MEROMORPHIC BRAIDED CATEGORY ARISING IN QUANTUM AFFINE ALGEBRAS

Y. Soibelman

April 30, 1999

Abstract: We construct meromorphic braided category with objects living on elliptic curve. It consists of flat vector bundles with an action of the affine quantum algebra on the fibers.

Key words: meromorphic braided category, affine quantum algebra, elliptic curve.

1 Introduction

Main result of this article consists in construction of what is called meromorphic braided (or tensor) category, see [So]. It arises in the representation theory of quantum affine algebras and lives on an elliptic curve.

There are many papers devoted to different aspects of finite-dimensional representations of quantum affine algebras. Surprisinly few of them consider categorical picture. Most of the authors study irreducible representations. Some fundamental questions remain unanswered. For example the fact that the universal R-matrix is meromorphic for any two finite-dimensional representations was not proved (to my knowledge) before [KS]. This fact is crucial for construction of meromorphic braided structure on the category of finitedimensional representations. The latter contains an interesting subcategory with objects naturally “localized” on an elliptic curve.

In this article we consider the simplest example related to the quantum affine algebra sl(2). Our constructions remind “chiral” objects from [BD]: we derive the braiding considering infinitesimal neighbourhood of the diagonal in the square of an elliptic curve.

The paper is organized as follows. Next two sections contain recollections from [So]. Last section contains construction and discussion. It can be read independently of the other part of the paper. In the last section we mainly concentrate on the case of quantum affine sl(2). It is explained at the very end of the paper how main construction can be generalized to higher ranks if one uses results of [FR].

Acknowledgements. Constructions of Section 4 have arisen from a number of discussions with Joseph Bernstein. I am grateful to Ed Frenkel for the discussions about q-characters and simple modules over quantum affine algebras.

I am also grateful to IHES and Max-Planck Institute of Mathematics for support and excellent working conditions. This work was also partially supported by an NSF grant.

2 Pseudo-braided categories

In this section we are going to recall (in a slightly revised form) main definitions from [So].

2.1

We denote by T the class of all planar trees, and by T (n) the subclass of trees having n tails (see [GK], [KM] about terminology). Then T (n) is a category (morphisms are identities and contraction of edges). For a given tree we orient edges and tails in such a way that there is a unique vertex with only one outgoing tail (this vertex is called the root of tree). We fix such an orientation for each tree. We also fix a numeration of tails : for each T ∈ T (n) they are numbered from 1 to n in such a way that the only outgoing tail is numbered by n.

An additional structure on T is given by the gluing operation: if T ∈ T (n + 1), T_{i} ∈ T (k_{i} + 1), i = 1, ..., n + 1 then one can construct a new tree T (T_{1}, ..., T_{n}) ∈ T (k_{1} + ... + k_{n} + 1) by gluing outgoing tail of T_{i} to the ith tail of T. The orientation of edges and numeration of tails for the new tree are defined in the natural way. The gluing operation is associative, and hence T becomes a strict monoidal 2-operad . The role of a unit object is played by the only tree e ∈ T (1).

Let S = (S_{T} , O_{S_{T}}) be a family of ringed spaces parametrized by trees from T . We say that S is a monoidal operad of spaces if the following conditions are satisfied:

1) for any morphism f : T′ → T ∈ T (n) we are given a morphism of ringed spaces l_{f} : S_{T′} → S_{T}.

2) For a gluing operation of trees T × T_{1} × ... × T_{n} → T (T_{1}, ..., T_{n}) we are given a morphism of ringed spaces (operadic composition) γ : S_{T} × S_{T_{1}} × ... × S_{T_{n}} → S_{T (T_{1},...,T_{n})} which is strictly associative with respect to the gluing of trees. It is also assumed to be functorial with respect to the morphisms of trees.

In particular if we put A = Γ(S_{e}, O_{S_{e}}), then all Γ(S_{T} , O_{S_{T}}) become A − A^{⊗n} bimodules, where T ∈ T (n).

2.2

Let X be a set, A a class. Its elements are called objects. A family of objects of A parametrized by X (or simply X-family) is an element of ∏_{X} A = A^{X}. Suppose that A is a category. We keep the same notation for the class of objects of A. Then X-families form a category F_{X} with Hom_{F_{X}}(M, N) = ∏_{X} Hom_{A}(M_{x}, N_{x}). If f : X → Y is a map the there is a pull-back functor f^{∗} : F_{Y} → F_{X}.

If (X, O_{X}) is a ringed space then we understand an X-family as a family of objects which are O_{X,x}-modules, where O_{X,x} denotes the fiber at x ∈ X . The pull-back functors exist in the case of ringed spaces as well. We do not impose any continuity condition on the family at the moment.

Suppose that S is a monoidal operad of spaces as in the previous subsection. We suppose that we are given a field k , and that all sheaves of rings are in fact k-algebras and all morphisms of sheaves of rings are morphisms of k-algebras.

Definition 1 A pseudo-monoidal category C of ringed spaces is given by the following data:

a) a class C called class of objects;

b) for every T ∈ T (n) , a sequence {X_{i}}, 1 ≤ i ≤ n of objects of C and an object Y ∈ C a family of k-vector spaces P_{T} ({X_{i}}, Y) over S_{T} (operations from {X_{i}} to Y);

c) for every morphism f : T′ → T a morphism ϕ_{f} : P_{T′}({X_{i}}, Y) → (l_{f})^{∗}P_{T} ({X_{i}}, Y) of families over S_{T′};

d) for an operadic composition of spaces γ : S_{T} × ∏_{i} S_{T_{i}} → S_{T (T_{1},...,T_{n})} we are given a morphism of families on

S_{T} × ∏_{i} S_{T_{i}} (composition of operations):

Φ_{γ} : P_{T} ({X_{i}}, Y) × ∏_{i} P_{T_{i}}({K_{j_{i}}}, X_{i}) → γ^{∗}P_{T (T_{1},...,T_{n})}({K_{j}}, Y) , Φ_{γ}(ϕ, (ψ_{i})) = ϕ(ψ_{i}). Here {X_{i}} are parametrized by the tails of T, {K_{j}} are parametrized by

the tails of T (T_{1}, ..., T_{n}) and {K_{j_{i}}} corresponds to the subsequence of objects parametrized by the tails of T_{i};

e) for every object X ∈ C there exists a family 1_{X} ∈ P_{e}({X}, X) on S_{e} such that ϕ(1_{X_{i}}) = γ^{∗}(ϕ) and 1_{X}(ϕ) = η^{∗}(ϕ).

Here ϕ ∈ P_{T} ({X_{i}}, X) and we use the notations γ and η for the natural operadic morphisms of spaces (see d)).

The composition morphisms Φ_{γ} from d) are required to be associative in the following sense:

Φ_{γ}(id × Φ_{δ}) = Φ_{δ}(Φ_{γ} × id) (when applied to the corresponding families).

Sometimes we will call C a pseudo-monoidal category over S. Suppose our spaces are k-schemes. We call a pseudo-monoidal category

algebraic if all families P_{T} ({X_{i}}, Y) are quasi-coherent sheaves and the corresponding morphisms are morphisms of quasi-coherent sheaves.

Suppose that k = C and our spaces are complex analytic. We call a pseudo-monoidal category analytic if the families P_{T} ({X_{i}}, Y) are complex analytic sheaves and the corresponding morphisms are morphisms of complex analytic sheaves.

Similarly, if these complex analytic spaces are irreducible, we can speak about meromorphic pseudo-monoidal categories (in this case P_{T} are analytic families on dense subsets and morphisms can be extended meromorphically to S_{T}).

In the case of schemes we obtain rational pseudo-monoidal categories. We are going to use this terminology without further discussion.

2.3

Here we recall what is it a representable pseudo-monoidal structure. Let S be a topological space, C a category, n ≥ 1 an integer.

We denote by F unct_{S}(C, n) the sheaf of categories on S such that for an open U in S we have F unct_{S}(C, n)(U)= category of families of functors {F_{x}}_{x∈U}, F_{x} : C^{n} → C. We will denote this sheaf of categories by F unct_{S}(n) if it will not lead to a confusion.

Suppose that in addition we have a sequence of topological spaces {S_{i}}, 1 ≤ i ≤ n , and a sequence of positive integers k_{1}, ..., k_{n} . Then we have a morphism of sheaves of categories F unct_{S}(n)×∏_{i} F unct_{S_{i}} → F unct_{S×∏_{i} S_{i}}(k_{1} + ... + k_{n}) (sheaves on S × ∏_{i} S_{i}) such that {F_{s}} × ∏_{i} F_{s_{i}}^{i} → F_{s}(F_{s_{1}}^{1}, ..., F_{s_{n}}^{n}).

If C is a k-linear category then for a family {F_{x}} ∈ F unct_{S}(n)(S) and a sequence {X_{i}}, 1 ≤ i ≤ n of objects of C and an object Y ∈ C we have a family of vectors spaces Hom_{C}(F_{x}({X_{i}}), Y) on S.

Suppose now that we have a monoidal operad of spaces S = (S_{T}) as before. Suppose that for every T ∈ T (n) we are given a family of functors {F_{x}^{T} }_{x∈S_{T}} ∈ F unct_{S_{T}} (n). We remark that there is a natural operadic composition F unct_{S_{T}} (n) × ∏_{i} F unct_{S_{T_{i}}}(k_{i}) → F unct_{S_{T} ×∏_{i} S_{T_{i}}}(k_{1} + ... + k_{n}) → γ^{∗}(F unct_{S_{T (T_{1},...,Tn)}}(k_{1} + ... + k_{n}) of sheaves of categories.

The last arrow corresponds to the operadic composition γ on the spaces which induces the obvious pull-back on the families. Suppose that we are given a pseudo-monoidal category on S as before.

Definition 2 We say that it is representable if for every T ∈ T (n) there exists a family F_{x}^{T} ∈ F unct_{S_{T}} (n)(S_{T}) and an isomorphism of families of kvector spaces P_{T} ({X_{i}}, Y) → Hom_{C}(F_{x}^{T} ({X_{i}}), Y) which is compatible with the operadic composition on both families.

It is also required that for the only tree e ∈ T (1) we have: Hom_{C}(F_{x}^{e}(Y), Y) corresponds to 1_{Y} under this isomorphism.

If C is an algebraic or analytic pseudo-monoidal category then representability is understood in the corresponding category. In particular it is assumed that families of functors define families of Hom’s in the corresponding category, and the isomorphisms of families must be isomorphisms of quasicoherent sheaves or analytic sheaves. Similarly one defines representable rational and meromorphic pseudo-monoidal categories. We skip ”pseudo” in the case when all morphisms in the Definition 1c) are isomorphisms. In this way we obtain for example the notion of meromorphic monoidal category discussed in [So].

It is easy to define the notion of a functor between two pseudo-monoidal categories A and B which live over different operads of spaces,say, S = (S_{T}) and R = (R_{T}) respectively. It consists of morphisms of ringed spaces h_{T} : S_{T} → R_{T} which produce a morphism of the monoidal operads of spaces, of the mapping of objects F : A → B, and of the morphisms of families l_{T} : P_{T}^{A}({X_{i}}, Y) → h_{T}^{∗} P_{T}^{B}({F (X_{i})}, F (Y)) which are compatible with the compositions and the unit family. It is clear how to specify this definition for the case of schemes or analytic spaces as well as to the rational or meromorphic case.

2.4

Let C be a pseudo-monoidal category over S. It is called pseudo − braided if for every element σ of the braid group B_{n} we are given a morphism of families μ_{σ} : P_{T} ({X_{i}}, Y) → P_{T} ({X_{σ(i)}}, Y) which is identical on S_{T}. Here σ acts on i as the corresponding permutation from the permutation group S_{n}.

These morphisms are required to satisfy various natural properties. Details can be found in [So]. Here we list them shortly:

a) μ_{στ} = μ_{σ}μ_{τ}, μ_{1} = id where 1 denotes the unit of the group;

b) for any σ ∈ B_{n} the morphism μ_{σ} commutes with morphisms in T (n), and μ_{1} preserves 1_{X} for any object X;

c) compatibility with the composition maps (this means natural commutative diagram, see [So]).

Now we can specify an operad S of spaces (schemes, analytic spaces, manifolds, etc.). Then we require that all μ_{σ} are morphisms in the corresponding category and arrive to various versions of pseudo-braded categories (algebraic, analytic, etc.). If the our spaces are irreducible (in the corresponding category) we can require that all μ_{σ} are defined in the generic point only. In this way we obtain the notions ot rational (in the case of schemes) and meromorphic (in the case of analytic spaces or complex manifolds) pseudo-braided category (see [So] for the details). The notion of a functor between such categories is defined in the natural way. The definitions in the representable case are also clear. Then we also have the notion of braided category, meromorphic braided category, etc. Sometimes we will simply call them tensor, meromorphic tensor, etc.

3 Examples

We recall here main examples.

Example 1

Let G be a complex analytic group (the condition on G can be weaker of course). We can define the following operad of spaces. For T ∈ T (n) we put S_{T} = G^{n}, G^{0} = id. Then the morphisms from the Definition 1c) are identities. We also have an operadic composition γ : G^{n} × G^{k_{1}} × ... × G^{k_{n}} → G^{k_{1}+...+k_{n}} such that γ((g_{i}) × ∏_{i}(g_{ij})) = (g_{i}g_{ij}).

Let C be a category equipped with the action of G on objects: an object M is transformed by g ∈ G into the object called M(g).

Then we say that C is a meromorphic monoidal G-category if for any T ∈ T (n) we are given a functor ⊗_{T} : C^{n} → C such that the families Hom_{C}(⊗_{T} X_{i}(g_{i}), Y) define a representable meromorphic pseudo-monoidal structure on C such that all morphisms from Definition 1c) are meromorphic isomorphisms. Subsequently we have the notions of G-braided category or meromorphic G-braided category, etc.

Example 2

This is a specialization of the Example 1.

We take as G the group C^{∗} of non-zero complex numbers. We take as C the category of finite-dimensional U_{q}(g)- modules where U_{q}(g) is the Drinfeld-Jimbo quantized enveloping algebra of an affine Kac-Moody Lie algebra g. We call it quantum affine algebra. The well-known fact explained in [So] is that C is a meromorphic C^{∗}-tensor category. We will discuss this one and related category later in the text.

Example 3

The usual notions of monoidal and braided (=tensor) categories are special cases. They can be described as representable pseudo-monoidal or pseudobraided structures. One can take either the trivial operad of spaces with all spaces being just one point. Or one can take the operad of moduli of stable punctured complex curves with tangent vector attached to the last point. Taking a connected stratum of the real points we get monoidal categories. In complex case we note that the fundamental groups of the punctured curves are pure braid groups. Thus we arrive to the description of tensor categories in terms of local systems on punctured curves due to Deligne ([De]). See [So] for details.

4 Quantum affine algebras

4.1

Let X be a complex manifold, A_{X} be a bundle of Hopf algebras on X which is equipped with a flat connection ∇. The latter means that ∇ has zero curvature and equalities ∇(ab) = ∇(a)b+a∇(b) and (∇⊗1+1⊗∇)(Δ(a)) = Δ(∇(a)) hold locally (here Δ is the comultiplication morphism for A_{X}).

We define a category C_{X} as a category of holomorphic vector bundles V on X of finite rank, equipped with a flat connection ∇_{V} which are (A_{X}, ∇)- modules. This means that V is a locally free sheaf of A_{X}-modules and the equality ∇_{V} (av) = ∇_{X}(a)v + a∇_{V} (v) holds locally for all sections a of A_{X} and v of V. Morphisms in C_{X} are morphisms of vector bundles compatible with the structures.

It is clear that naturally defined kernel and cokernel of a morphism (M, ∇_{M}) → (N, ∇_{N}) belong to C_{X}. The tensor product is defined fiberwise, and the unit object is defined as a trivial line bundle over X equipped with the trivial connection. This implies the following lemma.

Lemma 1 The category C_{X} is an abelian monoidal category.

Let (M_{i}, ∇_{i}) be a sequence of objects of C_{X}, 1 ≤ i ≤ n. We can make the tensor product M = ⊠_{i=1}^{i=n}M_{i} which is a holomorphic vector bundle on X^{n} equipped with the flat connection induced from tensor factors. The fiber over (x_{1}, ..., x_{n}) carries a structure of ⊗_{i}A_{X,x_{i}}-module. Let Z_{n} denotes infinitesimal neigbourhood of the diagonal {x_{1} = x_{2} = ... = x_{n}} with the union of all diagonals {x_{i} = x_{j}} being removed. Then we have functors j_{n∗} and j_{n}^{∗} in the category of D-modules on X^{n}, where j_{n} = j_{Z_{n}} is the canonical embedding of Z_{n} into X^{n}.

Let us consider j_{n∗}j_{n}^{∗}(M). Using the flat connection on M we can identify the fiber M_{x_{1},...,x_{n}} with the fiber M_{x_{1},...,x_{1}}. The latter carries the natural structure of an A_{X,x_{1}}-module.

Let us assume that for every such a fiber and every permutation σ ∈ S_{n} we are given an isomorphism of A_{X,x_{1}}- modules c_{σ} : j_{n∗}j_{n}^{∗}(⊠_{i}M_{i}) → j_{n∗}j_{n}^{∗}(⊠_{i}M_{σ(i)}) such that:

c_{1} = id, c_{στ} = c_{σ}c_{τ}.

Definition 3 We say that we are given an infinitesimal chiral braiding on C_{X} if

a) the above-mentioned isomorphisms are functorial with respect to M_{i};

b) they are compatible with the natural embeddings X^{n} → X^{m}, n ≤ m in the sense that if σ ∈ S_{m} permutes a subsequence of {1, ..., m} consisting of n elements then c_{σ}, σ ∈ S_{m} acts as cσ̄ ⊗id where σ̄ is the corresponding element of S_{n}.

If the above mentioned structure exists globally on the complement to each main diagonal {x_{1} = x_{2} = ... = x_{n}} then we say that C_{X} carries a chiral braiding.

Remark 1 This definition can be restated in such a way that it admits generalization to the case of families of objects of a braided category which are parametrized by X and equipped with automorphisms of infinitesimally close fibers. Then to any planar tree with tails numbered from 1 to n, and to a sequence of families M_{i}, 1 ≤ i ≤ n on X, one assigns a tensor product M_{T} = ⊠_{T} M_{i}. Although it actually depends on n, not on T, one needs to use trees to identify the fiber (M_{T})_{x_{1},...,x_{n}} with the tensor product ⊗_{T} M_{i,x_{1}} in the infinitesimal neighbourhood of the diagonal. Again, we get a family on X. We can now define chiral braiding as before (it is also similar to the definition of meromorphic braiding from 2.4 ). Notice that in this case we need additional compatibilies between trees with the same number of tails (chiral associativity) as well as compatibilities with the gluing operation on trees. We leave these details to the reader.

If C_{X} carries a chiral braided structure, then one can define meromorphic braided structure on global sections of the objects from C_{X}.

4.2

Let U be the quantized enveloping algebra of the affine Kac-Moody algebra { \atop sl(2)} corresponding to the parameter q such that |q| < 1. We denote by \overline{U} the quantized subalgebra U_{q}(sl(2)). We denote by A the category of finitedimensional U-modules of type 1 (see for example [ChP]). It follows from the results of [KS] that A is a meromorphic braided category (see discussion in [So] ).

To fix the notation, we recall that U is a Hopf algebra generated by X_{i}^{±}, K_{i}, K_{i}^{−1}, i = 0, 1 subject to relations

K_{i}X_{j}^{±}K_{i}^{−1} = q^{±a_{ij}}X_{j},

where (a_{ij}) is the Cartan matrix of { \atop sl(2)},

K_{i}K_{i}^{−1} = 1, K_{i}K_{j} = K_{j}K_{i},
[X_{i}^{+}, X_{i}^{−}] = (K − K^{−1})/(q − q^{−1}),

as well as quantized Serre relations for X_{i}^{+} and X_{i}^{−}, i = 0, 1. The coproduct Δ : U → U ⊗ U is defined by

Δ(K) = K ⊗ K, Δ(X_{i}^{±}) = X_{i}^{±} ⊗ K_{i} + K_{i}^{−1} ⊗ X_{i}^{±}, i = 0, 1.

Therefore the antipode is defined on generators by S(K) = K^{−1}, S(X_{i}^{±}) = −q^{±2}X_{i}^{±}, i = 0, 1.

The group C^{∗} acts on U via ϕ_{z}(X_{i}^{±}) = z^{±1}X_{i}^{±}, ϕ_{z}(K_{i}) = K_{i}, z ∈ C^{∗}.

4.3

One of our goals will be to construct a subcategory C of A and the extend it to a category C_{X} of the previous subsection, taking X to be the elliptic curve E = C^{∗}/q^{2Z}.

One way to do it is to construct a category C with the following properties:

1. C is a full monoidal rigid subcategory of A;

2. C is closed with respect to taking submodules and factor modules;

3. If V, W ∈ C then meromorphic brading in A gives rise to an isomorphism V (x) ⊗ W (y) → W (y) ⊗ V (x) as long as x/y does not belong to the set q^{2Z}.

Let C be a monoidal subcategory of A which satisfies Property 2 only.

Lemma 2 Property 3 holds for any two objects V, W ∈ C as long as it holds for simple V and W.

P roof. One case use induction by n = dimV · dimW. The Lemma holds for n = 1. Suppose that it holds for all k < n. Let us take V and W such that dimV ·dimW = n. If both objects are simple we are done. Suppose that V is not simple. Then there exists a non-trivial simple submodule V_{1} ∈ V. Then we have an exact sequence

0 → V_{1}(x) → V (x) → V (x)/V_{1}(x) → 0.

We can tensor it with W (y) from the left and from the right. Then using induction assumption, functoriality of meromorphic braiding and five-lemma we get the result. Q.E.D.

Suppose that we are given a category C which satisfies the properties 1-3 above. Every object V ∈ C gives rise to a trivial vector bundle V_{C^{∗}} on C^{∗} with the trivial connection.

We can assign to U a trivial bundle U_{C^{∗}} of Hopf algebras on C^{∗} equipped with the connection defined by the action ϕ_{z} of C^{∗}. Actually U_{C^{∗}} is an equivariant bundle. Then V_{C^{∗}} is a bundle of U_{C^{∗}}-modules such that the fiber U_{C^{∗},z} = U acts on the fiber V_{C^{∗},z} = V via automorphism ϕ_{z}.

To descent these data to E we need to define automorphisms between fibers at z and zq^{2} compatible with module structures. We define them to be id_{V} for V_{C^{∗}} (all fibers are canonically identified with V). We define an isomorphism γ_{z} : U = U_{C^{∗},z} → U_{C^{∗},zq^{2}} = U as γ_{z}(a) = ϕ_{q^{−2}}(a), a ∈ U. Since γ_{z}(ϕ_{z}(a)v) = ϕ_{zq^{−2}}(a)v = ϕ_{zq^{2}}(γ_{z}(a))v for any v ∈ V, a ∈ U we see that indeed all the structures are compatible and we obtain a bundle U_{E} of Hopf algebras on E equipped with a connection (in fact a structure of equivariant sheaf) as well as a bundle V_{E} of U_{E}-modules.

Let V_{E} and W_{E} be two U_{E}-modules. Then V_{E,x} ⊠ W_{E,y} carries a structure of U_{E,x} ⊗ U_{E,y}-module. For any x and y there is an isomorphism of the fiber U_{E,y} and the fiber U_{E,x} given by ϕ_{xy^{−1}} (to be more precise this is the formula on C^{∗} but it is compatible with all the structures so it descents to the elliptic curve). This makes V_{E,x} ⊠ W_{E,y} into U_{E,x} ⊗ U_{E,x}– module and via coproduct Δ into U_{E,x}-module.

If (x, y) ∈ E × E \ {diag} then the meromorphic braiding in A descents to an isomorphism of U_{E,x}– modules c_{x,y} : V_{E,x} ⊠ W_{E,y} → W_{E,y} ⊠ V_{E,x}.

Therefore we get a chiral braiding c_{V,W} : j_{∗}j^{∗}(V_{E} ⊠ W_{E}) ≃ j_{∗}j^{∗}(W_{E} ⊠ V_{E}). Here j is the embedding of the complement of the diagonal to E × E. It is easy to check that in this way we have obtained a chiral braided category (associativity constraint is trivial). Taking sections of the bundles we get a meromorphic braided category.

4.4

We are going to construct a subcategory C which satisfies Properties 1-3 .

To do this we recall that for any non-negative integer n and for any non-zero complex number a one has a simple object V_{n}(a) of A. It is constructed as evaluation representation of U corresponding to the point a and finite-dimensional simple (n + 1)-dimensional U_{q}(sl(2))-module V_{n}. We call ⊗_{i}V_{n_{i}}(a_{i}) a standard module corresponding to (a_{1}, a_{2}, ...). It is known (see [ChP]) that any simple object of A is a standard one with a_{1}, a_{2}, ... satisfy certain properties. Namely to every V_{n_{i}}(a_{i}) one assigns a finite set S_{n_{i}}(a_{i}) called q − string. It is a subset of a_{i}q^{2Z}. The condition mentioned above says that any two q-strings S_{n_{i}}(a_{i}) and S_{n_{j}}(a_{j}) are in a generic position (see [ChP] for precise definitions). If all a_{i}/a_{j} do not belong to q^{2Z} then the above-mentioned strings are in generic position.

We define C as a full rigid monoidal subcategory of A which is generated by V_{n}(q^{m}), m ∈ 2Z and closed under taking submodules and quotients. Clearly the trivial module 1 belongs to C. Since V_{n}(q^{m})^{∗} ≃ V_{n}(q^{2+m}) our category is rigid monoidal. Therefore it satisfies the Properties 1 and 2.

Theorem 1 Property 3 holds for the category C.

P roof. We will split the proof into several steps.

Step 1. Let V = ⊗_{i+1}^{i=n}V_{k_{i}}(q^{l_{i}}) and W = ⊗_{j=1}^{j=m}V_{r_{j}}(q^{s_{j}}). If x/y does not belong to q^{2Z} then two strings S_{k_{i}}(xq^{l_{i}}) and S_{r_{j}}(yq^{s_{j}}) are in generic position for any i and j. Consider the tensor product V (x) ⊗ W (y) = ⊗_{i}V_{k_{i}}(xq^{l_{i}}) ⊗ ⊗_{j}V_{r_{j}}(yq^{s_{j}}). We can use meromorphic bradings to interchange every module from the first group of tensor factors with every module from the second group of tensor factors. It is easy to deduce from [ChP], sections 4 , 5 and [KS], section 4, that meromorphic braidings do not have singularities and hence we obtain an isomorphism of U-modules V (x) ⊗ W (y) → W (y) ⊗ V (x).

Step 2. Let V and W be as on the Step 1. If M and N are either both submodules or factor modules of V and W then the Property 3 holds for M and N. This is clear from the Step 1 and the following functoriality of meromorphic braiding proven in [KS]: if meromorphic braiding c_{A(x),B(y)} :

A(x) ⊗ B(Y) → B(y) ⊗ A(x) is well-defined and f : A → A′, g : B → B′ are morphisms of U-modules then (f ⊗g)(c_{A(x),B(y)}) is a well-defined isomorphism A′(x) ⊗ B′(y) → B′(y) ⊗ A′(x).

Similarly one can prove the Property 3 in case if M is a submodule of V and N is a factor module of W. Using functoriality of meromorphic braiding once again, we can prove the Property 3 for any object of C. Q.E.D.

4.5

It is natural to ask for a description of simple objects of C. We start with the following elementary result.

Theorem 2 Any simple submodule or factor module of V = ⊗_{i=1}^{i=n}V_{k_{i}}(q^{l_{i}}) is of the form M = ⊗_{i=1}^{i=n}V_{r_{i}}(q^{s_{i}}).

P roof. Let us use induction by k = ∑_{i} k_{i}. For k = 0 the result is obvious. Suppose it holds for all k < m. Let us prove it for k = m. We prove it for submodules only. The case of factor modules easily follows if we take duals.

If V is simple we have nothing to prove. Let M ⊂ V be a non-trivial submodule. Since V is not simple there are two q-strings S_{k_{i}}(q^{l_{i}}) and S_{k_{j}}(q^{l_{j}}) which are not in generic position. We may assume that i = 1, j = 2 (otherwise use the bradings to get two consequitive strings not in generic position). Then according to [ChP], Section 4.9, there is a unique simple submodule of X = V_{k_{1}}(q^{l_{1}}) ⊗ V_{k_{2}}(q^{l_{2}}) of the type A = V_{d}(q^{e}) ⊗ V_{f}(q^{h}) with the factor module B of similar type. Then we have an exact sequence

0 → A ⊗_{i≥3} V_{k_{i}}(q^{l_{i}}) → V → B ⊗_{i≥3} V_{k_{i}}(q^{l_{i}}) → 0

If simple module M intersects A⊗_{i≥3} V_{k_{i}}(q^{l_{i}}) then it belongs to it and the result follows from the induction assumption. If M does not intersect this submodule then it is projected isomorphically to B ⊗_{i≥3} V_{k_{i}}(q^{l_{i}}) and again the result follows by induction. Q.E.D.

On ther other hand one can try to use the notion of q-character introduced in [FR] in order to describe the Grothendieck ring of the category C. For example it is natural to expect an affirmative answer to the following

Question Let χ_{q} be the q-character (notation from [FR]). Is it true that the Grothendieck ring K_{0} of our category C is isomorphic to the subring of the Grothendieck ring of U generated by t_{q^{n}}, where t_{q^{n}} is the class of V_{1}(q^{n}), n ∈ 2Z?

Clearly K_{0}(C) contain the subring Z[t_{q^{n}}], n ∈ 2Z. Thus the question is whether the q-character of an object from C belongs to this ring.

As was pointed to me by Ed Frenkel (private communication) the answer to the Question is positive, and it can be generalized to the higher rank case. We sketch his arguments below in the case of quantum affine algebra of sl(n). We denote by Γ_{n} the set 2Z if n = 2 and the set Z if n > 2.

Let us consider the monoidal category C_{n} generated by evaluation representations V_{ω_{i}}(q^{l}) of U_{q}({ \atop sl(n)}) where l ∈ Γ_{n} and ω_{i} is the fundamental weight of U_{q}(sl(n)) 1 ≤ i ≤ n − 1. Then the q-character of the tensor product of such representations belongs to A_{n} = Z[t_{i,q^{l}}] where t_{i,q^{l}} is the class of V_{ω_{i}}(q^{l}) in the representation ring of U_{q}({ \atop sl(n)}), and l ∈ Γ_{n}. Thus for n = 2 we have C_{2} = C and A := A_{2} is expected to be isomorphic to K_{0}(C).

Theorem 3 A simple object in C_{n} is isomorphic to a subquotient of a tensor product ⊗_{i}V_{ω_{i}}(q^{l_{i}}), l_{i} ∈ Γ_{n}.

In the case n = 2 it is isomorphic to a tensor product ⊗_{i}V_{n_{i}}(q^{l_{i}}) where V_{n_{i}}(q^{l_{i}}) is a standard module, l_{i} ∈ Z.

P roof. According to Chari and Pressley a simple U_{q}({ \atop sl(n)})-module is isomorphic to a subquotient of ⊗_{i}V_{ω_{i}}(a_{i}) (we have an isomorphism to a tensor product of standard modules for n = 2). Then one uses the fact that the q-character of the simple module contains the dominant term (terminology and notation from [FR], Section 4) equal to ∏_{i} Y_{i,a_{i}}, and is the sum with positive coefficients of monomilas in Y_{i,a_{i}q^{n_{i}}}^{±1}. This implies that all a_{i} ∈ q^{Γ_{n}}. Q.E.D.

Let V = ⊗_{i}V_{ω_{i}}(q^{l_{i}}).

Theorem 4 The q-character of every subquotient of V belongs to A_{n}.

P roof. Let L_{j}, 1 ≤ j ≤ m be the set of simple objects which appears in the composition series of V. Then χ_{q}(V) = ∑_{j} χ_{q}(L_{j}).

Every L_{j} is the highest weight module over U_{q}({ \atop sl(n)}). The q-character of such a module was computed in [FR]. It is equal to a sum of monomials in t_{i,a} with positive coefficients. It follows from the previous theorem that if t_{i,a} appears in such monomials we have a ∈ q^{Γ_{n}}. Hence χ_{q}(L_{j}) ∈ A_{n} for every j. This implies the theorem. Q.E.D.

Corollary 1 The Grothendieck ring K_{0}(C_{n}) is isomorphic to A_{n}.

P roof. It is easy to see that K_{0}(C_{n}) is generated by the isomorphism classes of subquotients of tensor products of the type V = ⊗_{i}V_{ω_{i}}(q^{l_{i}}) and then apply the previous theorem. Q.E.D.

There is little doubt about positive answer to the Question for an arbitary quantum affine algebra. One can try to use this fact in order to construct the corresponding meromorphic braided category on an elliptic curve.

References

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[GK] V. Ginzburg, M. Kapranov, Koszul duality for operads, Duke math. J.,76:1(1994), 203-272.

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[KS] D. Kazhdan, Y. Soibelman, Representations of quantum affine algebras, Selecta Math., New Series, 1:3(1995), 537-595.

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[So] Y. Soibelman, Meromorphic tensor categories, q-alg 9709030.