$ for all n > 1 and 1 ≤ i < j ≤ n form a basis in $A_g^1$.

Proof. It is sufficient to show that for a fixed n, the set $\{ \tau _{ij}^n\}$ for all 1 ≤ i < j ≤ n is a basis in Hom _g (V ^⊗n, V^⊗n-²). Linear independence is straightforward. To prove that $\{ \tau _{ij}^n\}$ span Hom _g (V ^⊗n, V^⊗n-²) consider the homomorphism

$$\rho :Ho{m_g}({V^{ \otimes n}},{V^{ \otimes n - 2}}) \to Ho{m_{{g_0}}}({({V^{ \otimes n}})^{n - 2}},{({V^{ \otimes n - 2}})^{n - 2}})$$

defined by restriction to the n - 2-th graded component. Since ${({V^{ \otimes n - 2}})^{n - 2}} = V_0^{ \otimes n - 2}$ generates the socle of V ^⊗n-², this homomorphism is injective. Write

$${({V^{ \otimes n}})^{n - 2}} = \mathop \oplus \limits_n^{k = 1} {M_{k,}}$$

where ${M_k} = V_0^{ \otimes k - 1} \otimes {V_1} \otimes {V_1} \otimes V_0^{ \otimes n - k}.$ Any ψ ∈ Hom_g0 $({({V^{ \otimes n}})^{n - 2}},V_0^{ \otimes n - 2})$ can be written in the form $\sum {{a_{kl}}\theta _k^l\,where} \,\theta _k^l:{M_k} \to V_0^{ \otimes n - 2}$ is the restriction of $\tau _{kl}^n$ to M _k if k < l or $\tau _{lk}^n$ if k > l. If ψ lies in the image of ρ, then ψ = ρ(ϕ) and therefore

$$\psi ({X_{{f_1},{f_2}}}({v_1} \otimes \cdots \otimes {v_n})) = {X_{{f_1},{f_2}}}\phi ({v_1} \otimes \cdots \otimes {v_n}) = 0$$

((9))

for any v ₁ ,…, v _n, f₁ , f ₂ ∈ V ₀. Choose v ₁ ,…, v _n ∈ V ₀ , f ₁ , f ₂ ∈ V ₁ so that (v _i, f₁) =0 for any i ≠ k, (v _i, f₂) = 0 for any i ≠ l, (v _k, f₁) = (v _l, f₂) = 1. Then (9) implies a _kl = alk. Therefore $\psi = \rho (\mathop \sum \limits_{k < l} {a_{kl}}\tau _{kl}^n).$. The result follows now from injectivity of ρ.

Lemma 16 Let λ ⁺ (resp. λ^-) denote the set of all µ obtained from λ by adding (resp. removing) one box. Then we have the following exact sequence

$$0 \to \mathop \oplus \limits_{v \in \lambda + } {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {V^v} \to {V^\lambda } \otimes V \to \mathop \oplus \limits_{\mu \in \lambda - } {\kern 1pt} {\kern 1pt} {\kern 1pt} {V^\mu } \to 0.$$

Proof Assume |λ| = n. From embedding V ^λ ⊗V ⊂ V ^⊗n+1 we have

$$R({V^\lambda }{\kern 1pt} {\kern 1pt} \otimes {\kern 1pt} {\kern 1pt} {\kern 1pt} V) = V_0^\lambda {\kern 1pt} {\kern 1pt} \otimes {\kern 1pt} {\kern 1pt} {\kern 1pt} {V_0} = \mathop \oplus \limits_{v \in \lambda + } {\kern 1pt} {\kern 1pt} {\kern 1pt} V_0^v.$$

Let M be the submodule in V ^λ ⊗ V generated by R (V ^λ ⊗ V). Then M = ⨁_v∈λ+ V ^v by Lemma 14 and Pierri rule. Let S = (V ^λ ⊗ V)/M and π : V ^λ ⊗ V → S be the natural projection. Then S is generated by $\pi (V_0^\lambda \otimes {V_1})$. Moreover, $\pi (V_0^\lambda \otimes {V_1}) \subset R(S)$. By Lemma 13(b) and [5]

$$\pi (V_0^\lambda \otimes {V_1}) \subset \mathop \oplus \limits_{\mu \in \lambda - } {\kern 1pt} {\kern 1pt} V_0^\mu .$$

To see that

$$\pi (V_0^\lambda \otimes {V_1}) = \mathop \oplus \limits_{\mu \in \lambda - } {\kern 1pt} {\kern 1pt} V_0^\mu $$

observe that the set $\{ \tau _{j,n + 1}^{n + 1}|1 \leqslant j \leqslant n\}$ spans $Ho{m_{{g_0}}}(V_0^\lambda \otimes {V_1},V_0^{n - 1})$. Since $\tau _{j,n + 1}^{n + 1}(M) = 0$ for all j ≤ n we have

$$M \cap (V_0^\lambda \otimes {V_1}) \subset so{c_{{g_0}}}(V_0^\lambda \otimes {V_1})$$

Now the statement follows from Lemma 14.

Lemma 17

$$soc{V^{ \otimes n}} = \bigcap\limits_{\varphi \in Ho{m_g}({V^{ \otimes n}},{V^{ \otimes n - 2}})} {Ker\varphi = \bigcap\limits_{1 \leqslant i \leqslant j \leqslant n} {Ker\tau _{ij}^n.} } $$

Proof. The inclusion

$$soc{V^{ \otimes n}} \subset \bigcap\limits_{1 \leqslant i \leqslant j \leqslant n} {Ker\tau _{ij}^n} $$

is trivial since $V_0^{ \otimes n} \subset Ker\,\tau _{ij}^n$ for all i, j and socV ^⊗n is generated by $V_0^{ \otimes n}$.

We prove equality by induction in n. Let ${X_n} = { \cap _{1 \leqslant i < j \leqslant n}}Ker\,\tau _{ij}^n$. By induction assumption we have

$${X_n} \subset {X_{n - 1}} \otimes V = (soc{V^{ \otimes n - 1}}) \otimes V.$$

Using the previous lemma one can easily see that there is an exact sequence

$$0 \to soc{V^{ \otimes n}} \to {X_{n - 1}} \otimes V \to Z \to 0,$$

where Z is a direct sum of some V ^µ with |µ| = n - 2. By the above exact sequence it is sufficient to check

$${(soc{V^{ \otimes n}})^{n - 2}} = \bigcap\limits_{1 \leqslant i \leqslant j \leqslant n} {Ker\tau _{ij}^n \cap {{({V^{ \otimes n}})}^{n - 2}}.} $$

Our calculation in the proof of Lemma 15 implies that

$$\bigcap\limits_{1 \leqslant i \leqslant j \leqslant n} {Ker\tau _{ij}^n \cap {{({V^{ \otimes n}})}^{n - 2}} = {g_{ - 1}}V_0^{ \otimes n}.} $$

Hence the statement.

Lemma 18 A _g is generated by $A_g^0$ and $A_g^1$.

Proof. Let φ ∈ Hom _g (V ^⊗n, V^⊗n-2k) for k > 1. Then socV ^⊗n ⊂ Ker φ. Let M = V ^⊗n/socV ^⊗n. By Lemma 17

$$\mathop \oplus \limits_{1 \leqslant i \leqslant j \leqslant n} \quad \tau _{ij}^n:{V^{ \otimes n}} \to {({V^{ \otimes n - 2}})^{ \oplus n(n - 1)/2}}$$

defines the embedding M ⊂ (V ^⊗n-2)^{⨁n (n-1})/ ². By injectivity of V ^⊗n-^2k there exists ψ ∈ Hom _g ((V ^⊗n-2)^⨁n(n-1)/2 , V ^{⊗n -2k}) such that

$$\varphi = \psi \;o\;(\mathop \oplus \limits_{1 \leqslant i \leqslant j \leqslant n} \;\tau _{ij}^n).$$

Hence

$$\varphi = \mathop \oplus \limits_{1 \leqslant i \leqslant j \leqslant n} \;{\psi _{ij}}\,o\;\tau _{ij}^n$$

for some ψ _ij ∈ Hom _g (V ^⊗n-2 , V ^⊗n-2k). Now the statement easily follows by induction in k.

Theorem 9 The graded algebras A _P(∞) and A _so(∞) are isomorphic. Hence the categories T _P(∞), T_so(∞), T_sp(∞) and T_osp are all equivalent.

It is an open problem to construct directly a functor T _P(∞) → T_so(∞) that preserves tensor product.

Finally, let us observe that we know very little about finite-dimensional representations of P(n). In particular, characters of simple modules and extensions between simple modules are unknown. On the other hand, in T _P(∞) these questions are easy to answer. Is it possible to use information about representations of P(∞) to make some progress in the finite-dimensional case?

5 The case of the queer Lie superalgebra Q(n)

5.1 Sergeev duality

In this section we assume g = Q(∞). Let us recall the analogue of Schur-Weyl duality result in this case. It is due to Sergeev [7].

Let Hn be the semidirect product of ℂ[S _n] and the Clifford algebra Cliff _n with generators pi,i= 1,…,n satisfying the relations

$$p_i^2 = - 1,{p_i}{p_j} + {p_j}{p_i} = 0.$$

Define the action of H _n on V ^⊗n as follows

$$\begin{gathered} {s_{i,j + 1}}({v_1} \otimes \cdots \otimes {v_i} \otimes {v_{i + 1}} \otimes \cdots \otimes {v_n}) = {( - 1)^{p({v_i})p({v_{i + 1}})}}({v_1} \otimes \cdots \otimes {v_{i + 1}} \otimes {v_i} \otimes \cdots \otimes {v_n}), \hfill \\ {p_i}({v_1} \otimes \cdots \otimes {v_i} \otimes \cdots \otimes {v_n}) = {( - 1)^{p({v_1}) + \cdots + }}^{p({v_{i - 1}})}({v_1} \otimes \cdots \otimes J({v_i}) \otimes \cdots \otimes {v_n}). \hfill \\ \end{gathered} $$

One can show (see [7]) that H _n ≃ U _n ⊗ Cliff _n for a certain finite-dimensional algebra U _n. The category of finite-dimensional representations of U _n is equivalent to the category of projective representations of S _n. Irreducible representations of U _n are enumerated by strict partitions.

Theorem 10 ([7]) H _n is the centralizer of Q(∞) in V ^⊗n. There is a decomposition

$${V^{ \otimes n}} = \oplus \,{2^{ - \delta (\lambda )}}{V^\lambda } \otimes {T_\lambda },$$

here summation is taken over all strict partitions of size n, T _λ is the irreducible H_n-module corresponding to the strict partition λ, δ (λ) is the parity of the length of λ.

The coefficient 2^-δ(λ) appears in the case when dimEnd _{Q (∞)}(V _λ) = (1|1). For example, the second tensor power of the standard representation V has a decomposition

$$V \otimes V = {S^2}(V) \oplus {\Lambda ^2}(V).$$

ButS ²(V) ≃Λ²(V) as Q(∞)-modules because p ₁ p ₂(S ²(V)) =A ²(V). There is only one strict partition λ = (2,0,…,0) of size 2, δ(λ) = 1. The reader will check that H ₂ has an irreducible 4-dimensional module (in the category of Z₂-graded modules).

5.2 The category TQ_(∞)

Let us consider the mixed tensor module

$$T = \mathop \oplus \limits_{m,n \geqslant 0} \;{V^{ \otimes n}} \otimes V_*^{ \otimes m}$$

We claim that T is an injective cogenerator in the category TQ _(∞) .

Lemma 19

$${V^{ \otimes n}} \otimes V_*^{ \otimes m} = \mathop \oplus \limits_{|\lambda | = n,|\mu | = m} {2^{ - \max (\delta (\lambda ),\delta (\mu ))}}{\tilde V^{\lambda ,\mu }} \otimes ({T_\lambda } \boxtimes T\mu ),$$

where ${\tilde V^{\lambda ,\mu }}$ is an indecomposable injective module in T_g with simple socle V ^λ,µ.

Define the graded subalgebra A _{Q (∞)} ⊂ End _{Q (∞)} (T) generated by ⨁ _{n,m ≥0} Hn ⊗ H _m and all contractions in $Ho{m_{Q(\infty )}}({V^{ \otimes n}} \otimes V_*^{ \otimes m}, \otimes V_*^{ \otimes m - 1}).$

Conjecture Let g = Q(∞).

• • V ^{λ, µ} (for pairs of strict partitions (λ,µ)) are all up to isomorphism simple objects in T_g.

• • ${\tilde V^{\lambda ,\mu }}$ are all up to isomorphism indecomposable injective objects in T_g.

• • A _g is a direct limit of self-dual Koszul rings.

• • T_g is antiequivalent to the category of finite dimensional (locally unitary) A _g -modules.

• • The socle filtration of ${\tilde V^{\lambda ,\mu }}$ is given by

  $$so{c^k}({\tilde V^{\lambda ,\mu }}) = \mathop \oplus \limits_{|\gamma | = k} \;R_{\gamma ,\lambda '}^\lambda R_{\gamma ,\mu '}^\mu {V^{\lambda ',\mu '}}.$$

  Here $R_{\gamma ,\lambda '}^\lambda$ stand for the Littlewood-Richardson coefficients for the Lie superal-gebra Q(∞).

Note that an analogue of Howe, Tan, Willenbring result for Q(N) for N >> 0 is difficult to get since there is no complete reducibility. On the other hand, even if we had such result, it is unclear how to proceed to ∞, because we “loose” some simple constituents at ∞.

For instance, SQ(n) has a one dimensional center for every n but

$$SQ(\infty ) = \mathop {\lim }\limits_ \to {\kern 1pt} {\kern 1pt} {\kern 1pt} SQ{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (n)$$

is simple.

Acknowledgements This work was supported by the NSF grants 0901554 and 1303301.