TENSOR PRODUCT IN CATEGORY O κ.

Transcription

1 TENSOR PRODUCT IN CATEGORY O κ. GIORGIA FORTUNA Let V 1,..., V n be ĝ κ -modules. Today we will construct a new object V 1 V n in O κ that plays the role of the usual tensor product. Unfortunately in fact V 1 V n is not in O κ (has already the wrong central charge nκ and moreover does not have reasonable finiteness properties. Before starting we will recall the main objects we are going to use. 1. Recap of definitions If S = {x 0,..., x n } is a finite set, we denote by ĝ κ S the Lie algebra fitting into the exact sequence 0 C ĝ κ S n i=0g((t i 0, corresponding to the quotient of ĝ κ ĝ }{{ κ by Ker(C } n+1 C. Note (n+1 times that for any collection of ĝ κ -modules V 1,..., V n, the tensor product V 1 V n carries a natural ĝ S κ -action (with central charge κ coming from the natural action of (ĝ κ S on it (in other words this action factors through the kernel of the map from C n+1 to C. Let now C be a connected projective curve isomorphic to P 1, and take a collection of n + 1 points S = {x 0,..., x n } on it. Assume that for every point x i in S, we are given charts γ i : P 1 C, such that γ i (0 = x i. If Γ(C {x 0,..., x n } is the algebra of regular functions on C {x 0,..., x n }, for any f Γ(C {x 0,..., x n } we denote by x i f the expansion at 0 of the rational function f γ i. From now on we will however focus on the case x 0 =, i.e. S = {, x 1,..., x n }. We will denote by ĝ κ,out the Lie algebra ĝ κ,out := Γ(P 1 {, x 1,..., x n } g. Remark 1. For f g ĝ κ,out, g g, the series Taylor expansion at each point in S gives a well defined map ĝ κ,out n i=0g((t i, Date: October 4,

2 2 GIORGIA FORTUNA moreover, because of the residue theorem, the above map lifts to a maps 0 C ĝ κ S n i=0 g((t i 0 ĝ κ,out hence the ĝ κ S -module structure on V 1 V n we introduced before, naturally gives a ĝ κ,out -action on this tensor product. 2. Existence of the tensor product Let S be the collection of points {, x 1,..., x n } and let V 1,..., V n be ĝ κ -modules. We will now construct the tensor product V 1 V n of these modules. One main property that we would like V 1 V n to satisfy is the following: < X, V 1 V n >= (X V 1 V n ĝκ,out, where the action of ĝ κ,out on the tensor product is the one coming from the ĝ κ S -action that we discussed before. By dualizing and rephrasing the expression above (using the fact that Hom Oκ (V, D(V =< V, V > this is equivalent to requiring Hom Oκ (X, D(V 1 V n = ((X V 1 V n ĝκ,out, hence to find such an object, it is enough to show that the functor is representable. F V1,...,V n : O op κ Vect X ((X V 1 V n ĝκ,out, Vect defined above is repre- Proposition 1. The functor F V1,...,V n : O op κ sentable by an element O κ. Proof. First of all we will show that F is left exact. Lemma 1. The functor defined above preserves finite limits (i.e. exact. is left Proof. First of all since finite limits in O op κ are finite colimits in O κ what we want to show is that if X = lim X I i, for a finite I, then ((lim X I i V 1 V n ĝκ,out = lim((x I i V 1 V n ĝκ,out. However, since right (resp. left exact functors commutes with finite colimits (resp. limits, and since tensor product and taking invariants are right exact,

3 we immediately get the equality: ( (lim I (lim I TENSOR PRODUCT IN CATEGORY O κ. 3 X i V 1 V n = lim(x I i V 1 V n (X i V 1 V n ĝκ,out = lim ((X I i V 1 V n ĝκ,out ( = lim (Xi V 1 V n ĝκ,out. I lim ((X i V 1 V n ĝκ,out I By a general theorem in category theory (see [B] and by Proposition 1, we know that F V1,...,V n is the inductive limit of representable functors, in other words it has the form: (1 F V1,...,V n ( = lim Hom I Oκ (, i, where i O κ. Moreover, since we are working in category O κ, we can rewrite (1 as In fact the following is true: Hom KLκ (, lim I i. Lemma 2. The category of KL κ -modules is the ind-completion of the category O κ. In other words (1 Objects in O κ are compact in KL κ. (2 Every object of KL κ is a filtred colimit of objects of O κ. Proof. Recall that we defined the category of KL κ -modules as the category of those ĝ κ -modules V such that: (1 V is smooth. (2 The action of tg[[t]] is locally nilpotent. (3 The action of g is locally finite. This is an abelian category and contains the abelian category O κ consisting of those modules V that are finitely generated. For part (1, if X is in O κ and lim A I i is a filtered colimit of objects A i KL κ, then the natural map lim Hom Oκ (X, A i Homĝκ mod(x, lim A I i I is an isomorphism. This follows from a general fact in category theory that asserts that a module is finitely presented if and only if Hom(X, commutes with filtered colimits. Since objects in O κ are finitely generated, and the category is noetherian, in particular they are finitely presented. Now if V

4 4 GIORGIA FORTUNA is a KL κ module, to prove part (2, we only have to show that V can be written as V = i V i, V i O κ. However every element v V we can simply consider the submodule W generated by this vector. By definition W is finitely generated, hence is in O κ. From the previous Lemma we know that F V1,...,V n is hence represented by a KL κ -module := lim I i. Now we want to show that is in O κ. To do this it is sufficient to show that Since (1 = ( tg[[t]] is finite dimensional. (1 = λ Λ + Homĝκ mod(v κ λ, M V1,...,Vn, (where Vλ κ denotes the Weyl module it is enough to show that only for finitely many λ s, Homĝκ mod(vλ κ, is non zero and moreover is finite dimensional. However, by definition we have Homĝκ mod(vλ κ, = ((Vλ κ V 1 V n ĝκ,out. Hence we want to show that ((Vλ κ V 1 V n ĝκ,out is finite dimensional. Because of the right exactness of tensor product and coinvariants, it λ Λ + is enough to show it in the case V i = Ni κ, where N i κ = Indĝκ g[[t]]+c N i, with dim(n i < ( every v O κ admits a surjection from such a module. In this case we have: (2 ((V κ λ N κ 2 N κ n ĝκ,out ((V λ N 2 N n g, and the last is by definition Hom g (V λ N 2 N n which is finite dimensional. Moreover, since only finitely many λ s can appear in the composition series of (N 2 N n we have that the sum over all dominant λ s is indeed a finite sum and we are done. Note that in the proof above we have used the following fact (proved in Sasha s notes: Proposition 2. For N κ i = Indĝκ g[[t]]+c N i, i = 1,..., n we have (N κ 1 N κ 2 N κ n ĝκ,out (N 1 N 2 N n g.

5 TENSOR PRODUCT IN CATEGORY O κ explicit construction of the tensor product Unfortunately what we have constructed is not quite the object V 1 V n that we wanted, but it is its dual. Moreover we don t have an explicit description of that allows us to take the dual. Hence we will now recover from the representability property. Proposition 3. The tensor product V 1 V n is equal to ( V 1 V n := D( = lim m (V 1 V n # G(m where G(m is the right ideal generated by those elements in U(ĝ κ,out of the form (f 1 c 1 (f m c m, with f i functions with zeros of order at least 1 Proof. To prove the proposition, we will first describe the module using the functor it represents. It would be nice to say that = Hom Oκ (U(ĝ κ, but unfortunately U(ĝ κ is not in our category. However we can take the pro-object lim U(ĝ κ /U(ĝ κ (tg i (tg m + U(ĝ κ J, where J runs over all left ideals in U(g of finite codimension. int, In fact Û(ĝ κ = lim U(ĝ κ and maps from Û(ĝ κ to a module V KL κ are by definition the same as maps from U(ĝ κ to it. Moreover if we denote by U(ĝ κ the term U(ĝ κ := U(ĝ κ /U(ĝ κ (tg i (tg m + U(ĝ κ J, then U(ĝ κ is in our category. Now we can see how looks like: = Homĝκ mod( lim U(ĝ κ, = lim Hom Oκ (U(ĝ κ, = lim((u(ĝ κ V 1 V n ĝκ,out = lim((v 1 V n G(m+J, where G(m is the right ideal generated by those elements in U(ĝ κ,out of the form (f 1 c 1 (f m c m, with f i functions with poles of order at least 1, and where J runs over the right ideals generated by the ideals of finite codimension in U(g. As we said before the module corresponds to the dual of the sought for V 1 V n. Hence to recover the definition of V 1 V n we need to compute D(. Recall that D(V := (V int. When we take

6 6 GIORGIA FORTUNA the dual we have: ( = Hom(lim((V 1 V n G(m+J, C = = lim((v 1 V n G(m+J. Lemma 3. The vector space (V 1 V n G(m+J is finite dimensional. Proof. This follows from the fact that is in O κ, hence in particular (m is finite dimensional. In fact we have the following injection into (m (m Hom(U(ĝ κ, = ((V 2 V n G(m+J hence ((V 1 V n G(m+J is finite dimensional too, which implies the lemma. Because of Lemma 3 we have ( = lim((v 1 V n G(m+J lim(v 1 V n G(m+J. Note that elements in U(g stabilize the ideal G(m, hence we can consider (V 1 V n G(m as a g-module and compute coinvariants in two steps: lim m+j (V 1 V n G(m+J = lim m lim J ((V 1 V n G(m J. The last term is equal to lim m (V 1 V n G(m since by the proposition below, also (V 1 V n G(m happens to be finite dimensional. Hence we have ( = lim m (V 1 V n G(m. applying and taking integrable vectors, now gives us ( V 1 V n = D( = lim(v 1 V n # G(m m as we wanted. Note that this definition coincides with the one that is given in [KL] Section 4. Proposition 4. For every V 2,..., V n in O κ the vector space is finite dimensional. (V 1 V n G(m Proof. Since the object (V 1 V n G(m is acted on by g integrably, it s enough to show that for every λ, the space of coinvariants ((V 1 V n G(m V λ g int,

7 TENSOR PRODUCT IN CATEGORY O κ. 7 are finite-dimensional, and non-zero for only finitely many λ s. In fact this would imply that ((V 1 V n G(m Hom g ((V 1 V n G(m, V λ Vλ λ is finite dimensional. Let J λ U(g be such that U(g/J λ = V λ. Consider the corresponding object U(ĝ κ λ. Then However ((V 1 V n G(m V λ g (U(ĝ κ λ V 1 V n ĝκ,out. ((U(ĝ κ λ V 1 V n ĝκ,out = Hom Oκ (U(ĝ κ λ, = = Hom(V λ, M(m, which is finite-dimensional and non-zero for only finitely many λ s since is in O κ (hence (m is finite dimensional. References [B] F. Borceux, Handbook of Categorical Algebra: Basic category theory. [KL] D. Kazhdan, G. Lusztig, Tensor Structures Arising from Affine Lie Algebras. I, JAMS, Vol 6, Num. 4, October 1993.