NOTES ON FACTORIZABLE SHEAVES. This is a preliminary version. Imprecisions are likely.

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1 NOTES ON FACTORIZABLE SHEAVES This is a preliminary version. Imprecisions are likely. 1. From Hopf algebras to factorizable sheaves 1.1. Configuration spaces. Let Λ be a lattice and Λ neg Λ a sub-semigroup, isomorphic to (Z 0 ) k. Let X be an algebraic curve (smooth, but not necessarily complete). For λ Λ neg, we will denote by X λ the algebraic variety assifying Λ-valued divisors D := Σ λ i x i with x i x j and λ i Λ neg. By definition, X 0 = pt. For a marked point x 0 X and an arbitrary λ Λ, let Xx λ 0 be the ind-scheme assifying Λ-valued divisors Σ λ i x i as above but with a weaker condition, namely, that λ i Λ neg for x i x 0. Let us denote by add λ1,λ 2 either of the maps X λ1 X λ2 X λ1+λ2 and X λ1 X λ2 x 0 X λ1+λ2 x 0. The map add λ1,λ 2 is finite. For a perverse sheaves F 1 on X λ1 and F 2 on X λ2 or X λ2 x 0, we shall denote by F 1 F 2 the perverse sheaf on X λ1+λ2 (or X λ1+λ2 x 0 ). (add λ1,λ 2 )! (F 1 F 2 ) (add λ1,λ 2 ) (F 1 F 2 ) Let (X λ1 X λ2 ) disj X λ1 X λ2 and (X λ1 Xx λ2 0 ) disj X λ1 Xx λ2 0 the open subschemes, corresponding to pairs of divisors (D 1, D 2 ) with the condition that the support of D 1 does not intersect the support of D 2 in the former case, and is also disjoint from {x 0 } in the latter case The construction. Let A be a Λ neg -graded Hopf algebra. For the following construction we will work over the ground field C and we take X to be the affine line A 1. (For the construction to work for any X, we need that the antipode on A be involutive.) We will assume that A 0 C and that its graded components are finite-dimensional. Theorem-Construction 1.3. (1) To A one can canonically attach a system Ω A of perverse sheaves Ω λ A on Xλ endowed with factorization isomorphisms (1.1) add λ 1,λ 2 (Ω λ1+λ2 A ) (X λ 1 X λ 2 )disj Ω λ1 A Ωλ2 A (X λ 1 X λ 2 ) disj. Moreover, the isomorphisms (1.1) are associative in a natural sense. (2) The *-stalk of Ω λ A at a point Σ λ i x i X λ is uasi-isomorphic to (Tor A(C, C)) λi, where the super-script refers to the corresponding graded component. i Date: February 26,

2 2 NOTES ON FACTORIZABLE SHEAVES 1.4. Verdier duality. For a perverse sheaf F on X λ let us view its Verdier dual D(F) as living over X λ (i.e., we change our semi-group to Λ pos := Λ pos ). For a Λ neg -graded Hopf algebra A as above, consider its graded linear dual A as a Λ pos - graded Hopf algebra. Proposition 1.5. There exists a natural isomorphism of perverse sheaves over X λ. D(Ω λ A) Ω λ A,oc, where the superscript oc denotes the Hopf algebra with reversed co-multiplication (and the old multiplication). The above isomorphisms for all λ Λ pos are compatible with the factorization isomorphisms (1.1). Note that D(D(Ω λ A )) Ω Aom,oc, where om denotes the Hopf algebra with reversed multiplication. However, A om,oc A, by means of the antipode. As a corollary, we obtain the following description of the!-stalk of Ω λ A at a point Σ λ i x i X λ. Namely, it is uasi-isomorphic to (Ext A (C,. C))λi i 1.6. Modules. We will now extend the construction of Sect. 1.2 to the case of modules. Let Dr(A) be the Drinfeld double of A. We will consider the category Dr(A)-mod of Λ-graded modules M over Dr(A), with finite-dimensional graded components and such that the set {λ M λ 0} is bounded from above (i.e., is contained in a set of the form λ + Λ neg for some λ Λ). Theorem-Construction 1.7. (1) To an object M Dr(A)-mod one can canonically associate system Ω A,M of perverse sheaves Ω λ A,M on Xλ x 0 for λ Λ endowed with following system of factorization isomorphisms with respect to Ω A : (1.2) add λ 1,λ 2 (Ω λ1+λ2 A,M ) (X λ 1 X λ 2 x 0 ) disj Ω λ1 A Ωλ2 A,M (X λ 1 X λ 2 x 0 ) disj. The isomorphisms (1.2) are associative in a natural sense with respect to (1.1). (2) The *-stalk of Ω λ A,M at a point Σ λ i x i X λ + λ 0 x 0 is uasi-isomorphic to (Tor A(C, C)) λi (Tor A(C, M)) λ0. i Note that linear duality M M defines a contravariant euivalence between Dr(A)-mod and Dr(A,oc )-mod, (reversing the braiding). Proposition 1.8. There is a natural isomorphism compatible with the isomorphisms (1.2). D(Ω λ A,M ) Ω λ A,oc,M, The proposition implies the following description of the!-stalks of Ω λ A,M. At a point Σ λ i x i X λ + λ 0 x 0, the!-stalk is isomorphic to (Ext A (C, (Ext C))λi A (C,. M))λ0 i

3 NOTES ON FACTORIZABLE SHEAVES Factorizable sheaves. Definition A factorizable sheaf with respect to Ω A is a system F of perverse sheaves F λ on Xx λ 0 euipped with an associative system of isomorphism (1.3) add λ 1,λ 2 (F λ1+λ2 ) (X λ λ 1 X 2 x 0 ) disj Ω λ1 A Fλ2 (X λ λ 1 X 2 x 0 ) disj, and such that the set {λ F λ 0} is bounded from above. Factorizable sheaves with respect to Ω A naturally form a category that we shall denote FS(Ω A ). Theorem The assignment M Ω A,M is an euivalence between the category Dr(A)-mod and FS(Ω A ) We begin with the following observation: 2. The commutative case Lemma 2.2. A Hopf agebra A is co-commutative (resp., commutative) if and only if the isomorphism (1.1) extends to a map respectively. add λ 1,λ 2 (Ω λ1+λ2 A ) Ω λ1 A Ωλ2 A or Ωλ1 A Ωλ2 A add! λ 1,λ 2 (Ω λ1+λ2 A ), Thus, for A which is co-commutative (resp., commutative) we have the maps (2.1) Ω λ1+λ2 A Ω λ1 A Ωλ2 A and Ωλ1 A Ωλ2 A Ωλ1+λ2 A, that we shall refer to as co-multiplication and multiplication, respectively The Bar-complex. Again, for A co-commutative (resp., commutative) we can form a complex of perverse sheaves on X λ s, denoted Bar(Ω A ) by taking A[±1]... A[±1], }{{} n 0 n with the appropriate Λ neg -grading and with ±1 = 1 in the co-commutative and ±1 = 1 in the commutative case. The system Bar(Ω A ) itself is endowed with factorization isomorphisms, and, moreover, an associative algebra (resp., co-associative co-algebra) structure with respect to. For a point x X, the direct sums ι λ x(bar(ω A )) and ι! λ x(bar(ω A )) λ Λ neg λ Λ neg thus acuire structures of a co-associative co-algebra and associative algebra, respectively (here ι λ x denotes the embedding of the corresponding point into X λ ). Lemma 2.4. We have canonical uasi-isomorphisms: λ Λ neg ι λ x(bar(ω A )) A, as associative algebras for A co-commutative, λ Λ neg ι! λ x(bar(ω A )) A, as co-associative co-algebras for A commutative.

4 4 NOTES ON FACTORIZABLE SHEAVES 2.5. Modules. Let A be co-commutative (resp., commutative). Then we can speak about systems F of perverse sheaves F λ on X λ (or X λ x 0 ) that are co-modules (resp., modules) with respect to Ω A and the operation. I.e., we have the maps F λ1+λ2 Ω λ1 A Fλ2 or Ω λ1 A Fλ2 F λ1+λ2, respectively, that are associative with respect to the maps (2.1). For a co-module (resp., module) F with respect to Ω A, the convolution (2.2) Bar(Ω A, F) := Bar(Ω A ) F acuires a natural differential, and as such is a module (resp., co-module) with respect to Bar(Ω A ). Moreover, the assignment (2.2) establishes an euivalence of suitably defined derived categories. The functor in the opposite direction sends a system of complexes F with an action (resp., co-action) of Bar(Ω A ) to with a canonical differential. Un-Bar(F ) := Ω A F 2.6. Modules vs. factorizable sheaves. Assume that A is co-commutative (resp., commutative). Definition 2.7. A factorizable sheaf F with respect to Ω A is said to be co-commutative (resp., commutative) if the factorization isomorphisms (1.3) extend to maps respectively. add λ 1,λ 2 (F λ1+λ2 ) Ω λ1 A Fλ2 and Ω λ1 A Fλ2 add! λ 1,λ 2 (F λ1+λ2 ), We note that in the derived category being co-commutative (resp., commutative) is not a property but an additional structure. Let F be a co-commutative (resp., commutative) factorizable sheaf with respect to Ω A. By adjunction, we obtain a co-associative co-action (resp., associative action) respectively. F Ω A F or Ω A F F, Thus, we obtain a functor from the category of co-commutative (resp., commutative) factorizable sheaves with respect to Ω A to that of Ω A -comodules (resp., modules). Proposition 2.8. (1) The above functor is a fully faithful embedding. (2) An Ω A -comodule (resp., module) F is a factorizable sheaf if and only if Bar(Ω A, F) is supported on λ x 0 Xx λ λ Λ 0. λ Λ

5 NOTES ON FACTORIZABLE SHEAVES Let us see how the notion of (co)commutative factorizable sheaf couples with Theorem- Construction 1.7. Let A be co-commutative (resp., commutative). Let A-comod (resp., A-mod) denote the category of graded A-comodules (resp. modules) N with finite-dimensional graded components and such that the set {λ N λ 0} is bounded from above. We have a natural functor from A-comod (resp., A-mod) to Dr(A)-mod, which makes A act (resp., co-act) via the augmentation. Proposition (1) For N A-comod (resp., N A-mod) the corresponding object N Dr(A)-mod has the property that Ω A,N is co-commutative (resp., commutative) with respect to Ω A. (2) The assignment N Ω A,N establishes an euivalence between the category of A-comodules (resp., A-modules) and that of co-commutative (resp., commutative) factorizable sheaves with respect to Ω A. (3) For N A-comod (resp., N A-mod), (Bar(Ω A, Ω A,N )) λ (ι λ x0 ) (N λ ). 3. The twisted case: preliminaries 3.1. Categories over a braided monoidal category. Let C be a braided monoidal category. We denote by S c1,c 2 the braiding isomorphism c 1 c 2 c 2 c 1. Let C denote the braided monoidal category, which euals C as a monoidal category, but with the reversed braiding := S 1 S c1,c 2 c 2,c 1. Let O be a monoidal category euipped with a central functor φ : C O, by which we mean a monoidal functor endowed with functorial isomorphisms S c,m : φ(c) M M φ(c) for c C and M O, compatible with tensor products and such that for c 1, c 2, S c1,φ(c 2) = φ(s c1,c 2 ). In this case we will say that O is a monoidal category over C. Note that the category O op, obtained from O by reversing the monoidal structure naturally receives a central functor from C The Drinfeld double. For (C, O, φ) as above, we can consider the category Z C (O) whose objects are V C, euipped with a system of isomorphisms R M,V : M V V M, M O, compatible with tensor products of M s and such that for M = φ(c), c C, R φ(c),v = S c,v. The tensor product in O euips Z C (O) with a monoidal structure. naturally braided by means of S V1,V 2 := R V1,V 2 : V 1 V 2 V 2 V 1. Moreover, Z C (O) is Note, however, that unless C is symmetric, we do not have any naturally defined functor C Z C (O). In other words, Z C (O) is not a category over C (unless C is symmetric). Note that we have a naturally defined monoidal functor Z C (O op ) (Z C (O)) op, which reverses the braiding, and makes the following diagram commute: Z C (O op ) (Z C (O)) op O op Id O op.

6 6 NOTES ON FACTORIZABLE SHEAVES 3.3. Hopf algebras. Assume that we also have a faithful monoidal functor F : O C such that F φ = Id C, as monoidal functors, and such that F (S c,m ) = S c,f (M). In this case, we will say that O is a monoidal category in C. Modulo representability, a datum of a monoidal category in C is euivalent to that of a Hopf algebra A in C. Note that O op is then naturally a category over C. The corresponding Hopf algebra may be thought of as A oc (which it literally is, if C is symmetic). As was mentioned above, the category Z C (O), although endowed with a monoidal functor to C, does not correspond to a Hopf algebra in C. However, it corresponds to an algebra in this category, denoted Dr(A). By definition, Dr(A)-mod := Z C (O). The forgetful functor Z C (O) O defines a homomorphism of algebras A Dr(A) The Koszul dual category. For (O, C, φ, F ) as above consider the category O = Funct O (C, C) of functors C C that commute in a natural sense with the action of O, the latter being M O, c C F (M) c. Composition makes O into a monoidal category. We have a natural monoidal functor where the last arrow comes from the braiding. F : O Funct C (C, C) C op C, In addition, since C maps to Z(O) by means of φ, we have a natural monoidal functor φ : C Z(O ). I.e., we obtain that O is naturally a monoidal category over C. The functor F makes O into a monoidal category in C. If O corresponds to a Hopf A in C, and assume that A is dualizable as an object of C. Then O corresponds to the Hopf algebra A,oc in C Relationship between Drinfeld doubles. Note that we have a natural functor Z C (O) Z C (O ) op, which reverses the braiding and makes the following diagram commute Z C (O) Z C (O ) C C, where the bottom horizontal line is the identity functor. In the case when O corresponds to a Hopf algebra A in C, we obtain an isomorphism of algebras Dr(A) Dr(A,oc ); in particular, Dr(A) receives a homomorphism from A,oc.

7 NOTES ON FACTORIZABLE SHEAVES 7 4. The twisted case: construction 4.1. Gerbes. Recall that on a topological space, a data of a C -gerbe allows to twist the notion of a sheaf. Given a C -gerbe P over Y, we will refer to the corresponding category as P-twisted sheaves on Y. If Y is stratified, we can talk about P-twisted constructible (or perverse, if we are given a perversity function) sheaves on Y. Let us recall the following constriction. Let L be a complex line bundle over Y, and C. We define the gerbe L log( ) over Y as follows: its sections (over an open subset of Y) is the category of 1-dimensional local systems on the corresponding cire bundle (over the given open subset) with monodromy along the fiber. The corresponding category of L log( ) -twisted sheaves identifies with that of sheaves on the cire bundle, which are monodromic with monodromy along the fiber Gerbes attaches to a pairing. Let be a symmetric pairing Λ Λ C. The following mimics [CHA], Sect Assume that for every λ Λ we are given a C -gerbe P λ X, over X plus the following data: for every λ 1, λ 2 an identification c λ1,λ2 : P λ1 X, Pλ2 X, Pλ1+λ2 X, ω 2log((λ1,λ2)) X, where ω X is the canonical line bundle on X. We need the data of c λ1,λ2 to be symmetric in a natural sense. For three elements λ 1, λ 2, λ 3 we need a data that identifies the three arising isomorphisms of gerbes, P λ1 X, Pλ2 X, Pλ3 X, Pλ1+λ2+λ3 X, ω 2log((λ1,λ2)) X ω 2log((λ2,λ3)) X ω 2log((λ3,λ1)) X, such that the corresponding identity holds for 4-tuples of λ s. Given a pairing, it is easy to see that the data of (P λ X,, ) exists (see below for a cλ1,λ2 construction). Having two pieces of data like this, their ratio is canonically a Ť -gerbe over X, where Ť is the torus C Λ. Z Given, we will refer to a data P := {(P λ X,, )} as above as a -twisted Ť -gerbe on X. cλ1,λ The canonical gerbe attached to a pairing. We define a -twisted Ť -gerbe on X by setting: P λ can,x, := (ω X ) log((λ,λ)), and c λ1,λ2 to be the natural identification (ω) log((λ1+λ2,λ1+λ2)) (ω X ) log((λ1,λ1)) (ω X ) log((λ2,λ2)) (ω X ) 2 log((λ1,λ2)). In what follows we will refer to this choice as a canonical -twisted Ť -gerbe on X, and denote it P,can Gerbes over configuration spaces. A datum of a -twisted Ť -gerbe on X gives rise to the following construction: For every n-tuple λ 1,..., λ n of elements of Λ we obtain a C -gerbe P λ1,...,λn X n, over X n, whose restriction to the diagonal φ : X m X n corresponding to a surjection φ : {1,..., n} {1,..., m} identifies with P µ1,...,µm X m,, where µ j = Σ λ i. i,φ(i)=j

8 8 NOTES ON FACTORIZABLE SHEAVES Namely, we set P λ1,...,λn X n, ( ) := P λ1 X,... Pλn X, O( i,j) 2log((λi,λj)). i j For n = n 1 + n 2 and {λ 1,..., λ n } = {λ 1 1,..., λ 1 n 1 } {λ 2 1,..., λ 2 n 2 } we have an identification of gerbes P λ1,...,λn X n, (X n 1 X n 2 )disj P λ1 1,...,λ1 n 1 X n 1, P λ2 1,...,λ2 n 2 X n 2, (X n 1 X n 2 )disj. In particular, for a semi-group Λ neg Λ, the above datum gives to a C -gerbe denoted P X λ, over X λ and a C -gerbe denoted P X λ x0, over Xx λ 0 endowed with factorization isomorphisms (4.1) P X λ 1 +λ 2, (X λ 1 X λ 2 )disj P X λ 1, P X λ 2, (X λ 1 X λ 2 )disj and (4.2) P X λ 1 +λ 2 x 0, (X λ 1 X λ 2 x 0 ) disj P X λ 1, P X λ 2 x 0, (X λ 1 X λ 2 x 0 ) disj, which are associative in the natural sense The braided category attached to. Let be as above. We consider a new braiding on the category of Λ-graded vector spaces. Namely, for two such vector spaces M 1 and M 2 and elements m i M i od degrees λ i, respectively, we set We denote this category by Vect Λ. m 1 m 2 (λ 1, λ 2 ) m 2 m 1. Let now A be a Hopf algebra in Vect Λ. Assume that as a Λ-graded vector space, A satisfies the same assumptions as in the non-twisted case. We take our curve X to be A 1 with a fixed coordinate. Let P be P,can as defined in Sect (Note that the choice of a coordinate trivializes the gerbes P λ,can,x for every λ, however, the gerbe P Xλ,,can over X λ is non-trivial 1.) The following generalizes Theorem-Construction 1.3 to the present context: Theorem-Construction 4.6. (1) To A one can associate a system Ω A of P X λ,-twisted perverse sheaves Ω λ on X λ, endowed with factorization isomorphisms as in (1.1), (the latter make sense in view of (4.1)), which are associative in the natural sense. (2) The *- and!-stalks of Ω A are calculated by the same formula as in the non-twisted case. (This makes sense, since according to our choice of the -twisted Ť -gerbe, its restriction to the strata of the diagonal stratification of X λ is trivial, so *- and!-fibers of a twisted sheaf are (non-twisted) vector spaces ). 1 The choice of the coordinate allows to trivialize each individual PX λ,,can as well, which is how it is done in [BFS], but this will not be compatible with the factorization isomorphisms.

9 NOTES ON FACTORIZABLE SHEAVES Factorizable sheaves in the twisted situation. Due to (4.2), the definition of factorizable sheaf given in Sect. 1.9 transfers to the twisted context, where the F λ s are now P X λ x0,-twisted perverse sheaves. Thus, for A as above, we have a well-defined category FS(Ω A ). Recall now the category (which is in fact braided monoidal) Dr(A)-mod. Theorem-Construction 1.7 and Theorem 1.11 we have: Generalizing Theorem-Construction 4.8. (1) To an object M Dr(A)-mod one can canonically associate an object Ω A,M in FS(Ω A ). Moreover, the assignment M Ω A,M is an euivalence of categories. (2) The *- and!-stalks of Ω A,M are calculated by the same formula as in the non-twisted case Verdier duality in the twisted case. Given a C -gerbe P over Y we have the Verdier duality functor that maps the (derived category of) P-twisted sheaves to that of P 1 -twisted sheaves. Recall that we can think of A,oc as a Hopf algebra in Vect Λ 1, and there is an isomorphism Dr(A)-mod Dr(A,oc )-mod, as monoidal categories that reverses the braiding. The rigidity functor on Dr(A)-mod can be thought of as a contravariant functor which underlies the dualization functor We have: Dr(A)-mod Dr(A,oc )-mod, M M : Vect Λ Vect Λ 1. Proposition D(Ω A ) Ω A,oc and D(Ω A,M ) Ω A,oc,M. 5. The uantum group 5.1. Lusztig s algebra. Let Λ be the weight lattice of a reductive group G. We denote by I the Dynkin diagram, and for i I by α i the corresponding root. Let Λ pos be the positive span of the α i s. Let be as above. Our primary interest is the category of modules over Lusztig s uantum group L U (with divided powers). We will denote by L U -mod the category of Λ-graded modules with weights bounded from above. This uantum R-matrix makes L U -mod into a braided monoidal category. Along with L U we have the Kac-De Concini version of the uantum group, denoted KD U, and the corresponding category of modules KD U -mod. We have a homomorphism of Hopf algebras φ : KD U L U, compatible with the the fiber functor to Vect. The morphism φ is known to be an isomorphism away from roots of unity, i.e., when (α i, α i ) is not a root of unity for any i I, and at = 1. We remark, however, that KD U -mod fails to be a braided category at roots of unity.

10 10 NOTES ON FACTORIZABLE SHEAVES 5.2. Positive and negative parts. Let U [B + ] be either Lusztig s are the Kac-De Concini sub-hopf algebra of U, corresponding to the uantum Borel. The projection on the torus and the forgetful functor to Vect Λ define monoidal functors Vect Λ U [B + ]-mod, and it is easy to see that U [B + ]-mod becomes a monoidal category in Vect Λ 1. We shall denote by L U + and KD U +, respectively, the resulting Hopf algebras in Vect Λ 1. As associative algebras, they coincide with the same-named sub-algebras of L U and KD U, respectively. Similarly, we consider the Hopf algebra U [B ], and the category U [B ]-mod with the corresponding functors Vect Λ U [B ]-mod. We change the monoidal structure on the above functors by multiplying it by (λ, µ) on the (λ, µ) graded component for the functor, and by its inverse for the functor. This makes U [B ]-mod into a monoidal category over Vect Λ. We shall denote by L U and KD U, respectively, the resulting Hopf algebras in Vect Λ. As associative algebras, they coincide with the same-named sub-algebras of L U and KD U, respectively. The Drinfeld doubles make sense as Hopf algebras in Vect. Dr( L U ), Dr( KD U ), Dr( L U + ) and Dr( KD U + ) The uantum R-martix on U has the property that the monoidal restriction functors L U -mod L U [B ]-mod KD U [B ] naturally lift to a braided monoidal functors (5.1) L U -mod Z Vect Λ ( L U [B ]-mod) and L U -mod Z Vect Λ ( KD U [B ]-mod) Moreover, the above functors are euivalences away from roots of unity. Similarly, when we consider L U -mod with the inverted braiding, we have the functors (5.2) L U -mod Z Vect Λ ( L U [B + ]-mod) and L U -mod Z Vect Λ ( KD U [B + ]-mod), which are euivalences away from roots of unity. We have the corresponding homomorphisms of Hopf algebras (5.3) Dr( L U ) L U, Dr( KD U ) L U, Dr( L U + ) L U and Dr( KD U + ) L U. The categories (5.4) Z Vect Λ ( L U [B ]-mod), Z Vect Λ ( KD U [B ]-mod), Z Vect Λ ( L U [B + ]-mod), Z Vect Λ ( KD U [B + ]-mod) will serve as approximations for L U -mod. In the seuel we will make an assumption that (α i, α i ) 2 1 for any i I, and will define categories that approximate L U -mod even better.

11 NOTES ON FACTORIZABLE SHEAVES The free algebras. We consider the braided monoidal category Vect Λ, and we let fr U be the free associative algebra in it, generated by the elements F i, i I, where deg(f i ) = α i. We define a Hopf algebra structure on it by F i F i F i. Similarly, we consider the braided monoidal category Vect Λ 1, and we let fr U + be the free associative algebra in it, generated by the elements E i, i I, where deg(e i ) = α i. We define a Hopf algebra structure on fr U + by E i E i E i. We let cofr U be ( fr U + ),oc, which is a Hopf algebra in Vect Λ, and we let cofr U + be ( fr U ),oc, which is a Hopf algebra in Vect Λ 1. From now on, we will assume that (α i, α i ) 2 1 for any i I. Then the formulas (F i, E i ) = define canonical maps of Hopf algebras 1 1 (α i,α i) 2 (5.5) fr U cofr U and fr U + cofr U Lusztig s and Kac-De Concini algebras. By definition, the Kac-De Concini version KD U of U is a uotient Hopf algebra of fr U given by the uantum Serre relations, and similarly for KD U +. Under the above assumption on, Lusztig s algebra L U is a Hopf sub-algebra of cofr U, and similarly for L U +, such that the isomorphisms cofr U ( fr U + ),oc and cofr U + ( fr U ),oc induce isomorphisms The maps L U ( KD U + ),oc and L U + ( KD U ),oc. KD U L U and KD U + L U +, induced by (5.5) coincide with those induced by the map φ : KD U L U. and i.e., In particular, we have: Z Vect Λ ( L U [B ]-mod) Z Vect Λ 1 (KD U [B + ]-mod) Z Vect Λ ( KD U [B ]-mod) Z Vect Λ 1 (L U [B + ]-mod), Dr( L U ) Dr( KD U + ) and Dr( L U + ) Dr( KD U ), reversing the braiding, and compatible with the homomorphisms (5.3) The small uantum group. Let u (resp., u + ) be the images of the maps respectively. These are Hopf algebras in Vect Λ By the above, Let U, + 2 L U and, U +, 2 φ : KD U L U and φ : KD U + L U +, u + (u ),oc. and Vect Λ 1, respectively. L U be the sub-hopf algebras, eual to the images of Dr( L U ) L U and Dr( L U + ) L U,

12 12 NOTES ON FACTORIZABLE SHEAVES respectively. They are generated as algebras by ( L U, u + ) and (u, L U + ). I.e., the corresponding categories of modules, denoted U, + 2 -mod and U, 2 -mod are monoidal, and braided since the subalgebras U, + 2 L U and, U +, 2 L U contain the R-matrix of L U. We have a commutative diagram of homomorphisms of Hopf algebras: Dr( L U ) Dr( D U + ) U, + 2 L U Dr(u ) Dr(u + ) U +, 2 The above diagram is compatible with the R-matrices of respectively. The intersection Dr( L U ), Dr(u ), Dr( KD U ) and L U, u := U, + 2 U +, 2 L U Dr( KD U ) Dr( L U + ). is the small uantum group. The above diagram shows that u identifies both with Dr(u ) (as a Hopf algebra with an R-matrix), and with the image of φ : KD U L U The uantum Frobenius. Let Λ Λ be the sub-lattice consisting of elements {λ (λ, µ) 2 = 1, µ Λ}. Note that the corresponding braided monoidal subcategory Vect Λ Vect Λ is symmetric. We denote the corresponding category simply by Vect Λ. For i I let l i be the minimal integer such that (α i, α i ) li = 1. Then, it is easy to see that α i, := l i α i Λ, and ˇα i, := ˇαi l i (Λ ). According to [Lu], (Λ, (Λ ), I, α i,, ˇα i, ) form a root datum. We shall denote by U the resulting uantum enveloping algebra. The braided monoidal category U -mod is in fact symmetric. Let G denote the reductive group, corresponding to the above root datum. By loc. cit., the category U -mod is euivalent, as a braided monoidal category, to O the category O corresponding to G. In addition, we have a homomorphism of Hopf algebras (the uantum Frobenius): Frob : L U U, and the corresponding braided monoidal functor The composition factors as Frob : U -mod L U -mod. U -mod L U -mod u -mod U -mod Vect Λ u -mod, where elements of Vect Λ are considered as 1-dimensional representations of u -mod with the E i s and F i s acting trivially.

13 NOTES ON FACTORIZABLE SHEAVES 13 Moreover, we have an exact seuence of Hopf algebras: (5.6) 1 u L U U 1 (see, [Lu], Theorem or [AG]). 6. Factorizable sheaves attached to the uantum group 6.1. The algebras. Denote (6.1) L Ω := Ω L U, L Ω + := Ω L U +, KD Ω := Ω KD U and KD Ω + := Ω KD U + and also (6.2) Ω,small := Ω u, Ω+,small := Ω u +. We have canonical maps between factorization algebras: L Ω KD Ω and L Ω + KD Ω +. By the previous section, the categories in (5.4) are euivalent to FS( L Ω ), FS( KD Ω ), FS( L Ω + ) and FS( KD Ω + ), respectively, and (6.3) FS(Ω,small ) u -mod Free factorization algebras. Assume now that (α i, α i ) 2 1 for any i I. Denote (6.4) fr Ω := Ω fr U, fr Ω + := Ω fr U +, cofr Ω := Ω cofr U, cofr Ω + := Ω cofr U +. We obtain the following isomorphisms: L Ω D( KD Ω + ), KD Ω D( L Ω + ), fr Ω D( cofr Ω + ), cofr Ω D( fr Ω + ) and Ω,small D(Ω+,small ). We have canonical maps (6.5) fr Ω KD Ω Ω,small L Ω cofr Ω. The Verdier dual of this seuence is fr Ω + KD Ω + Ω +,small L Ω + cofr Ω +. In what follows we will describe the above factorization algebras more explicitly.

14 14 NOTES ON FACTORIZABLE SHEAVES 6.3. Change of gerbe. Recall that the factorization algebras of (6.5) were in the category of perverse sheaves twisted by the gerbes P,can. Consider a different -twisted Ť -gerbe P,can on X, where we put P λ X,,can := ωlog((λ,λ)) log((λ,2ρ)) X, where 2ρ Λ is the sum of positive roots. The data of c λ1,λ2 for P,can follows from that of P,can since the function λ (λ, 2ρ) : Λ C is linear. Let us assume now that the C -gerbes ω (λ,2ρ) X have been trivialized in a way compatible with isomorphisms ω (λ1+λ2,2ρ) X ω (λ1,2ρ) X ω (λ2,2ρ) X. For example, such a trivialization arises from a trivialization of ω X, and for X = A 1 there is a canonical choice for one. This data defines an euivalence of -twisted Ť -gerbes P,can = P,can. Thus from now on, we will think of the factorization algebras in (6.5) as being in P,can -twisted perverse sheaves. Remark. The above replacement of the gerbe has no significance for X = A 1, but would have one if we worked with arbitrary curves. The necessity of the replacement can be explained as follows: In order for the construction A Ω A to be defined for any X, we need to consider Vect Λ as a ribbon category, and we need that the suare of the antipode on A be eual to the balancing on Vect Λ. The modification of the gerbe effects the reuired modification of the balancing on Vect Λ Description on the open part. A crucial property of P,can is that the C -gerbe P λ X,,can is canonically trivial for λ = α i for i I. Indeed, we have: and α i = s i (ρ) ρ. (λ, λ) = (λ, 2ρ) for λ = w(ρ) ρ, w W, For λ Λ neg let X λ be the open subset of X λ corresponding to multiplicity-free divisors i.e., to points of the form Σλ k x k with x k x k and each λ k of the form α i for some i I. By the above, the gerbe P X λ,,can, restricted to X λ is canonically trivial. Hence, we can think of P Xλ,,can -twisted sheaves on X λ as ordinary sheaves. Let? Ω?? denote the restriction of any of the twisted perverse sheaves appearing in (6.5) to X λ. By the above, we can think of it as ordinary perverse sheaves. Proposition 6.5. The maps fr Ω,λ KD Ω,λ Ω,λ,small L Ω,λ cofr Ω,λ are isomorphisms. The corresponding perverse sheaf is canonically the sign local system on X λ. Proof. The assertion follows from the fact that for all of the algebras U involved (Tor U (C, C)) αi is 1-dimensional and is concentrated in the cohomological degree 1. We will denote the resulting local system simply by Ω,λ.

15 NOTES ON FACTORIZABLE SHEAVES Description of extensions. Let j λ denote the open embedding X λ X λ. We shall denote by j? λ,? =!,,! the corresponding functors on the category of P X λ,,can -twisted perverse sheaves. (Note that although the twisting was trivial on X λ and can be non-canonically trivialized over the entire X λ, each of the above extension functors in the twisted and the non-twisted contexts is different.) Proposition 6.7. We have: fr Ω,λ j! λ ( Ω,λ ) and cofr Ω,λ j λ ( Ω ), compatibly with factorization isomorphisms. The map fr Ω,λ corresponds to the canonical map j λ! j λ. cofr Ω,λ Proof. It is enough to prove the first isomorphism, since the second one would follow by Verdier duality and replacing by +. Our assertion is euivalent to ( Tor fr U (C, C) ) λ = 0 for λ being not one of the negative simple roots. However, the latter follows from the fact that fr U is the free associative algebra with the generators in the specified degrees. The assertion about factorization follows from the corresponding property over X λ, and likewise the assertion about the map between Ω s. Corollary 6.8. We have: compatibly with factorizations. Ω,λ,small jλ! ( Ω,λ ) Proof. This follows by combining the previous proposition with (6.5). Combining the above corollary with (6.3), we reprove the main result of [BFS] Description of L Ω and KD Ω. By (6.5), as perverse sheaves, L Ω,λ j λ ( Ω,λ ) and KD Ω,λ receives a surjective map from j! λ ( Ω,λ ). injects into Let λ denote the main diagonal on X λ. Let j λ denote the open embedding of its complement. By factorization and induction on λ, we can assume that we know the restrictions of L Ω,λ and KD Ω,λ to X λ λ, and we can assume that λ α i, i I. Proposition (1) The perverse sheaf L Ω,λ X λ λ, i.e., j ( L Ω,λ ) admits the following description in terms of its restriction to for λ w(ρ) ρ, w W, for λ = w(ρ) ρ, l(w) 3, for λ = w(ρ) ρ, l(w) = 2, L Ω,λ L Ω,λ L Ω,λ j (j L Ω,λ )), H ( 0 j (j ( L Ω,λ )) ), j! (j ( L Ω,λ )).

16 16 NOTES ON FACTORIZABLE SHEAVES (2) The perverse sheaf KD Ω,λ X λ λ, i.e., j ( KD Ω,λ ) for λ w(ρ) ρ, w W, for λ = w(ρ) ρ, l(w) > 2, for λ = w(ρ) ρ, l(w) = 2, admits the following description in terms of its restriction to KD Ω,λ KD Ω,λ KD Ω,λ j! (j L Ω,λ )), H ( 0 j! (j ( KD Ω,λ )) ), j! (j ( KD Ω,λ )). Proof. Again, by Verdier duality, it suffices to prove statement (2). By Theorem- Construction 4.6(2), the *-restriction of KD Ω,λ to λ is the constant sheaf tensored with Tor KD U (C, C), and the latter satisfies 0 for λ w(ρ) ρ, w W, is concentrated in cohomlogical degees < 2 for λ = w(ρ) ρ, l(w) > 2, is concentrated in cohomlogical degee 2 for λ = w(ρ) ρ, l(w) = 2, since the Kac-De Concini algebra has the same homology as the assical U(n ). This implies the assertion regarding KD Ω,λ except in the case λ = w(ρ) ρ, l(w) = 2. In the latter case, we have to show that the!-restriction of KD Ω,λ to λ is concentrated in the perverse cohomological degrees 1. Again, by Verdier duality, it suffices to show that the *-restriction of L Ω,λ to λ is concentrated in the perverse cohomological degrees 1, i.e., ( ) λ that Tor KD U (C, C) is concentrated in degrees 2. However, Tor 1 KD (C, C) is spanned by elements whose weights are proportional to the U negative simple roots, implying our assertion. 7. Geometric incarnation of the uantum Frobenius 7.1. For λ Λ let us denote by X λ the corresponding space taken with respect to the lattice Λ, and by X λ the corresponding space taken with respect to the lattice Λ. We have a natural osed embedding Let Ω, Ω+, Ω, λ Frob : X λ X λ. D(Ω ), Ω+, D(Ω + ) be the factorization algebras corresponding to the lattice Λ and the assical uantum group U with the sub-algebras U and U+. These are P X λ,,can-twisted perverse sheaves on the spaces X λ, λ Λ. Note that the corresponding -twisted Ť-gerbe P,can has the property that the identifications (4.1) extend to identifications of gerbes over the entire X λ1 X λ2. Thus, we can speak about factorization algebras that are commutative or co-commutative as in the non-twisted case. Lemma 7.2. The factorization algebras Ω and Ω+ are co-commutative and the factorization algebras Ω, and Ω +, are commutative.

17 NOTES ON FACTORIZABLE SHEAVES Note that the C -gerbe P X λ,,can on Xλ identifies with the pull-back of P X λ,,can by means of λ Frob. Note also that for λ Λ and µ Λ we have natural identifications of gerbes: P X λ+µ,,can X λ X µ P X λ,,can P X µ,,can and also P X λ+µ x 0,,can X λ X P µ X λ,,can P X x µ 0,,can, compatible with the factorization property of the P X λ,,can s. Therefore, given a co-commutative (resp., commutative) factorization algebra Ω on the X λ, we can speak about co-modules (resp., modules) with respect to it on the X µ s. The maps of Hopf algebras L U U and L U + U + define maps ( ) (7.1) λ Frob ( L Ω,λ ) Ω,λ and ( λ ) Frob ( L Ω +,λ ) Ω +,λ and hence, by duality, a map ( λ )! Frob ( KD Ω,λ ). Ω +,,λ Let add λ,µ denote the corresponding maps X λ X µ X λ+µ and X λ X µ x 0 and let ( X λ X µ) disj Xλ X µ X λ+µ x 0,, denote the corresponding open subset. Lemma 7.4. The map add λ,µ( L Ω,λ+µ ) (X λ Xµ ) Ω,λ L Ω,µ disj (X λ Xµ ), disj which results from (7.1) and (1.1), extends canonically onto the entire X λ X µ. Similarly, the map Ω +,,λ KD Ω,µ (X λ Xµ ) add! λ disj,µ( KD Ω,λ+µ ) (X λ Xµ ) disj extends canonically onto the entire X λ X µ. Thus, we obtain that L Ω is naturally a co-module with respect to Ω, and KD Ω is naturally a module with respect to Ω +, Recall the natural maps KD Ω Ω,small and Ω,small L Ω, as well as the notation regarding the bar-construction. Proposition 7.6. The natural map is a uasi-isomorphism. Likewise, the map is a uasi-isomorphism. Ω,small Bar(Ω ) L Ω Bar(Ω +, ) KD Ω Ω,small

18 18 NOTES ON FACTORIZABLE SHEAVES Proof. The assertion of the proposition is the short exact seuence of Hopf algebras (5.6), i.e., that ( ) Tor L U (C, C) Tor U C, Tor u (C, C). Remark. From the above proposition, we obtain that Ω,small, as an object of the derived category, carries an action of Bar(Ω ), which is an incarnation of U, and a co-action of Bar(Ω +, ), which incarnates an action of U +. We would like to be able to phrase in what way these two structures are compatible, i.e., how to speak about an action on Ω,small of the entire U. Morally, this should correspond to the outer action of G on u coming from the short exact seuence of Hopf algebras 1 u L U U Let F be a factorizable sheaf with respect to L Ω. We shall say that it is co-commutative with respect to Ω if the map add λ,µ(f λ+µ ) (X λ Xµ ) disj Ω,λ (which follows from (7.1) and (1.3)) has been extended to a map on the entire X λ X µ. add λ,µ(f λ+µ ) Ω,λ F µ F µ (X λ Xµ ) disj Similarly, if F is a factorizable sheaf with respect to KD Ω +, we shall say that it is commutative with respect to Ω +, if the map has been extended to a map on the entire X λ X µ. Ω +,,λ F µ (X λ Xµ ) disj add! λ,µ(f λ+µ ) (X λ Xµ ) disj Ω +,,λ F µ add! λ,µ(f λ+µ ) We remark that in both cases above, if F consists of twisted perverse sheaves, than an extension of the reuired map, if exists, is uniue. This is no longer the case in the derived category, where such an extension is an additional structure. Proposition 7.8. (1) There exists an euivalence between the category of factorizable sheaves with respect to L U co-commutative with respect to Ω and + U, 2 of functors U, + 2 -mod, such that we have a commutative diagram -mod Dr( L U )-mod FS( L Ω ) Ω cocom FS(L Ω ). (2) The assignment F Bar(Ω ) F defines a functor FS( L Ω ) Ω cocom FS(Ω,small )

19 NOTES ON FACTORIZABLE SHEAVES 19 that makes the following diagram of categories commutative: Similarly, we have: U, + 2 -mod u -mod FS( L Ω ) Ω co com FS(Ω,small ). Proposition 7.9. (1) There exists an euivalence between the category of factorizable sheaves with respect to KD Ω commutative with respect to Ω +, and U +, 2 -mod, such that we have a commutative diagram of functors U +, 2 -mod Dr( KD U )-mod FS( KD Ω ) Ω +, com FS(KD Ω ). (2) The assignment F Bar(Ω +, ) F defines a functor FS( KD Ω ) Ω +, com FS(Ω,small ) that makes the following diagram of categories commutative: U +, 2 -mod u -mod FS( KD Ω ) Ω +, co com FS(Ω,small ). Remark. Let M be a module over L U. Consider its restriction to u and let F be the corresponding factorizable sheaf with respect to Ω,small. We obtain that it carries an action of Bar(Ω ) and a co-action of Bar(Ω+, ). We would like to be able to spell out the compatibility of these two structures, which would determine which factorizable sheaves with respect to Ω,small correspond to modules that came by restriction from L U.

20 20 NOTES ON FACTORIZABLE SHEAVES [AG] [BFS] [CHA] [Lu] References