Representations of quantum affine algebras

Representations of Quantum Affine Algebras arXiv:q-alg/9503022v1 31 Mar 1995 D. Kazhdan Dept. of Mathematics Harvard University Cambridge, MA 02138 Y. Soibelman Dept. of Mathematics Kansas State University Manhattan, KS 66506 Introduction Let G be a finite dimensional simple Lie algebra and G the corresponding affine Kač-Moody algebra. The notion of the fusion in the category O of representations of affine Kač-Moody algebras G was introduced ten years ago by physicists in the framework of Conformal Field Theory. This notion was developed in a number of mathematical papers (see, for example, [TUY]), where the notion of fusion is rigorously defined and [D1] where the relation between the fusion and quantum groups in the quasiclassical region was established). This line of development was extended in [KL] to a construction on equivalence between the “fusion” category for an arbitrary negative charge and the category of representations of the corresponding quantum group (for simply-laced affine Kač-Moody algebras). It is natural to try to define the notion of fusion for the category O for affine quantum groups. One can hope that it might be useful for a deformed CFT (see [FR]). The usual construction does not work since it is based on the existence of a subalgebra Γ ⊂ G×G×G which is a central extension of the Lie algebra Γ(P1 − {0, 1, ∞}, G). Unfortunately such subalgebra Γ does not have a natural generalization for the quantum case (see [D3] where the problem was stated). In the remarkable paper [FR], I. Frenkel and N. Reshetikhin found a way to describe the fusion between finite-dimensional representations and representations from the category O for affine quantum groups. In the present paper we reconsider this problem from the point of view of [KL]. Our main result is the construction of the quasi-associativity constraints (see Sect.3 and 5). Main phenomena which should be mentioned in connection with the problem are the appearance of elliptic curves instead of genus zero curves 1 and Z-sheaves instead of bundles with flat connections. From a general point of view this reflects the fact that the categories we are considering do not carry monoidal structure. Nevertheless those categories of representations of affine quantum algebras carry some other interesting structures discussed in Sect.1 in the general situation. These structures explain the categorical meaning of the so-called quantum Knizhnik-Zamolodchikov equations introduced in [FR]. Among other results we can mention meromorphicity of the quantum R-matrix for any two finite-dimensional representations (see Sect. 4) which has also general categorical origin. Acknowledgements. It is a pleasure for us to thank P. Deligne, V. Drinfeld, P.Etingof and N.Reshetikhin for the useful discussions on the topic. Y.S. is grateful to the Department of Mathematics of Harvard University, the Institute for Advanced Study and the Max-Planck Institute für Mathematik for the support and hospitality during various stages of this work. 2 §1. Monoidal categories 1.1 Definitions and basic properties. e ⊗, a) where Ce is a category, 1.1.1. Definition. A monoidal category C is a triple C = (C, ⊗ : Ce × Ce → Ce is a functor and a is the natural transformation between the functors e a = {aX,Y,Z : (X ⊗ Y ) ⊗ Z −→ X ⊗ (Y ⊗ Z)}, ⊗(⊗ × id) and ⊗(id × ⊗) from Ce× Ce× Ce to C, X, Y, Z in Ce such that all aX,Y,Z are isomorphisms, the pentagon axiom is satisfied, and ∼ there exists an object U in Ce and an isomorphism u : U ⊗ U −→U such that the functors X → X ⊗ U and X → U ⊗ X are autoequivalences of Ce (see [DM]). We call such a pair (U, u) the identity object of C. (It is clear that the identity object is determined uniquely up to a unique isomorphism.) e ⊗, A) be a monoidal category, and (U, u) the identity object in C. Let C = (C, ∼ Lemma. a) For any object X in Ce there exist unique isomorphisms rX : X −→X ⊗ U , ℓX : X → U ⊗ X such that aX,U,U ◦ (rX ⊗ idU ) = idX ⊗ u and idU ⊗ ℓX = aU,U,X ◦ (u ⊗ idX ). b) rU = u. c) For any morphism α : Y → X in Ce the diagrams Y  yrY Y ⊗U α −→ X  y rX Y  yℓY α⊗1 U ⊗Y −→ X ⊗ U α −→  X yℓX α⊗1 −→ U ⊗ X are commutative. Proof: Well known. (See [DM].) e ⊗, a) is strict if for any 1.1.2 Definition. We say that a monoidal category C = (C, X, Y, Z in Ce we have (X ⊗ Y ) ⊗ Z = X ⊗ (Y ⊗ Z), aX,Y,Z = id, there exists an object 11 e and for any X, Y in Ce the composition in Ce such that 11 ⊗ X = X ⊗ 11 = X for all X in C, X ⊗ Y = (X ⊗ 11) ⊗ Y = X ⊗ (11 ⊗ Y ) = X ⊗ Y 3 is equal to idX⊗Y . Mac Lane’s theorem says that any monoidal category is equivalent to a strict one (see [M, Chapter 7]). To simplify formulas we will from now on assume that all monoidal categories are strict. We will also assume that the category Ce is abelian, contains inductive def limits and that ⊗ is an additive exact functor. Let FC = EndCe(11). It is easy to see that FC is a commutative ring and for any X, Y in Ce the group HomCe(X, Y ) has a natural structure of an FC -module. From now on we will denote the category Ce simply as C. We hope that this will not create any confusion. 1.1.3. Assume until the end of this section that FC is a field. Any FC -vector space R can be written as an inductive limit of finite dimensional spaces R = limRi . We define → def RC = limRi ⊗ 11. It is clear that the object RC in C does not depend on a choice of a → presentation R = limRi . → We denote by VecC the category of FC -vector spaces and for any object X in C we denote by LX the functor from VecC to itself such that def LX (R) = HomC (X, RC ) for all R ∈ VecC . It is easy to see that the functor LX satisfies the conditions of Theorem 5.6.3 in [M]. Therefore the functor LX is representable. Definition. We denote by hXi the object in VecC which represents the functor LX . As follows from the definition of hXi, for any X in C we have a natural isomorphism def hXi∨ ∼ = HomC (X, 11) where L∨ = Hom(L, FC ) for any L in VecC . 1.1.4. Let C be a strict monoidal category. We denote by C the set of isomorphism classes of objects in C. For any X in Ce we denote by [X] ∈ C the equivalence class of X. Definition. We say that an object X in C is rigid if there exists an object Y in C and a pair of morphisms iX : 11 → X ⊗ Y eX : Y ⊗ X → 11 4 such that the compositions i ⊗id id⊗e X = 11 ⊗ X X −→ X ⊗ Y ⊗ X −→X X e ⊗id id⊗i are equal to idX and idY X Y = Y ⊗ 11 −→X Y ⊗ X ⊗ Y −→ Y correspondingly. In this case we say that Y is dual to X. It is easy to see that such a triple (Y, iX , eX ) if it exists, is unique up to a unique isomorphism. For any rigid X in C we denote by [X]∗ ∈ C the isomorphism class of objects Y as in Definition 1.1.4. 1.1.5 Definition. We say that the category C is rigid if all its objects are rigid and for any [Y ] in C there exists X in C such that [X]∗ = [Y ]. If C is a rigid category, then there exists an equivalence ∗ : C → C op of categories such that X ∗ is dual to X for all X in C, α ∈ HomC (X, Y ) the morphism α∗ ∈ HomC (Y ∗ , X ∗ ) is the composition id⊗i id⊗α⊗id e ⊗id Y Y ∗ −→ Y ∗ ⊗ 11 −→X Y ∗ ⊗ X ⊗ X ∗ −→ Y ∗ ⊗ Y ⊗ X ∗ −→ 11 ⊗ X ∗ = X ∗ . Moreover, such an equivalence is unique up to a unique isomorphism. We will fix such an equivalence between the categories C and C op . Then the functor X −→ X ∗∗ is an auto-equivalence of C. 1.1.6. Example. Let (H, m, ∆, i, ε, S) be a Hopf algebra over a ring F , CH be the category of H-modules X = (ρX , X) and ⊗ : CH × CH → CH be the tensor product over F . That def is, X ⊗ Y = ((ρX ⊗ ρY ) ◦ ∆, X ⊗ Y ), and 11 = (F, ǫ). Then (CH , ⊗, 11) is a strict monoidal category. In this case, FC = F . Let H0 ⊂ H be the kernel of the counit ε. For any X = (ρX , X) in CH we define def H H = ρX (H0 )X. Often we will write X(0) instead of X(0) X(0) . In the case when F is a field, def we can identify hXi (see Definition 1.1.3) with the quotient hXi = X/X(0) . Assume that F is a field. An object X = (ρX , X) is rigid if dimF X < ∞. In this case, def X ∗ = (ρX ∗ , X ∗ ) where X ∗ = X ∨ (= HomF (X, F )), and ρX ∗ (h) = (ρX (S(h)))∗ , where for any α ∈ End X we denote by α∨ the endomorphism of X ∨ dual to α. In this case, the morphism eX : X ∗ ⊗ X → 11 is induced by the natural pairing X ∗ ⊗ X → F . This pairing defines a canonical isomorphism of the linear space X ∗∗ with X and the action ρX ∗∗ of H on X ∗∗ = X is given by the rule ρX ∗∗ (h) = ρ(S 2 (h)). 5 1.1.7 Proposition. For any X in C and ϕ ∈ Hom(X, X) the diagrams iX 11  iX y ∗ −→ X  ⊗X  ϕ ⊗ id y id⊗ϕ∗ eX X∗ ⊗ X −→ X ⊗ X ∗ X ⊗ X∗ ϕ∗ ⊗id ∗ X  ⊗X  1 ⊗ ϕ y ∗ −→ X ⊗X  eX y −→ 11 are commutative. Proof: We prove the commutativity of the first diagram. The proof of the commutativity of the second diagram is completely analogous. Let def def a = (ϕ ⊗ idX ∗ ) ◦ iX , b = (idX ⊗ ϕ∗ ) ◦ iX ∈ HomC (11, X ⊗ X ∗ ). By the definition of iX and eX , a is equal to the composition i X 11 −→ X ⊗ X∗ ϕ⊗idX ∗ −→ X ⊗ X ∗ iX ⊗idX⊗X ∗ −→ X ⊗ X ∗ ⊗ X ⊗ X ∗ idX ⊗eX ⊗idX ∗ −→ X ⊗ X ∗ . But the composition (iX ⊗ idX⊗X ∗ ) ◦ (ϕ ⊗ idX ∗ ) is equal to the composition (idX⊗X ∗ ⊗ ϕ ⊗ idX ∗ ) ◦ (iX ⊗ idX⊗X ∗ ). Since the compositions i X 11−→ X ⊗X ∗ idX⊗X ∗ ⊗iX i X −→ X ⊗X ∗ ⊗ X ⊗ X ∗ and 11−→ X ⊗X ∗ iX ⊗idX⊗X ∗ −→ X ⊗X ∗ ⊗X ⊗X ∗ i ⊗i are equal (and both coincide with the composition 11 = 11 ⊗ 11 X−→X X ⊗ X ∗ ⊗ X ⊗ X ∗ ) we see that α is equal to the composition i X 11−→ X⊗X ∗ idX⊗X ∗ ⊗iX −→ X⊗X ∗ ⊗X⊗X ∗ idX⊗X ∗ ⊗ϕ⊗idX ∗ −→ X⊗X ∗ ⊗X⊗X ∗ idX ⊗eX ⊗idX ∗ −→ X⊗X ∗ . But the composition of the last three arrows is equal to b = idX ⊗ ϕ∗ . Lemma 1.1.7 is proved. 1.1.8. For any X, Y in C the morphisms i, e defined as the compositions id⊗iY ⊗id i X i : 11−→X ⊗ X ∗ = X ⊗ 11 ⊗ X ∗ e : (Y ∗ ⊗ X ∗ ) ⊗ (X ⊗ Y ) −→(X ⊗ Y ) ⊗ (Y ∗ ⊗ X ∗ ) id⊗eX ⊗id e Y −→Y ∗ ⊗ 11 ⊗ Y = Y ∗ ⊗ Y −→1 1 satisfy the conditions of Definition 1.1.4. Therefore they define isomorphisms of (X ⊗ Y )∗ with Y ∗ ⊗ X ∗ such that i = iX⊗Y and e = eX⊗Y . We will freely use this identification and will therefore identify the second dual (X ⊗ Y )∗∗ with X ∗∗ ⊗ Y ∗∗ . Remark: As follows from ([DM], 1.17) the rigidity of C implies the semisimplicity of 11. We will always assume that 11 is simple. In this case FC is a field. 6 1.1.9. Definition. Let C be a monoidal category, D ⊂ C a full monoidal subcategory such that any object of D is rigid. ∨ ∗∗ For any V, W in C and X, Y in D we define the maps αX ⊗ V i∨ V : hV ⊗ Xi → hX and ∗ ∗ ϕX,Y V,W : HomC (V ⊗ X, Y ⊗ W ) → HomC (Y ⊗ V, W ⊗ X ) as the compositions ∨ ∗∗ αX ⊗ V ⊗ X ⊗ X ∗ , X ∗∗ ⊗ X ∗ ) V : hV ⊗ Xi = HomC (V ⊗ X, 11) −→ HomC (X idX ∗∗ ⊗V ⊗iX ⊗eX ∗ −→ HomC (X ∗∗ ⊗ V, 11) = hX ∗∗ ⊗ V i∨ . id⊗i X ∗ ∗ ∗ ϕX,Y V,W (a) : Y ⊗ V −→ Y ⊗ V ⊗ X ⊗ X id⊗a⊗id −→ e Y Y ∗ ⊗ Y ⊗ W ⊗ X ∗ −→ W ⊗ X∗ for all a ∈ HomC (V ⊗ X, Y ⊗ W ). Remark: We will write ϕX,Y instead of ϕX,Y V V,V . V ∗∗ Lemma. If FC is a field, then there exist FC -linear maps βX ⊗ V i → hV ⊗ Xi ∗∗ : hX V ∨ such that αX V = (βX ∗∗ ) . Proof: For any R in VecC we can define a FC -linear map αX V (R) : Hom(V ⊗ X, RC ) → Hom(X ∗∗ ⊗ V, RC ) exactly in the same way as we have defined the map αX V . Then maps αX V (R) define a morphism from the functor LV ⊗X to the functor LX ∗∗ ⊗V . The correV sponding morphism between the representing objects is βX . 1.1.10. For any V in C we denote by EV the ring of endomorphisms of V and by EVop the opposite ring. If V is rigid, the map f → f ∗ defines an isomorphism of EV with EVop∗ . X,Y V Lemma. a) The linear maps αX V , βX ∗∗ and ϕV,W are isomorphisms. (X⊗Y ) b) For any V in C, X, Y in D the linear map αV is equal to the composition ∗∗ V ⊗X Y ⊗V Y V . αX Y ∗∗ ⊗V ◦ αV ⊗X and the map β(X⊗Y )∗∗ is equal to the composition βY ∗∗ ◦ βX ∗∗ c) For any fX ∈ EX , fY ∈ EY , fW ∈ EW ,fV ∈ EV and a ∈ HomC (V ⊗ X, Y ⊗ W ) we have X,Y ∗ ∗ ϕX,Y V,W ((fY ⊗ fW ) ◦ a ◦ (fV ⊗ fX )) = (fW ⊗ fX ) ◦ ϕV,W (a) ◦ (fY ⊗ fV ). 7 Proof: a) Consider the map from hX ∗∗ ⊗ V i∨ to hV ⊗ Xi∨ defined as the composition hX ∗∗ ⊗ V i∨ = HomC (X ∗∗ ⊗ V, 11) −→ HomC (X ∗ ⊗ X ∗∗ ⊗ V ⊗ X, X ∗ ⊗ X) iX ∗ ⊗idV ⊗X ⊗eX −→ HomC (V ⊗ X, 11) = hV ⊗ Xi∨ . As follows immediately from Definition 1.1.4, this map is the inverse to αX V . The construction of the inverse to ϕX,Y V,W is completely analogous. b) Follows immediately from the definitions. c) We have to show that for any a ∈ HomC (V ⊗ X, X ⊗ V ) the following equalities hold: X,Y i) ϕX,Y V,W (a ◦ (fV ⊗ idX )) = ϕV,W (a)(idY ∗ ⊗ fV ) for any fV ∈ EV X,Y ∗ ii) ϕX,Y V,W (a ◦ (idW ⊗ fX )) = (idW ⊗ fX ) ◦ ϕV,W (a) for all fX ∈ EX . X,Y iii) ϕX,Y V,W ((idY ⊗ fW ) ◦ a) = (fW ⊗ idX ∗ ) ◦ ϕV,W (a) X,Y ∗ iv) ϕX,Y V,W ((fY ⊗ idW ) ◦ a) = ϕV,W (a) ◦ (fY ⊗ idV ). The proofs of i) and iii) are straightforward. We will show how to prove ii). The proof of (iv) is completely analogous. By the definition the map ϕX,Y V,W (a ◦ (idV ⊗ fX )) is equal to the composition 11⊗i Y ∗ ⊗ V −→X Y ∗ ⊗ V ⊗ X ⊗ X ∗ id⊗fX ⊗id −→ Y ∗ ⊗ V ⊗ X ⊗ X ∗ eY ⊗id id⊗a⊗id −→ Y ∗ ⊗ Y ⊗ W ⊗ X ∗ −→ W ⊗ X ∗ . As follows from Lemma 1.1.7 a) this composition is equal to the composition 11⊗i Y ∗ ⊗ V −→X Y ∗ ⊗ V ⊗ X ⊗ X ∗ ∗ id⊗idX ⊗fX −→ Y ∗ ⊗ V ⊗ X ⊗ X ∗ e ⊗id id⊗a⊗id Y −→ Y ∗ ⊗ Y ⊗ W ⊗ X ∗ −→ W ⊗ X ∗. ∗ But the last composition is equal to (idW ⊗ fX ) ◦ ϕX,Y V,W (a). Lemma 1.1.10 is proved. 1.1.11. Let H be a Hopf algebra, C = CH , X = (ρX , X), Y = (ρV , V ) be H-modules such that dimC X < ∞. Let PVX : X ⊗ V → V ⊗ X be the permutation PVX (x ⊗ v) = v ⊗ x. We can consider PVX as a linear map from X ∗∗ ⊗ V to V ⊗ X. 8 1.1.12 Lemma. PVX maps the subspace (X ∗∗ ⊗ V )(0) into (V ⊗ X)(0) and induces the V linear map from hX ∗∗ ⊗ V i to hV ⊗ Xi equal to βX ∗∗ . Proof: Let (PVX )∗ : (V ⊗X)∗ → (X ∗∗ ⊗V )∗ be the linear map dual to PVX . It is sufficient to show that for any λ ∈ hV ⊗ Xi∨ ⊂ (V ⊗ X)∗ we have (PVX )∗ (λ) = (αX V )(λ). Consider first the case when V = X ∗ and λ = eX . It is easy to check that αX X ∗ (eX ) = eX ∗ and the validity of Lemma 1.1.12 follows from definition of eX . e ⊗ idX ) where For any λ ∈ hV ⊗ Xi∨ = HomCH (V ⊗ X, 11) we have λ = eX ◦ (λ e ∈ HomC (V, X ∗) and Lemma 1.1.12 follows from the functoriality of αX and P X . Lemma λ H V V 1.1.12 is proved. 1.1.13 Definition. A strict endomorphism T of C is a functor from the category C to itself such that a) T (11) = 11, b) T (X⊗Y ) = T (X)⊗T (Y ) for all X, Y in C and those identifications are compatible with morphisms in C, and c) for any rigid object X in C we have T (X ∗ ) = (T (X))∗ , d) T (iX ) = iT (X) , T (eX ) = eT (X) . We say that T is a strict automorphism if it is an equivalence of categories. If T is a strict automorphism of C, then for any X in C, T defines an isomorphism from HomC (X, 11) = hXi∨ to HomC (T (X), 11) = hT (X)i∨ . It is easy to see that this ∼ ∼ isomorphism hXi∨ −→ hT (X)i∨ comes from an isomorphism hXi−→ hT (X)i. 1.1.14 Example: For any rigid monoidal category D the functor X → X ∗∗ is a strict automorphism. We denote the inverse to this automorphism by T∗ . It is clear that any strict automorphism T of D commutes with T∗ . V 1.1.15 Remark: We can interpret the linear map βX defined in Lemma 1.1.9 as the linear map from hX ⊗ V i to hV ⊗ T∗ (X)i. 1.1.16. Let C be a monoidal category, D and O full subcategories of C such that D is rigid and T a strict automorphism of D. 9 Definition. A right T -braiding of D on O is a functorial isomorphism sV,X : V ⊗ X → T (X) ⊗ V, defined for all V in O and X in D such that sV,11 = idV and for all X, Y in D we have sV,X⊗Y = (idT (X) ⊗ sV,Y ) ◦ (sV,X ⊗ idY ). 1.1.17 Proposition. For any X in D and V in O the diagram idT (X) ⊗sV,X T (X)∗ ⊗ V ⊗ X    −1 sV,X ∗ ⊗ idX y −→ eX V ⊗ X∗ ⊗ X −→ T (X ∗ ) ⊗ T (X) ⊗ V    eT (X) y V is commutative. Proof: As follows from the functoriality of sV,X and the equality sV,11 = id, the diagram V ⊗ (X ∗ ⊗ X)   eX y V sV,X ∗ ⊗X = −→ (X ∗ ⊗ X) ⊗ V   eX  y V is commutative. On the other hand, we know that sV,X ∗ ⊗X = (idT (X ∗ ) ⊗ sV,X ) ◦ (sV,X ∗ ⊗ idX ). Therefore the diagram T (X)∗ ⊗ V ⊗ X x   sV,X ∗ ⊗ idX  V ⊗ X∗ ⊗ X idT (X ∗ ) ⊗sV,X −→ eX −→ is commutative. Proposition 1.1.17 is proved. 10 T (X ∗ ) ⊗ T (X) ⊗ V    eT (X) y V 1.1.18 Proposition. For any V in O and any X in D X,T (X) ϕV (sV,X ) = s−1 V,X ∗ . X,T (X) Proof: By the definition the map ϕV (sV,X ) ∈ HomC (T (X)∗ ⊗ V, V ⊗ X ∗ ) is defined as the composition T (X)∗ ⊗ V sV,X id⊗iX −→T (X)∗ ⊗ V ⊗ X ⊗ X ∗ −→T (X)∗ ⊗ T (X) ⊗ V ⊗ X ∗ id⊗eT (X) ⊗id −→ V ⊗ X ∗ . As follows from Lemma 1.1.17, this composition is equal to the composition T (X)∗ ⊗ V id⊗iX −→T (X)∗ ⊗ V ⊗ X ⊗ X ∗ ⊗id s−1 V,X ∗ −→V ⊗ X ∗ ⊗ X ⊗ X ∗ idV ⊗eX ⊗idX ∗ −→ V ⊗ X ∗ . As follows from the definition of rigidity, the composition of the last two morphisms is equal to s−1 V,X ∗ . Proposition 1.1.18 is proved. 1.1.19 Definition. Let C be a strict monoidal category. A braiding s on C is a functorial system of isomorphisms sX,Y ∈ Isom(X ⊗ Y, Y ⊗ X) for X, Y in C such that a) for any X, Y, Z in C we have sX⊗Y,Z = (sX,Z ⊗ idY ) ◦ (idX ⊗ sY,Z ), sX,Y ⊗Z = (idX ⊗ sX,Z ) ◦ (sX,Y ⊗ idZ ); b) sX,11 = s11,X = idX for all X in C. 1.2 KZ-data. 1.2.1 Definition. Let D be a rigid monoidal category. A weak braiding b = (D (2) , s) on D is 1) A choice of a subset D (2) of D × D such that a) For any X, Y, Z in D such that ([X], [Y ]), ([X], [Z]) are in D ([X], [Y ⊗ Z]) is in D (2) (2) the pair . b) For any X, Y, Z in D such that ([X], [Z]) and ([Y ], [Z]) in D ([X ⊗ Y ], [Z]) is in D (2) . 11 (2) the pair c) (X, 11) ∈ D (2) and (11, X) ∈ D (2) , for all X ∈ D. 2) A functorial isomorphism sX,Y between the restrictions of functors (X, Y ) → X ⊗ Y, (X, Y ) → (Y ⊗ X) on D (2) such that a) s11,X = sX,11 = id for all X ∈ D b) sX,Y ⊗Z = (1Y ⊗ sX,Z ) ◦ (sX,Y ⊗ 1Z ) for all X, Y, Z ∈ D such that (X, Y ), (X, Z) ∈ D (2) c) sX⊗Y,Z = (sX,Z ⊗ 1Y ) ◦ (1X ⊗ sY,Z ) for all X, Y, Z ∈ D such that (X, Z), (Y, Z) ∈ D (2) , where we denote by D (2) the full subcategory of D × D of pairs (X, Y ) such that ([X], [Y ]) ∈ D (2) . 1.2.2 Definition. A KZ-data consists of a monoidal category C, its full subcategories D, C ± , strict automorphisms T± of D, right T± braidings s± of D on C ± , and a weak braiding (D (2) , s) compatible with the automorphisms T± . 1.2.3. We say that KZ-data (C, D, C ± , s± , T± , D (2) , s) is rigid if D is rigid. In this case we denote by T the automorphism of D which is the composition T = T− T∗ T+ . 1.2.4 Definition. Let K = (C, D, C ± , s± , T± , D (2) , s) be a rigid KZ-data. a) We say that a pair (X, Y ) ∈ D × D is T -generic (or simply generic) if for any r ∈ Z, (T r (X), Y ) ∈ D (2) . b) For any n ∈ Z we denote by Sn = SnT the set of n-tuples (X1 , . . . , Xn ) ∈ D n such that all the pairs (Xi , Xj ), 1 ≤ i 6= j ≤ n are generic. T c) For any i, 1 ≤ i ≤ n − 1 we denote by pi : SnT → Sn−1 the map (X1 , . . . , Xn ) → (X1 , . . . , Xi−1 , Xi ⊗ Xi+1 , Xi+2 , . . . , Xn ). d) We define an action of the group Zn on SnT by the rule Ei (X1 , . . . , Xn ) = (X1 , . . . , Xi−1 , T (Xi ), Xi+1 , . . . , Xn ), where {Ei }, 1 ≤ i ≤ n, is the standard set of generators of the group Zn . 12 e) For any pair V ∈ C + , W ∈ C − and a point x = (X1 , . . . , Xn ) ∈ Sn we define a vector def (n) (n) space FV,W (x) = hV ⊗X1 ⊗· · ·⊗Xn ⊗W i. We will consider FV,W as a set-theoretical vector bundle over Sn . It is clear that for any i, 1 ≤ i ≤ n − 1 we have a canonical isomorphism (n) ∼ (n−1) FV,W −→ p∗i (FV,W ). 1.2.5 Definition. We denote by θn : Sn → Sn an automorphism given by the rule θn (X1 , . . . , Xn ) = (X2 , . . . , Xn , T (X1 )) (n) (n) and by θbn its lifting θbn : FV,W → θn∗ (FV,W ) to the bundle defined as the composition (n) θbn : FV,W (x) = hV ⊗ X1 ⊗ · · · ⊗ Xn ⊗ W )i = hV ⊗ X1 ⊗ Y ⊗ W i → s+ V,X 1 + −→ hT (X1 ) ⊗ V ⊗ Y ⊗ W i s− W,T ∗ T+ (X1 ) −→ V ⊗Y ⊗W βT (X ) + 1 −→ hV ⊗ Y ⊗ W ⊗ T∗ T+ (X1 )i → (n) hV ⊗ Y ⊗ T (X1 ) ⊗ W i = FV,W (θn (x)) for all x = (X1 , . . . , Xn ) ∈ Sn where Y = X2 ⊗ · · · ⊗ Xn and the isomorphism ∼ βZU : hZ ⊗ U i −→ hU ⊗ T∗ (Z)i is defined in 1.1.15. (n) (n−1) It is clear that pn−1 ◦ θn2 = θn−1 ◦ pn−1 . Since FV,W = p∗n−1 (FV,W ) we can consider (n) θbn2 and p∗n−1 (θbn−1 ) as automorphisms of FV,W lifting the transformation θn2 of Sn . Anal(n) ogously for any i, 1 ≤ i < n − 1 we can consider p∗i (θbn−1 ) as an automorphism of FV,W over θn . 1.2.6 Proposition. a) θbn commutes with endomorphisms of Y = X1 ⊗ · · · ⊗ Xn−1 and for any endomorphism a of Xn we have (θn ◦ a) = T (a) ◦ θn . b) θbn2 = p∗n−1 (θbn−1 ). c) θbn = p∗i (θbn−1 ) for all i, 1 ≤ i < n − 1. Proof: Follows immediately from the definition and the properties of s± and s. 13 (n) 1.2.7 Definition. For any i, 1 ≤ i ≤ n we denote by δi (n) (n) : FV,W (x) → FV,W (Ei (x)) a (n) lifting of Ei to FV,W defined by a composition (n) δi (n) : FV,W (x) = hV ⊗ X + ⊗ Xi ⊗ X − ⊗ W i −→ s−1 b θn hV ⊗ X + ⊗ X − ⊗ T (Xi ) ⊗ W i −→ hV ⊗ Xi ⊗ X + ⊗ X − ⊗ W i −→ Xi ,X + sX − ,T (X −→ i) (n) hV ⊗ X + ⊗ T (Xi ) ⊗ X − ⊗ W i = FV,W (Ei (x)) for all x = (X1 , . . . , Xn ) ∈ Sn where X + = X1 ⊗ · · · ⊗ Xi−1 , X − = Xi+1 ⊗ · · · ⊗ Xn . 1.2.8 Proposition. (n) (n) a) δi+1 δi (n) (n) (n−1) ) (n−1) ) = p∗i (δi b) δi δi+1 = p∗i (δi (n) c) If j < i, then δi (n−1) = p∗j (δi−1 ) (n) d) If j > i + 1, then δi (n−1) = p∗j (δi ). Proof: a) For simplicity we consider the case i = 1. The proof in the general case is completely analogous. Let x = (V1 , . . . , Vn ) ∈ Sn . We define Y = X3 ⊗ · · · ⊗ Xn . To prove a) we consider the composition (n) (n) δ2 δ1 (x) : hV ⊗ X1 ⊗ X2 ⊗ Y ⊗ W i → hV ⊗ Y ⊗ T (X1 ) ⊗ T (X2 ) ⊗ Y ⊗ W i. Using the equality sX2 ⊗Y,X1 = (sX2 ,X1 ⊗ idY ) ◦ (idX2 ⊗ sY,X1 ) we can write this map as the composition b θn hV ⊗ X1 ⊗ X2 ⊗ Y ⊗ W i −→ hV ⊗ X2 ⊗ Y ⊗ T (X1 ) ⊗ W i sY,T (X1 ) −→ b θn hV ⊗ X2 ⊗ T (X1 ) ⊗ Y ⊗ W i −→ hV ⊗ T (X1 ) ⊗ Y ⊗ T (X2 ) ⊗ W i sY,T (X2 ) −→ −→ hV ⊗ T (X1 ) ⊗ T (X2 ) ⊗ Y ⊗ W i. By Proposition 1.2.6 a) this composition is equal to the composition b b θn θn hV ⊗ X1 ⊗ X2 ⊗ Y ⊗ Y ⊗ W i −→ hV ⊗ X2 ⊗ Y ⊗ T (X1 ) ⊗ W i −→ hV ⊗ Y ⊗ T (X1 ) ⊗ T (X2 ) ⊗ W i sY,T (X2 ) −→ sY,T (X1 ) −→ hV ⊗ T (X1 ) ⊗ Y ⊗ T (X2 ) ⊗ W i −→ hV ⊗ T (X1 ) ⊗ T (X2 ) ⊗ Y ⊗ W i 14 Part a) follows now from Proposition 1.2.6 b) and the definition of weak braiding. To prove part b) we consider the composition (n) (n) δ1 δ2 (x) hV ⊗ T (X2 ) ⊗ Y ⊗ T (X1 ) ⊗ W i −→ 2 : hV ⊗ X1 ⊗ X2 ⊗ Y ⊗ W i −→ hV ⊗ X1 ⊗ Y ⊗ T (X2 ) ⊗ W i sT (X2 ),T (X1 ) s−1 X ,X b θn 1 −→ hV ⊗ X2 ⊗ X1 ⊗ Y ⊗ W i −→ b θn −→ hV ⊗ X1 ⊗ T (X2 ) ⊗ Y ⊗ W i −→ sY,T (X2 ) sY,T (X1 ) −→ hV ⊗ T (X2 ) ⊗ T (X1 ) ⊗ Y ⊗ W i −→ hV ⊗ T (X2 ⊗ X1 ) ⊗ Y ⊗ W i T (sX2 ,X1 ) −→ hV ⊗ T (X1 ⊗ X2 ) ⊗ Y ⊗ W i. (n) (n) (n−1) It follows now from Proposition 1.2.6 a) that δ1 δ2 (x) = p∗1 (δ1 (n) 1.2.9 Proposition. The isomorphisms δi (n) (n) : FV,W −→ Ei∗ (FV,W ), 1 ≤ i ≤ n commute. (n) (n) Proof: It follows from Proposition 1.2.8 that δi−1 and δi (n) (n) δi−2 and δi ). commute. To prove that (n−2) commute we observe that (by Proposition 1.2.8) δi−2 (n−1) and δi−1 commute. Therefore (n−1) (n−1) (n−1) (n−1) p∗n−2 (δi−2 )p∗n−2 (δi−1 ) = p∗n−2 (δi−1 )p∗n−2 (δi−2 ). By Lemma 1.2.8 we can rewrite this equality in the form (n) (n) (n) δi−2 δi−1 δi (n) (n) (n) = δi δi−2 δi−1 . (n) Since the transformation δi−1 is invertible and, by the same Proposition 1.2.8 it commutes (n) (n) with both δi−2 and δi (n) (n) that δi , δj (n) (n) we see that δi−2 and δi commute. Analogous arguments show commute for all i, j, 1 ≤ i, j ≤ n. Proposition 1.2.9 is proved. 1.3. A useful formula. def 1.3.1. Let (H, 1, ε, m, ∆) be a Hopf algebra. H0 = ker ε ⊂ H and (H ⊗ H)0 ⊂ H ⊗ H be the subgroup of linear combinations of elements of the form ∆(x0 )a, x0 ∈ H0 , Let S be the antipode of H. 15 a ∈ H ⊗ H. 1.3.2 Lemma. For any x ∈ H such that ∆(x) = n X x′r ⊗ x′′r we have r=0 x⊗1= n X ∆(x′r )(1 ⊗ S(x′′r )). r=0 Proof: The right side is equal to (id⊗m)(id⊗id⊗S)(∆⊗1)∆(x) = (id⊗m)(id⊗id⊗S)(1⊗ ∆)∆(x). By the definition of the antipode this expression is equal to (id ⊗ ǫ)∆(x) = x ⊗ 1. Lemma 1.3.2 is proved. 1.3.3 Lemma. For any x ∈ H we have x ⊗ 1 − 1 ⊗ S(x) ∈ (H ⊗ H)0 . Proof: Since (ǫ⊗1)∆(x) = 1⊗x we can write ∆(x) in the form ∆(x) = 1⊗x+ where x′r ∈ H0 , x′′r ∈ H, 1 ≤ r ≤ n. Then it follows from Lemma 1.3.2 that x⊗1 = n X ∆(x′r )(1 ⊗ S(x′′r )) + 1 ⊗ S(x). r=1 Lemma 1.3.3 is proved. 16 n X r=1 x′r ⊗x′′r §2. Quantum affine algebras In this section we will use freely notations from [L]. 2.1 Basic definitions. 2.1.1. Let (I, ·) be an affine irreducible Cartan datum and (Λ∨ , Λ, h , i) be a simply connected root datum of type (I, ·) (see [L] 2.1 and 2.2) which is an affinization of a finite root datum. Any such datum is either a symmetric root datum (see [L] 2.1) or is obtained as a quotient of a symmetric one by a finite group of automorphisms which preserve some special vertex (see [L] 14.1.5). We denote by i0 ∈ I this special vertex. As follows from [L] 14.1.4 such a vertex is defined uniquely up to an automorphism of the root datum and def (i0 ·i0 ) = 2. We define I = I −{i0 }. (In the terminology of [K] we consider the non-twisted affine case). Let Z[I] → Λ be the group homomorphism such that i 7−→ i′ for all i ∈ I (see [L] 2.2.1). This homomorphism is not injective, it has a kernel isomorphic to Z. As follows X ni · i of this kernel such that ni0 = 1. We define from [K] 6.2 there exists a generator i∈I ∨ the dual Coxeter number h def as the sum h∨ = X ∨ and denote by Λ∨ ni (i·i) 0 ⊂ Λ the 2 i∈I subgroup generated by an element Σni (i·i) 2 i. ∨ Let Λ ⊂ Λ∨ and Λ ⊂ Λ be the subgroups generated by elements i and i′ correspond∨ ingly for i ∈ I. Since ni0 = 1 we have a direct sum decomposition Λ∨ = Λ ⊕ Λ∨ 0 which def induces a direct sum decomposition Λ = ′ Λ ⊕ Λ0 , where Λ0 = {λ ∈ Λ|hi, λi = 0 ∀i ∈ I} and ′ Λ = Λ ⊗Z Q ∩ Λ. Then Λ is a subgroup of finite index d in ′ Λ. As follows from the definition of a root datum we have (i · i)hi, λ′ i = 2(i · λ) for all i ∈ I, λ ∈ Λ∨ . The map ∨ λ 7−→ λ′ defines an imbedding Λ ֒→ ′ Λ and there exists unique symmetric bilinear form ∨ [ , ] : ′ Λ ×′ Λ → 1/dZ such that [λ′ , µ′ ] = (λ · µ) for all λ, µ ∈ Λ . We denote by ρ′ ∈ [i′ , ρ′ ] = (i·i) 2 ′ Λ the unique element such that hi, ρ′ i = 1 for all i ∈ I. Then for all i ∈ I. Remark: In the terminology of [K], Λ is the weight lattice, Λ∨ is the coroot lattice, ′ Λ is the weight lattice of finite-dimensional root datum (I, ·). 17 def def ∨ 2.1.2. We fix a number qe ∈ C∗ such that |e q | < 1 and define q = qedh , qi = q (i·i)/2 for all i ∈ I. def qin −qin qi −qi−1 For any n ∈ N, i ∈ I we define [n]i = def and [n]i ! = Qn s=1 [s]i . 2.1.3. We denote by Ǔ the C-algebra generated by elements Ei , Fi , Kµ , i ∈ I, µ ∈ Λ∨ and relations (a)-(e) below. (a) K0 = 1, Kµ Kµ′ = Kµ+µ′ , µ, µ′ ∈ Λ∨ . ′ (b) Kµ Ei = q hµ,i i Ei Kµ , i ∈ I, µ ∈ Λ∨ . ′ (c) Kµ Fi = q −hµ,i i Fi Kµ , i ∈ I, µ ∈ Λ∨ . ei − K e −i K (d) Ei Fj − Fj Ei = δi,j for i, j ∈ I, where qi − qi−1 X (e) e ±i = K (i·i) K i ± 2 ′ (p) ′ (p ) (−1)p Ei Ej Ei (p) def = 0 for all i 6= j ∈ I, where Ei = Eip /[p]i ! p+p′ =1−2i·j/(i·i) e def 2.1.4. Let Z = Y e lies in the center of Ǔ we define U = Ǔ[Z] where Z is e ni . Then Z K i i∈I ∨ e defined as a central element such that Z dh = Z. def 2.1.5. Let ΛC∗ be the tensor product ΛC∗ = Λ ⊗ C∗ and Λ ֒→ ΛC∗ be the imbedding induced by the imbedding Z ֒→ C∗ : n 7−→ q n . We extend an imbedding Z ֒→ C∗ to an imbedding d1 Z ֒→ C∗ in such a way that 1 Λ d ֒→ ΛC∗ . 1 d ∨ 7−→ qeh . This imbedding defines an imbedding We denote the group structure on ΛC∗ as +. The pairing h , i : Λ∨ ×Λ → Z, (µ, λ) 7−→ hµ, λi defines the pairing d1 Λ∨ × ΛC∗ → C∗ which we will also denote by (µ, λ) 7−→ hµ, λi. For any representation (ρ, V ) of U and λ ∈ ΛC∗ we define V λ = {v ∈ V Kµ v = hµ, λiv} for all µ ∈ Λ∨ . Definition. We denote by C the category of representations (ρ, V ) of U such that V = M V λ . For any commutative ring B containing C we keep the notation C for the category λ∈ΛC∗ of UB -modules (where UB = U ⊗C B) obtained from C by changing the scalars. For any complex-valued function F on ΛC∗ and any V = (ρ, V ) in C we denote by F the linear endomorphism of V which preserves the direct sum decomposition V = ⊕λ∈ΛC∗ Vλ and such that 18 F |Vλ = F (λ)IdVλ for all λ ∈ ΛC∗ . def For any V = (ρ, V ) in C we define V ∗ as the direct sum V ∗ = ⊕λ∈ΛC∗ Hom(V λ , C). Then V ∗ has a natural structure of a U-module. We denote this U-module as V ∗ . By the definition V ∗ belongs to C. In this paper by an expression “a U-module” we will always understand “a U-module from the category C”. 2.1.6. For any z ∈ C∗ we denote by Cz the full subcategory of C of representations (ρ, V ) such that ρ(Z) = zIdV . The full subcategory of C1 consisting of finite-dimensional representations is denoted by D. def 2.1.7. Let ΛC∗ = Λ ⊗ C∗ and r : ΛC∗ → ′ ΛC∗ be the projection induced by the direct sum decomposition Λ = ′ Λ ⊕ Λ0 . By the definition hi, λi = 0 for all i ∈ I, λ ∈ ker r. Therefore ∨ we obtain a pairing ′ Λ × ΛC∗ → C∗ which we also denote as h , i. It is clear that this ∨ pairing coincides with the restriction of the pairing h , i : d1 Λ∨ × ΛC∗ → C∗ to ′ Λ × ΛC∗ . The imbedding Λ ֒→ ΛC∗ induces the imbedding Λ ֒→ ΛC∗ . We denote by S the torus ΛC∗ /Λ and by Π : ΛC∗ → S the natural projection of ΛC∗ to S. Definition. a) For any s ∈ S we denote by s C the full subcategory C of U-modules (ρ, V ) such that V λ = {0} for all λ ∈ ΛC∗ such that Π(r(λ)) 6= s. b) For any s ∈ S, z ∈ C∗ we define s Cz as the intersection of s C and Cz . Remark: In this paper, we will almost always assume that our U-modules are in the subcategory [0] def C = [ s C, s∈Π(′ Λ) The reason to consider the more general category C is to have a possibility to prove results for s C, where s is “generic” first and then to “deform” [0] C to −s C for “generic” s (see, for example, the proof of Theorem 3.2.2). 2.1.8. Let U0 ⊂ U be the subalgebra generated by Z and Kµ , µ ∈ Λ∨ and Uf0 ⊂ U0 its ∨ subalgebra generated by Kµ , µ ∈ Λ . For any ν ∈ N[I] we denote by Uν+ the subspace of U spanned by elements of the form x+ where x ∈ fν (see [L] 3.1.1). We define Un+ as a 19 direct sum M def Un+ = Uν+ ν∈N[I] trν=n ′ ≥1 > n and denote by U≥n + the direct sum ⊕n′ ≥n U+ . We denote U+ as U+ and define U+ ⊂ U f to be the subalgebra generated by U0 and U> + and U+ ⊂ U+ as the subalgebra generated f −1 by Uf0 and U> ]. + . Then U+ = U+ [Z, Z ∨ 2.1.9. We denote by U ⊂ U the subalgebra generated by Kµ , µ ∈ Λ and Ei , Fi for i ∈ I. ∨ Then U is the quantum algebra corresponding to the root datum (Λ , ′ Λ). 2.1.10. Let ∆ : U → U → U ⊗ U be the comultiplication such that e i ⊗ Ei , ∆(Ei ) = Ei ⊗ 1 + K i∈I ∆(Kµ )= Kµ ⊗ Kµ , µ ∈ Λ∨ i∈I e −i + 1 ⊗ Fi , ∆(Fi ) = Fi ⊗ K ∆(Z) = Z ⊗ Z (see [L] 23.1.5) . We define a co-unit ǫ on U as the unique homomorphism ǫ : U → C such that ǫ(Ei ) = ǫ(Fi ) = 0 i ∈ I, ǫ(Kµ ) = 1, µ ∈ Λ∨ and ǫ(Z) = 1. Then U is a Hopf algebra where the antipode S is given e −i Ei , S(Fi ) = −Fi K e i , S(Kµ ) = K−µ and S(Z) = Z −1 . by the formulas S(Ei ) = −K This comultiplication defines a strict monoidal structure on the category C. 2.1.11. Let ω be the involution of the algebra U such that ω(Ei ) = Fi , ω(Fi ) = Ei , i ∈ I, ω(Kµ ) = K−µ , µ ∈ Λ∨ , ω(Z) = Z −1 (see [L] 3.1.3). f) of U on the For any U-module (ρ, M ) we denote by ω M the representation (e ρ, M same space M such that ρe(x) = ρ(ω(x)). 2.1.12. Let F be a field containing C, UF = U ⊗C F . For any central invertible u ∈ UF we define two automorphisms ϕ and ψ of the Hopf algebra UF where 20 ∨ ∨ ϕu (Ei0 ) = udh Ei0 , ϕu (Fi0 ) = u−dh Fi0 , ϕu (Ei ) = Ei , ϕu (Fi ) = Fi , i ∈ I, ϕu (Kµ ) = Kµ , ψu (Ei ) = u d(i·i) 2 µ ∈ Λ∨ , ϕu (Z) = Z, Ei , i ∈ I, ψu (Fi ) = u− d(i·i) 2 Fi , i ∈ I, ψu (Kµ ) = Kµ , µ ∈ Λ∨ , ψu (Z) = Z. Remark: The automorphisms ϕu and ψu of UF define strict automorphisms of the category C which we denote as Tuϕ and Tu respectively. 2.1.13. Fix a point s ∈ S and an element a ∈ Π−1 (s) ⊂ ΛC∗ . For any central invertible element u ∈ UF we denote by Lau the function on Π−1 (s) with values in UF such that Lau (a) = 1 and Lau (λ + i′ ) = ud(i·i)/2 Lau (λ) for any λ ∈ Π−1 (s), i ∈ I. We extend Lau to ΛC∗ in such a way that Lau (λ) = 0 if λ ∈ / Π−1 (s), λ ∈ ΛC∗ and Lau is constant on the fibers of r. Definition. For any V = (ρ, V ) ∈ s C we denote by Lau a linear endomorphism of V corresponding to the function Lau (see 2.1.5). ′ ′′ Remark: For any two a′ , a′′ ∈ Π−1 (s) the operators Lua and Lau are proportional and the coefficient of proportionality does not depend on a choice of V in s C. Therefore we can consider Lau as a section of a line bundle Ľu on S. 2.1.14. Proposition. For any V = (ρ, V ) in s C the map Lau defines an isomorphism of the UF -module Tuϕ (V ) with the UF -module Tu (V ). Proof: We have to show that for any x ∈ UF , we have Lau ϕu (x) = ψu (x)Lau . This equality is obvious if x = Kµ , µ ∈ Λ∨ or x = Z. In the case when x = Ei or Fi , i ∈ I the claim follows immediately from the definition of Lau and the equalities x+ V λ ⊂ V λ+ν , x− V λ ⊂ V λ−ν for x ∈ fν . In the case when x = Ei0 or Fi0 the claim follows from the definition of the dual Coxeter number h∨ . Proposition 2.1.14 is proved. 2.1.15 Proposition. For any finite-dimensional X = (ρX , X) ∈ C the identity map X −→ X defines an isomorphism X ∗∗ −→ Te ∨ (X) of UF -modules. q −2h Proof: By the definition X ∗∗ = (ρX ∗∗ , X) where ρX ∗∗ (a) = ρX (S 2 (a)), a ∈ UF and S : UF → UF is the antipode. The claim follows now from the formulas for the antipode S in 2.1.10. 21 2.2 Sugawara operators. 2.2.1. For any V = (ρ, V ) in C and n ∈ N we define a subspace V (n) of V def V (n) = n o v ∈ V av = 0 ∀a ∈ U≥n . + It is clear that V (1) ⊂ V (2) ⊂ · · · ⊂ V (n) ⊂ · · ·. We define a subspace V (∞) of V as the union of all V (n), n > 0. Proposition. For any i ∈ I, n ∈ N we have (Ei )V (n) ⊂ V (n) and (Fi )V (n) ⊂ V (n + 1). Proof: The first part follows immediately from the definitions and the second follows from [L] 3.1.6. Corollary. V (∞) is a U-invariant subspace of V . We denote the corresponding U-module by V (∞). Let C + be the full subcategory of C of modules V such that V (∞) = V and let C − be the subcategory of C of U-modules W such that w W ∈ C + . For any W in C − we define D(W ) = W ∗ (∞) where W ∗ is as in 2.1.5. We define s C ± as s C ∩ C ± and Cz± as C ± ∩ Cz . Remark: For any V in C the submodule V (∞) ⊂ V is the maximal subobject of V contained in C + . 2.2.2. For any V in C + and W in C − we define a linear map HomU (V, D(W )) → HomU (V ⊗ W, C) as in [KL] 2.30. As follows from 1.1.6 we can identify the linear space HomU (V ⊗ W, C) with hV ⊗ W i∨ . Lemma. The map HomU (V, D(W )) → hV ⊗ W i∨ is an isomorphism. Proof: Analogous to the proof of Lemma 2.3.1 in [KL]. 2.2.3. Let Ω≤p be elements defined in [L] 6.1.1. Then for any M = (ρ, M) in C + there exists an operator Ω on M such that Ω(m) = Ω≤p (m) for any m ∈ M and all sufficiently e −i Ei Ω = K e i ΩEi , ΩFi = Fi K e i ΩK e i and ΩKµ = Kµ Ω, i ∈ I, µ ∈ Λ∨ . big p. We have K 22 2.2.4. Fix a point s ∈ S and an element a ∈ Π−1 (s) ⊂ ΛC∗ . Let Ga : Π−1 (s) → C∗ be the function such that Ga (a) = 1 and for any λ ∈ Π−1 (s), i ∈ I we have Ga (λ)Ga (λ − i′ )−1 = hi, λi(i·i) . We continue Ga to ΛC∗ in the same way as we did with Lau in 2.1.13. For any V = (ρ, V ) in s C we denote by Ga the corresponding endomorphism of V (see (2.1.5)). Remark: As in the case of Lau we can consider Ga as a section of a line bundle on S. Definition. For any s ∈ S, V = (ρV , V ) ∈ s C + and a ∈ Π−1 (s) we define linear endodef morphisms TVa (or simply T a ) of V as composition T a = La(Ze ΩGa . We will call all ∨ q h )−2 of them Sugawara operators. Proposition. T a ∈ HomU (V, T(Ze (V )). ∨ q h )−2 Proof: We have to check that for any x ∈ U we have T a x = ψ(Ze (x)T a . Let ∨ q h )−2 ′ def T a = Ω · Ga . It is sufficient to prove that α) ′ T a Kµ = Kµ ′ T a , ′ T a Z = Z ′ T a β) ′ T a Ei = Ei ′ T a , ′ T a Fi = Fi ′ T a for i ∈ I ∨ ∨ ∨ γ) ′ T a Ei0 = (Z qeh )−2dh Ei0 ′ T a , ′ T a Fi0 = (Z qeh )2dh Fi0 ′ T a . The equalities α) are obviously true. The proof of equalities β) is completely analogous to the proof of Proposition 6.1.7 in [L]. So we give only the proof of equalities γ). Actually ∨ ∨ we only give a proof of the equality ′ T a Ei0 = (Z qeh )2dh Ei0 ′ T a . The proof of the second part of γ) is completely analogous. def If v ∈ V λ then Ei0 v ∈ V λ+i′0 . Let ν = X ni i ∈ Λ. Since λ + i′0 = λ − ν ′ we have i∈I ′ −(i0 ·i0 ) T a Ei0 v = Ga (λ − ν ′ )ΩEi0 v = Ga (λ − ν ′ )Ki0 Ei0 Ωv = Y a ′ e−2 e 2ni Ei Ωv = K = Ga (λ − ν ′ )Ki−2 E Ωv = G (λ − ν ) Z i 0 0 i 0 i∈I = Z −2dh ∨ Y e 2ni Ei ′ T a v = (Z qeh∨ )−2dh∨ Ei ′ T a v. Ga (λ − ν ′ )Ga (λ)−1 K 0 0 i i∈I Proposition 2.2.4 is proved. 23 2.2.5. In the case when V = (ρV , V ) lies in [0] C we can give a more explicit formula for the operators Lu and G on V . Let Lu be the function on ′ Λ with values in the center of U and let G be the complex-valued function on ′ Λ such that ′ Lu (λ) = ud[λ,ρ ] ′ ′ ′ ′ and G(λ) = q ([λ+ρ ,λ+ρ ]−[ρ ,ρ ]) for λ ∈ ′ Λ where the bilinear form [ , ] is as 2.1.1. They define Lu and G in the obvious way. Let (M ⊗ N )0 ⊂ M ⊗ N be the subspace defined as in 1.1.6. 2.2.6. For any V = (ρV , V ) in s C − we define an endomorphism Ť a of V by the rule def Ť a = (TωaV )−1 , where the module ω V ⊂ s C + is defined as in 2.1.11. Remark: We can consider T a and Ť a as elements in appropriate completions of U. Then Ť a = ω(T a ). 2.2.7 Proposition. For any i ∈ I we have ∨ a) T a Ei = (Z qeh )−(i·i)d Ei T a , b) c) d) ∨ Ť a Ei = (Z qe−h )−(i·i)d Ei Ť a , ∨ T a Fi = (Z qeh )(i·i)d Fi T a , ∨ Ť a Fi = (Z qe−h )−(i·i)d Fi Ť a ,. Proof: Parts a) and c) are corollaries of Proposition 2.2.4. We prove b). The proof of d) is completely analogous. We have ∨ ∨ Ť a Ei = ω[((T a )−1 Fi )] = ω[(Z qeh )−(i·i)d Fi (T a )−1 ] = (Z qe−h )(i·i)d Ei Ť a , where the second equality is a restatements of c). Proposition 2.2 is proved. 2.2.8. For any V = (ρV , V ) in s C + and W = (ρW , W ) in −s − C the linear map T a ⊗ Ť a of V ⊗ W does not depend on a choice of a ∈ Π−1 (s) and we denote this operator as T ⊗ Ť . 24 Proposition. (T ⊗ Ť ) preserves the subspace (M ⊗ N )0 of M ⊗ N . Proof: It is sufficient to show that for any m ∈ M, n ∈ N , i ∈ I and any µ ∈ Λ∨ , we have a) (T ⊗ Ť )(Ei )(m ⊗ n) ∈ (M ⊗ N )0 , b) (T ⊗ Ť )(Fi )(m ⊗ n) ∈ (M ⊗ N )0 , c) (T ⊗ Ť )[(Kµ )(m ⊗ n) − (m ⊗ n)] ∈ (M ⊗ N ). Proof: of a): By the definition of the action of U on M ⊗ N we have e i m ⊗ Ť Ei n. (T ⊗ Ť )Ei (m ⊗ n) = T Ei m ⊗ Ť n + T K Therefore it follows from Proposition 2.2.7 that ∨ ∨ e i T m ⊗ (Z qe−h )d(i·i) Ei Ť n (T ⊗ Ť )Ei (m ⊗ n) = (Z qeh )−d(i·i) Ei T m ⊗ Ť n + K ∨ = qe−d(i·i)h [∆(Ei )(Z −d(i·i) T m ⊗ Ť n) − (Z −d(i·i) A ⊗ B − A ⊗ Z d(i·i) B) ∨ = qe−d(i·i)h ∆(Ei )(Z −d(i·i) T m ⊗ Ť n) − (∆(Z −d(i·i) ) − ǫ(Z −d(i·i) ))(A ⊗ Z d(i·i) B), e i T m and B = Ť n. where A = K The inclusion a) is proved. The proof of the inclusion is completely analogous and the proof of c) is obvious. Proposition 2.2.7 is proved. Remark: We will show later (see 3.2) that T ⊗ Ť induces the identity map on the quotient def hM, N i = M ⊗ N /(M ⊗ N )0 . We extend Lu and G to functions on ΛC∗ which are constant on fibers of r and are zero outside of r −1 (′ Λ) and denote by Lu and G the corresponding automorphism of V for any V = (ρ, V ) in [0] C. Then Lu and G are sections of the line bundles Ľu and Ǧ over the finite set Π(′ Λ) ⊂ S. For any V = (ρV , V ) in in [0] − C def [0] + C def ( = C + ∩ [0] C) and W = (ρW , W ) ( = C − ∩ [0] C) we denote by T ∈ End V and Ť ∈ End W the Sugawara operators corresponding to the functions Lu and G. 25 2.2.9 Proposition. Let M = (ρM , M) ∈ [0] Cz+ be a U-module, λ ∈ d1 Λ and let m ∈ M λ , be such that Ei m = 0 for all i ∈ I. Then ′ ′ ′ ′ ′ ′ ′ T m = q [[(λ+ρ ),(λ+ρ )]−[ρ ,ρ ]] z −2d[λ,ρ ] q −2[λ,ρ ] m = q [λ,λ] z −2d[λ,ρ ] m. Proof: Follows immediately from the definition of T . 2.2.10. Let M = (ρM , M ) be a U-module in [0] C. For any n ∈ N we denote by M(n) ⊂ M the subspace generated by vectors of the norm xm, x ∈ U>n − , m ∈ M . For any N = (ρN , N) in Cz− we define N(n) = (ω N )(n) ⊂ N where we identify the spaces N and ω N . We def def define Mn = M /M(n) , Nn = N /N(n) and denote by π the natural projections M → Mn and N → Nn . 2.2.11. For any p ∈ N, we define def T(p) = L(Ze Ω G, q h∨ )−2 ≤p where Ω≤p ∈ U is as in [L] 6.1.1. Proposition. a) For any M in [0] C and any pair m > n ∈ N we have T(m) M(n) ⊂ M(n) . b) The induced operator Tn on Mn does not depend on a choice of m > n. c) The system {Tn ∈ EndMn } is compatible with the natural projections Mn → Mn−1 . Proof: a) and b) follow from [L] 6.1.1, and c) is obvious. c def c → Mn the natural 2.2.12. For any M in C we define M = limMn and denote by π bn : M ← c(∞) ⊂ M c be the submodule as in 2.2.1. Since M c(∞) ∈ C + we can projection. Let M c define a linear transformation TM b ∈ End M . Proposition. π bn ◦ TM bn for all n ∈ N. b = Tn ◦ π c(∞) and n ∈ N such that U≥n m Proof: Fix m b ∈M + b = 0. We may assume that p > n. Let m ∈ M be such that πn (m) = πn (m). b We have π bn (TM b =π bn (T(p) m) b = πn (T(p) m) = b m) Tn πn (m) = Tn π bn (m). b Proposition 2.2.12 is proved. 26 2.3. The R-matrix. ′ ′′ ′ ′′ 2.3.1. Let Ξ be the complex-valued function on ′ Λ ×′ Λ such that Ξ(λ , λ ) = q −[λ ,λ ] . For any U-modules V ′ = (ρ′ , V ′ ), V ′′ = (ρ′′ , V ′′ ) in [0] C we denote by Ξ the endo- morphism of V ′ ⊗ V ′′ which preserves subspaces Vλ′′ ⊗ Vλ′′′′ for all λ′ , λ′′ ∈ ΛC∗ and such that ′ ′′ Ξ|V ′ ′ ⊗V ′′′′ = Ξ(λ , λ )Id, λ λ ′ ′′ r(λ′ ) and λ def r(λ′′ ). where λ def = = 2.3.2 Definition. We say that a pair V ′ , V ′′ , V ′ = (ρ′ , V ′ ) ∈ s′ C, V ′′ = (ρ′′ , V ′′ ) ∈ s′′ C, is admissible if for any v ′ ∈ V ′ , v ′′ ∈ V ′′ we have Θν (v ′′ ⊗ v ′ ) = 0 for almost all ν ∈ Z[I] def where Θν ∈ U(2) = U ⊗ U are as in [L], 4.1.1. If v ′ = (ρ′ , V ′ ), V ′′ = (ρ′′ , V ′′ ) is an admissible pair we denote by Θ an endomorphism of V ′′ ⊗ V ′ such that Θ(v ′′ ⊗ v ′ ) = X Θν (v ′′ ⊗ v ′ ) for all v ′ ∈ V ′ , v ′′ ∈ V ′′ . ν∈Z[I] Remark: If V ′ , V ′′ ∈ C are such that either V ′ ∈ C + or V ′′ ∈ C − then the pair (V ′ , V ′′ ) is admissible. 2.3.3. If V ′ , V ′′ is an admissible pair we define linear map ′ sV ′ ,V ′′ and sV ′ ,V ′′ from V ′ ⊗V ′′ to V ′′ ⊗ V ′ as compositions: ′ def sV ′ ,V ′′ = Θ ◦ Ξ ◦ σ, def sV ′ ,V ′′ = ′ sV ′ ,V ′′ ◦ (LZ −1 ⊗ LZ −1 ) where σ : V ′ ⊗ V ′′ → V ′′ ⊗ V ′ is the isomorphism such that σ(v ′ ⊗ v ′′ ) = v ′′ ⊗ v ′ for all v ′ ∈ V ′ , v ′′ ∈ V ′ . 2.3.4. Let ϕ(2) be the automorphism of U ⊗ U, such that  x⊗y         e−1 ⊗ Ei Z 0 (2) ϕ (x ⊗ y) = −1 e  Ei0 ⊗ Z     e ⊗ Fi  Z  0   e Fi0 ⊗ Z if x ⊗ y ∈ {Kλ ⊗ Kµ , Ei ⊗ 1, 1 ⊗ Ei , Fi ⊗ 1, 1 ⊗ Fi } for λ, µ ∈ Λ∨ , i ∈ I; if x ⊗ y = 1 ⊗ Ei0 ; if x ⊗ y = Ei0 ⊗ 1; if x ⊗ y = 1 ⊗ Fi0 ; if x ⊗ y = Fi0 ⊗ 1. 27 Proposition. If V ′ = (ρ′ , V ′ ), V ′′ = (ρ′′ , V ′′ ) ∈ [0] C is an admissible pair, then ′ sV ′ ,V ′′ ◦ ϕ(2) (∆(x)) = ∆(x) ◦ ′ sV ′ ,V ′′ for all x ∈ U. 2.3.5 Proof: As in [L] 32.1, we have to show that Θ · Ξϕ(2) (t ∆(x)) = ∆(x)ΘΞ when x runs through a system of generators of U. Let α be an automorphism of U(2) as in [L] 32.1. Then we have ∆(x)Θ = Θα(t ∆(x)) for all x ∈ U (see [L] 32.1). We have to show that ϕ(2) (t ∆(x)) = Ξ−1 α(t ∆(x))Ξ for a system of generators x in U. For any λ ∈ ′ Λ we define def V ′λ = M def V ′λ , V ′′λ = λ∈r−1 (λ) M V ′′λ . λ∈r−1 (λ) Then it is sufficient to prove the equality ϕ(2) (t ∆(x))(v ′′ ⊗ v ′ ) = Ξ−1 α(t ∆(x))Ξ(v ′′ ⊗ v ′ ) (*) for all v ′ ∈ V ′λ′ , v ′′ ∈ V ′′λ′′ , ′′ λ′ , λ ∈ ΛC∗ and a system of generators x of U. If x ∈ U0 , then the validity of (*) is obvious. If x = Ei or Fi for i ∈ I then the validity of (*) is proven in [L] 32.1. So it is sufficient to prove (*) in the case when x = Ei0 and x = Fi0 . We prove (*) in the case when x = Ei0 . The case x = Fi0 is completely analogous. 2.3.6. If we compute the right side of (*) for x = Ei0 we find that it is equal to ′ ′ e−1 v ′′ ⊗ Ei v ′ . On the other hand, q [i0 ,λ ] Ei0 v ′′ ⊗ v ′ + Z 0 e−1 ⊗Ei = Ei ⊗K −1 +Ze−1 ⊗Ei . ϕ(2) (t ∆(E0 )) = ϕ(2) (Ei0 ⊗Ki0 +1⊗Ei0 ) = Ei0 ⊗Ze−1 Ki0 +Z 0 0 ν 0 where ν = X ni (i·i) 2 i. and we see that the left side of (*) is equal to the right side. i∈I Proposition 2.3.4 is proved. 28 2.3.7 Corollary. Let V ′ , V ′′ be an admissible pair of U-modules such that V ′ ∈ Cz′ , V ′′ ∈ Cz′′ , z ′ , z ′′ ∈ C∗ . Then ′ sV ′ ,V ′′ : V ′ ⊗ V ′′ → V ′′ ⊗ V ′ is an U-module isomorphism between U-modules Tzϕ′′ −1 (V ′ ) ⊗ Tzϕ′ −1 (V ′′ ) and V ′′ ⊗ V ′ . 2.3.8. Let V ′ , V ′′ be as in 2.3.7. Corollary. The linear map sV ′ ,V ′′ defines a U-module isomorphism between U-modules Tz′′ −1 (V ′ ) ⊗ Tz′ −1 (V ′′ ) and V ′′ ⊗ V ′ . 2.3.9 Lemma. Let V ′ , V ′′ , V ∈ [0] C be such that the pairs (V, V ′ ), (V, V ′′ ) and (V, V ′ ⊗V ′′ ) are admissible. Then sV,V ′ ⊗V ′′ = (IdV ′ ⊗ sV,V ′′ ) ◦ (sV,V ′ ⊗ IdV ′′ ). Proof: Completely analogous to the proof of Proposition 32.2.4 in [L]. 2.4. Quantum algebras over C[t]. 2.4.1. Let A = C[t], F = C(t), An = A/tn A, n ∈ N. For any U-module M and n ∈ N we def define n M = M ⊗A An . Let U be as in 2.1.4, def UA = U ⊗C A, (2) def def UF = U ⊗C F, UA = UA ⊗A UA and n n U = U ⊗C An , (2) U(2) = UA ⊗A An . We will consider U as a subalgebra in UA and Un . The comultiplications ∆ : U → U ⊗ U (2) defines A-linear comultiplications UA → UA and n U → n U(2) which we also denote by ∆. 2.4.2. Let def def ψ = ψt be the automorphism of UF as in 2.1.12 and (2) Γ = {x ∈ UA |(1 ⊗ ψ)(∆(x)) ∈ UA }. As follows from [L], Proposition 3.2.4, we have a def direct sum decomposition U = ⊕ν Uν , where Uν = U+ (Uν− ). We consider the C-linear map η : U → UA such that η(u) = t|ν|d u for u ∈ Uν and extend it to an A-linear map n X (i ·i ) ηA : U ⊗C A → UA . Here |ν| = νp p2 p if ν = (ν1 , · · · , νn ). This is called a degree of ν. p=1 29 Proposition. a) Γ is a subalgebra of UA , b) Im η ⊂ Γ c) The map ηA : U ⊗C A → Γ is an isomorphism of A-modules. d) The algebra Γ is generated by U+ and tdi Fi , i ∈ I, where ti = t(i·i)/2 . e) ∆(Γ) ⊂ Γ ⊗ Γ. 2.4.3 Proof: Part a) follows immediately from the definition of Γ since (1 ⊗ ψ) ◦ ∆ is an algebra homomorphism from UF to UF ⊗ UF . Part b) follows from a) and the inclusions (2) (1 ⊗ ψ) ◦ ∆(U+ ) ⊂ (1 ⊗ ψ)(U+ ⊗ U+ ) ⊂ U+ ⊗ U+ ⊂ UA and e −i ) + td (1 ⊗ Fi )) = td (Fi ⊗ K e −i ) + 1 ⊗ Fi ∈ U(2) . (1 ⊗ ψ) ◦ ∆(tdi Fi ) = (1 ⊗ ψ)(tdi (Fi ⊗ K i i A 2.4.4. To prove part c) we choose a basis B in f consisting of homogeneous elements and containing 1 (see [L] 3.2.4). For any b ∈ B we denote by |b| its degree, i.e., |b| = |ν| if b ∈ fν . As follows from Proposition 3.2.4 in [L] we can write x ∈ Γ as a sum x = P ′ + − |b| ′ ′ ′ b′ ,µ,b cb ,µ,b b Kµ b with cb ,µ,b ∈ A. We have to show that cb ,µ,b ∈ t A. Since the map ∆ : U → U ⊗ U is a monomorphism, we see that the inclusion ′ (1 ⊗ ψ) ◦ ∆(x) ∈ UA ⊗ UA implies the inclusion cb′ ,µ,b (1 ⊗ ψ)∆(b + Kµ b− ) ∈ UA ⊗ UA for ′ all b, b′ ∈ B, µ ∈ Λ∨ . Therefore we may assume that x = ab + Kµ b− for b′ , b ∈ B, µ ∈ Λ∨ and a ∈ A. (2) 2.4.5. Assume that a ∈ / t|b| A. We want to show that (1 ⊗ ψ) (∆(x)) ∈ / UA . Choose def n ∈ N such that a ∈ tn A − tn+1 A and define x e = t|b|−n x. Let − (2) : UA → U ⊗ U be the (2) natural projection (= reduction mod t). As follows from part b), (1 ⊗ ψ)∆(e x) ∈ UA . Let x be the image of this element in U ⊗ U. Since |b| − n > 0 it is sufficient to show that x 6= 0. We have ′ (1 ⊗ ψ) ◦ ∆(e x) = t|b|−n a(1 ⊗ ψ)∆(b + )(∆(Kµ ))(1 ⊗ ψ) · (∆(b− )). Therefore it follows from formulas in 2.1.10 and the definition of ψ that ′ x = ab + Kµ ⊗ Kµ b− 30 where a ∈ C is the reduction of at−n ∈ A mod t. By the definition of n, we have a 6= 0. Therefore, x 6= 0. Part c) of Proposition 2.4.2 is proved. Part d) follows immediately from c) and part e) follows from d) and explicit formulas for ∆ (see 2.1.12). Proposition 2.4.2 is proved. (2) 2.4.6 Corollary. Γ = {x ∈ UA |(ψ ⊗ 1)(∆(x)) ∈ UA }. (2) Proof: Let Γ′ = {x ∈ UA |(ψ ⊗ 1)∆(x) ∈ UA }. It follows from Proposition 2.4.2 that Γ ⊂ Γ′ . An analogous argument shows that Γ′ ⊂ Γ. So Γ = Γ′ . 2.4.7. The map ∆ : U0 → U0 ⊗ U0 ⊂ U ⊗ U defines an imbedding of U0 in U ⊗ U. Since U0 ⊂ Γ and (1 ⊗ ψ) ◦ ∆|U0 = ∆|U0 we can consider U0 as a subalgebra of Γ. Therefore (2) (2) the imbeddings (1 ⊗ ψ) ◦ ∆ : Γ ֒→ UA and U− ⊗ U+ ֒→ UA define an A-linear map e (2) → U(2) , where U e (2) def α:U A A A = Γ ⊗U0 (U− ⊗ U+ ). Analogously for any n ∈ N we can e (2) → define an An -linear map αn : n U n def e (2) n e (2) U(2) , where n U(2) = U A /t UA . Theorem. The map α is an isomorphism. 2.4.8 Proof: We start the proof with the following general result. Lemma. Let M, N be free An -modules, let α : M → N be a morphism such that the induced map α : M/tM → N/tN is an isomorphism. Then α is an isomorphism. Proof of Lemma: Follows from Nakayama’s lemma. 2.4.9 Proposition. The maps αn : n ∈ N are isomorphisms. e (2) = Γ1 (U− ⊗ U+ ) Proof: Consider first the case n = 1. We have 1 U(2) = U ⊗ U, 1 U where the subalgebra Γ1 ⊂ U ⊗ U is generated by elements x+ ⊗ 1, 1 ⊗ x′− and Kµ ⊗ Kµ for x, x′ ∈ f and µ ∈ Λ∨ . Therefore in the case when n = 1 Proposition 2.4.9 follows from the triangular decomposition for U (see [L] 3.2). The general case follows now from Lemma 2.4.8. Proposition 2.4.9 is proved. 2.4.10 Corollary. The map α is a monomorphism. e (2) be such that α(x) = 0. It follows from Proposition 2.4.9 that the Proof: Let x ∈ U A e (2) is equal to zero for all n ∈ N. Therefore x is divisible in U e (2) by tn image of x in tn U A A e (2) is free. Therefore x = 0. The Corollary is proved. for all n ∈ N. But the A-module U A 31 2.4.11. For any m ∈ N we denote by U(2) (m) the subspace of U ⊗ U spanned by elements ′ of the forms x+ Kµ x′− ⊗ y − Kµ′ y + , where µ, µ′ ∈ Λ∨ and x, x′ , y, y ′ are homogeneous (2) (2) elements such that |x| + |y| ≤ m and we define UA (m) = U(2) (m) ⊗C A ⊂ UA . (2) Lemma. For all m ∈ N we have UA (m) ⊂ Im(α). 2.4.12 Proof: We will prove Lemma 2.4.11 by the induction in m. If m = 0, then the result follows from the inclusion U(2) (0) ⊂ U− ⊗ U+ . Assume that the lemma is true for m − 1. It is sufficient to show that for any µ, µ′ , x, x′ , y, y ′ as in 2.4.11 there exists γ ∈ Γ ′ and u e ∈ U− ⊗ U+ such that x+ Kµ x − ⊗ y − Kµ y ′ + (2) − (1 ⊗ ψ)∆(γ)e u ∈ UA (m − 1). But we ′ ′ can take γ = t|y| x+ y − and u e = K−δ Kµ x − ⊗ K−δ′ Kµ′ y + , where ∆(x+ ) = x+ ⊗ Kδ′ + · · · , ∆(y − ) = Kδ ⊗ y − + · · · . Lemma 2.4.12 is proved. 2.4.13. Now we can finish the proof of Theorem 2.4.7. It follows from the triangular S (2) (2) decomposition that UA = m UA (m). Therefore Lemma 2.4.11 implies the surjectivity of α. On the other hand, the injectivity of α follows from Corollary 2.4.10. Theorem 2.4.7 is proved. 2.4.14. We will use a following version of Theorem 2.4.7. As follows from Corollary 2.4.6 (2) (2) we have (ψ ⊗ 1)∆(γ) ∈ UA for all γ ∈ Γ. Let β : Γ ⊗U0 (U+ ⊗ U− ) → UA be the morphism such that β(γ ⊗ u) = (ψ ⊗ 1)∆(γ)u for all γ ∈ Γ and u ∈ U+ ⊗ U− . theorem. β is an isomorphism of A-modules. 2.4.15 Corollary. Let β (1) be the linear map from U ⊗ U+ ⊗ U− to U ⊗ U such that β (1) (x ⊗ u) = ∆(x)u. Then β (1) is an isomorphism of linear spaces. Proof: Let ev 1 : A → C be the morphism of evaluation at t = 1. Since β (1) = β ⊗A C, where A acts on C by ev 1 . Theorem 2.4.14 implies the validity of the corollary. 2.4.16. As follows from part e) of Proposition 2.4.2 Γ has a natural structure of a Hopf A-algebra. Let Γ0 ⊂ Γ be the kernel of the counit ǫA : UA → A to Γ. We denote by (2) (2) (2) U0 ⊂ UA the span of elements of the form (ψ ⊗ 1)(∆(γ))(x), γ ∈ Γ0 , x ∈ UA . (2) Proposition. For any x ∈ fν there exists y, z ∈ U0 (2) and (x− ⊗ 1) − z ∈ t|ν| UA . 32 (2) such that (1 ⊗ x+ ) − y ∈ t|ν| UA 2.4.17 Proof: We prove the first part of the Proposition. The proof of the second part is completely analogous. Since U+ is a Hopf algebra, it follows from Lemma 1.3.3 that there exists ar ∈ U+ , P br ∈ U+ ⊗U+ , 1 ≤ r ≤ R such that ǫ(ar ) = 0 and S −1 (x+ )⊗1−1⊗x+ = 1≤r≤R ∆(ar )br , where S ∈ Isom(U+ , Uop + ) is the antipode. Therefore ψ(S −1 (x+ )) ⊗ 1 − 1 ⊗ x+ = X (ψ ⊗ 1) ◦ ∆(ar ) · (ψ ⊗ 1)br . 1≤r≤R (2) Since ψ(U+ ) ⊂ U+ ⊗ A, we see that ar ∈ Γ and (ψ ⊗ 1)br ∈ UA for all r, 1 ≤ r ≤ R. On the other hand, the inclusion x ∈ (U+ )ν implies that S −1 (x) ∈ (U+ )ν and therefore ψ(S −1 (x+ )) = t|ν| S −1 (x+ ). Proposition 2.4.16 is proved. def 2.4.18. For any U-module M = (ρ, M) the A-module M A = M ⊗C A has a natural UA module structure. Therefore two imbeddings id and ψ of Γ into UA define two Γ-module structures on M A . We denote the first one as M [t]0 (or simply M if this does not create def def confusion) and the second as M [t]. We define n M = M [t]0 ⊗A An and n M [t] = M [t]⊗A An . For any M ′ = (ρ′ , M ′ ), M ′′ = (ρ′′ , M ′′ ) in C we denote by n (M ′ [t]⊗M ′′ ) the Γ-module which is the tensor product of n M ′ [t] and n M ′′ over An . Corresponding representation is denoted by n (ρM ′ [t] ⊗ ρM ′′ ). def (2) (2) 2.4.19. For any ν ∈ N[I] we define Θψ are ν = (1 ⊗ ψ)(Θν ) ∈ UA where Θν ∈ U (2) d|ν| as in 2.3.2. Then Θψ UA for all ν ∈ N[I]. Therefore for any M ′ = (ρM ′ , M ′ ), ν ∈ t M ′′ = (ρM ′′ , M ′′ ) in [0] C and any n ∈ N we have n (ρM ′′ ⊗ ρM ′ )(Θψ ν ) = 0 for almost all ν ∈ N[I] and we can define an An -linear map n sM ′ [t],M ′′ from M ′ ⊗C M ′′ ⊗C An to M ′′ ⊗C M ′ ⊗C An as a finite sum n def sM ′ [t],M ′′ = X n (ρM ′′ ⊗ ρM ′ )(Θψ ν )Ξσ ◦ (LZ −1 ⊗ LZ −1 ) ν∈N[I] where Ξ, σ and LZ −1 are as in 2.3.3. 2.4.20 Proposition. For any M ′ , M ′′ ∈ [0] C such that M ∈ Cz′ M ′′ ∈ Cz′′ , z ′ , z ′′ ∈ C∗ and any n ∈ N the map n sM ′ [t],M ′′ is a Γ-module morphism from n (Tz′′ −1 (M ′ )[t] ⊗ Tz′ −1 (M ′′ )) to n (M ′′ ⊗ M ′ [t]). Proof: Clear. 33 2.4.21 Proposition. For any M ′ , M ′′ , M in n [0] C and n ∈ N we have sM [t],M ′ ⊗M ′′ = (IdM ′ ⊗ n sM [t],M ′′ ) ◦ (n sM [t],M ′ ⊗ IdM ′′ ). Proof: Analogous to the proof of Proposition 32.2.4 in [L]. 2.4.22. For any M, N in C and u ∈ C∗ we denote by u e the automorphism of A such that u e(t) = ut and by u b=u bM,N the C-linear endomorphism of the space M ⊗C N ⊗C A such b is a that u b(m ⊗ n ⊗ a) = m ⊗ n ⊗ u e(a) for all m ∈ M , n ∈ N , a ∈ A. It is clear that u u e-linear automorphism of M ⊗C N ⊗ A. Proposition. For any M, N in C and u ∈ C∗ we have u b(M [t]⊗N )(0) ⊂ (Tu (M )[t]⊗N )(0) , where the subspaces (M [t] ⊗ N )(0) and (Tu (M )[t] ⊗ N )(0) of M ⊗ N ⊗ A are as in 1.1.6. Proof: Follows immediately from the definitions. Corollary. The u e-morphism u b defines a u e-linear isomorphism between A-modules ∼ hM [t] ⊗ N i −→ hTu (M )[t] ⊗ N i, where hV i denote hV iΓ . Proof: Follows from Proposition 2.4.22. We denote the induced isomorphisms from hM [t] ⊗ N i to hTu (M )[t] ⊗ N i also by u b. 34 2.5 Γ-coinvariants. 2.5.1. For any Γ-module M we denote by hM iΓ (or simply hM i) the A-module of Γcoinvariants (see 1.1.6). 2.5.2. Let M = (ρM , M) be a U+ -module, N = (ρN , N ) be a U− -module and X = (ρX , X) a U-module. Then M ⊗ X ⊗ N has a natural structure of a U0 -module and we denote by hM ⊗ X ⊗ N iU0 the space of U0 -coinvariants of M ⊗ X ⊗ N . def def Let M = U ⊗U+ M and N = U ⊗U− N be the induced U-modules and let j : (M⊗N ) → M ⊗N ⊂ M [t]⊗N be the natural imbedding, that is, j(m⊗n) = (1⊗m)⊗(1⊗n) for m ∈ M, n ∈ N . Then j is a morphism of U0 -modules and it induces a morphism of A-modules j : hM ⊗ N iU0 ⊗ A −→ hM [t] ⊗ N i. Proposition. The map j is an isomorphism. Proof: Follows from Theorem 2.4.15. Corollary. For any n ∈ N the natural map h(M [t] ⊗ N ) ⊗A An i → hn (M [t] ⊗ N )i is an isomorphism. Proof: Clear. 2.5.3 Lemma. The map hM ⊗ N iU0 −→ hM ⊗ N iU induced by the imbedding j is an isomorphism. Proof: Using the same arguments as in the proof of Corollary 2.4.15 we deduce Lemma 2.5.3 from Proposition 2.5.2. 2.5.4. Let V, W be U-modules such that there exist exact sequences M1 → M0 → V → 0 and N1 → N0 → W → 0 of U-modules such that Mi = U ⊗U+ Mi , Ni = U ⊗U− Ni , i = 0, 1, where Mi (resp. Ni ) are U+ (resp. U− ) modules. 35 Proposition. For any n ∈ N the natural An -morphism hV [t] ⊗ W i ⊗A An −→ hn (V [t] ⊗ W )i is an isomorphism. Proof: The functor ⊗C is exact and therefore the sequence of (M1 [t] ⊗ N0 ) ⊕ (M0 [t] ⊗ N1 ) → M0 [t] ⊗ N0 → V [t] ⊗ W → 0 (*) of Γ-modules is exact. Since h , i is a right exact functor we see that the sequence hM1 [t] ⊗ N0 i ⊕ hM0 [t] ⊗ N1 i −→ hM0 [t] ⊗ N0 i −→ hV [t] ⊗ W i −→ 0 is exact. Analogously one shows that for any n ∈ N the sequence hn (M1 [t] ⊗ N0 )i ⊕ hn (M0 [t] ⊗ N1 )i −→ hn (M0 [t] ⊗ N0 )i −→ hn (V [t] ⊗ W )i −→ 0 is also exact. Therefore Proposition 2.5.4 follows from Corollary 2.5.2 and the five- homomorphism lemma. 2.5.5. Assume that V, W are U-modules as in 2.5.4 and dim Mk < ∞, dim Nk < ∞ for k = 0, 1. Lemma. hV [t] ⊗ W i is a finitely generated A-module. Proof: Follows from the exactness of the sequence (*) and Proposition 2.5.2. 2.5.6. Let D be the rigid category of finite-dimensional U-modules X such that Z acts trivially on X. Then D is a subcategory of [0] C. Given z ∈ C∗ , V ∈ [0] Cz+ , X ∈ D and W ∈ [0] Cz−−1 we denote by n δV,X,W the An -linear map from hn ((V ⊗ X)[t] ⊗ W )i to hn ((V ⊗ Tz2 e ∨ (X)[t] ⊗ W )i defined as the composition q 2h hn ((V ⊗ X)[t] ⊗ W )i sV,X ⊗idW −→ hn ((Tz (X) ⊗ V )[t] ⊗ W )i = β = hn ((Tz (X)[t]) ⊗An n V [t] ⊗An n WA i −→ idn V [t] ⊗s−1 T (X),W e −→ ∨ z q 2h −→ hn V [t] ⊗An n WA ⊗An n (Tze ∨ (X)[t])i q 2h hn ((V [t] ⊗An n (T (X)[t]) ⊗An n WA i = hn ((V ⊗ T (X))[t] ⊗ W )i, def n V [t]⊗ An where β = βn Tz (X)[t] n WA def is as in 1.1.15 and T = T(ze . ∨ q h )2 (3) Remark: The map n δV,X,W is an analog of the map δa 36 in 1.2.7. 2.5.7. We have assumed that W lies in [0] − Cz−1 . Then the composition (T −1 ⊗ id ⊗ Ť −1 ) ◦ n δV,X,W defines a Γ-module morphism ϕV,X,W from hn (V ⊗ X)[t] ⊗ W i to hn ((T (V ) ⊗ n ((T (X))[t] ⊗ T (W ))i = hT (n ((V ⊗ X)[t] ⊗ W ))i. As follows from 1.1.3 we can identify the An -module hT (n ((V ⊗ X)[t] ⊗ W ))i with hn ((V ⊗ X)[t] ⊗ W )i. Therefore we can consider ϕV,X,W as an endomorphism of the An -module hn ((V ⊗ X)[t] ⊗ W i. Theorem. ϕV,X,W = id. We will prove this theorem at the end of section 3.2. 37 §3. The category of smooth representation 3.1 The category Oz+ . 3.1.1 Definition. A finite dimensional Uf+ -module N is called a nil-module if there exists n ∈ N such that the U≥n + N = 0 and N = ⊕λ∈ΛC∗ Nλ (see 2.1.5). For any nil-module N and a number z ∈ C∗ we extend the action of Uf+ on N to an action of U+ on N in such a way that Z acts as zId and define def N z = U ⊗U+ N . The map n → 1⊗n defines a U0 -covariant imbedding N ֒→ N z and we will always consider N as a U0 -submodule of N z . 3.1.2 Definition. N z is called a generalized Verma module. Remark: N z lies in Cz+ . For any a ∈ ΛC∗ we denote by Va the one-dimensional representation of Uf+ such that ∨ Kµ acts as a multiplication by hµ, ai for all µ ∈ Λ and U> + acting as zero. In this case the U-module N z is denoted by Vaz ; it is called a Verma module. 3.1.3 Proposition. a) For any generalized Verma module N z there exists a finite filtration by submodules such that the successive quotients are Verma modules. b) Let V be an object in Cz+ . Then V is a quotient of a generalized Verma module if and only if there exists n ≥ 1 such that dim V (n) < ∞ and V (n) generates V as a U-module. c) Any object V in C + is a union of subobjects which are isomorphic to quotients of generalized Verma modules‘. Proof: Analogous to the proof of Proposition 2.5 in [KL]. 3.1.4. The following result concerns the action of the Sugarawa operator T a on a Verma module Vaz (see 2.2.4). For any ℓ ∈ C∗ we set ℓ Vaz = {v ∈ Vaz T a (v) = ℓv}. 38 Proposition. M z a) Vaz = ℓ Va . ℓ∈C∗ h∨ b) If |e q z| = 6 1, then ℓ Vaz is a finite-dimensional vector space, for all ℓ ∈ C. c) If ℓ Vaz 6= 0, then ℓ = (z d q)2n for some n ∈ N. d) 1 Vaz = Va . e) Vaz = Vaz (∞). f) The U-module Vaz has unique irreducible quotient Lza . ∨ g) If |e q h z| > 1 then Vaz has finite length for any a ∈ ΛC∗ . Proof: The proof of a) - f) is analogous to the proofs of Propositions 2.7 - 2.9 in [KL] when one uses Proposition 2.2.7. In order to prove g) one should observe that if Vaz′ → Vaz ∨ is a non-trivial homomorphism of U-modules then La h∨ −2 (a′ )Ga (a′ ) = (ze q h )2nd for (ze q ) n X such n ∈ N that a − a′ = i′p . Under our assumptions the LHS is bounded and the RHS p=1 increases when n → ∞. This implies that there are finitely many such a′ for a given a. Then one can finish the proof along the lines of [KL], 2.22. Definition. a) Complex numbers z1 , · · · , zn ∈ C∗ are multiplicatively independent if for any ~r = (r1 , · · · , rn ) ∈ Zn , ~r 6= 0 we have z1r · · · znrn 6= 1. b) A pair (s, z), s ∈ S, z ∈ C∗ is generic if for any a ∈ Π−1 (s) the complex numbers def a(i) = hi, ai, i ∈ I, z and q are multiplicatively independent. c) A nil-module N is z-generic if for any λ ∈ ΛC∗ such that Nλ 6= {0} the pair (λ, z) is generic. 3.1.6 Proposition. a) If a pair (s, z) ∈ S × C∗ is generic, then the Verma module Vaz is irreducible for a ∈ Π−1 (s). b) If a nil-module N is z-generic then the corresponding generalized Verma module N z is a direct sum of Verma modules. Proof: Follows from the same arguments as Propositions 9.9 and 9.10 in [K]. 3.1.7 Definition. We denote by Q≥0 ⊂ C∗ the set of numbers such that |z| ≤ 1. 39 ∨ In the remainder of this paper we assume z ∈ C∗ is such that qeh z ∈ / Q≥0 . 3.1.8 Proposition. Let V be an object in Cz+ . The following conditions are equivalent: a) There exists a finite composition series of V with subquotients of the form Lza for various a ∈ NI . b) V is a quotient of a generalized Verma module. c) There exists n ≥ 1 such that V (n) generates V as a U-module and dimV (n) < ∞. d) dimV (1) < ∞. Proof: Analogous to the proof of Theorems 2.22 and 3.2 in [KL]. 3.1.9 Definition. Oz+ is the full subcategory of Cz+ consisting of modules satisfying the conditions of Proposition 3.1.8. 3.1.10 Corollary. Any object in Cz+ can be represented as an inductive limit of objects from Oz+ . 3.1.11. In this subsection A denotes the ring of regular functions on ΛC∗ × C∗ , AU def = U ⊗C A, A U+ = U+ ⊗C A, etc. We denote by Ȧ the A U+ -module which is isomor- phic to A as an A-module and such that A U> + acts trivially, Kµ acts as a multiplication by the function hµ, λi if µ ∈ Λ∨ 0 and Z acts as a multiplication by z, where (λ, z) are natural coordinates on ΛC∗ × C∗ . We denote by V the induced A U-module V = A U ⊗A U+ Ȧ. 3.1.12. For any z0 ∈ C∗ we define by evz0 : A → C∗ the homomorphism of the evaluation at the point (0, z0 ) ∈ ΛC∗ × C∗ . This homomorphism defines an algebra homomorphism evz0 : AU → U. Given a Uf+ -nil-module N and z0 ∈ C∗ we can use evz0 to define a structure of A U+ -module a tensor product AN on N . We denote by = N ⊗C V. Let A Ň AN the A U+ -module be the induced module A Ň which is defined as def = AU ⊗A U+ AN . For any point (λ, z) ∈ ΛC∗ × C∗ we denote by mλ,z ⊂ A the maximal ideal of functions equal to zero at (λ, z). Define (λ,z) Ň def = A Ň /mλ,z . of a U-module. 40 Then (λ,z) Ň has a natural structure 3.1.13 Proposition. a) A Ň is a free A-module. b) For all (λ, z) ∈ ΛC∗ × C∗ the U-module obtained from the nil-module (λ,z) N def = (λ,z) Ň is a generalized Verma module A N /mλ,zA N . c) For almost all (λ, z) ∈ ΛC∗ ×C∗ the module (λ,z) Ň is a direct sum of Verma modules. Proof: a) and b) follow from definitions and c) follows from Proposition 3.1.6 b). 3.1.14 Definition. We denote by Oz− ⊂ Cz−−1 the category of U-modules M such that ω M lies in Oz+ . def ω For any nil-module N and z ∈ C∗ we define N−z = (N z ). Then N−z lies in C − and, as before, we have a nature imbedding ω N ֒→ N−z of U0 -modules where we identify ω N with N as a vector space and the action of U0 on ω N is given by the map (x, n) → ω(x)n, x ∈ U0 , n ∈ N . 3.1.15. For any Hopf algebra H and two H-modules M = (ρM , M ) and N = (ρN , N) we def define hM, N iH = hM ⊗ N i (see 1.1.6) and denote by prM,N (or simply pr) the natural projection pr : M ⊗ N → hM, N iH . i 3.1.16. Let M, N be nil-modules z ∈ C∗ and M ⊗ ω N ֒→ Mz ⊗ N−z be the natural imbedding. Then we have a linear map pr ◦ i : (M ⊗ ω N ) −→ hMz , N−z iU which factorizes through the map i : hM, ω N iU0 −→ hMz , N−z iU . Proposition. The map i is an isomorphism. Proof: Using Theorem 2.4.15 one can immediately apply the arguments of the proof of Proposition 9.15 in [KL]. 3.2 The action of Sugawara operators on coinvariants. 3.2.1. In this section we prove Theorem 2.5.7. We start with the special case when X = 11. So V = (ρV , V ), W = (ρW , W ) be U-modules such that V ∈ Cz+ and W ∈ Cz−−1 . Let T ⊗ Ť ∈ End(V ⊗ W ) be the linear map as in 2.2.8. As follows from Proposition 2.2.8 the T ⊗ Ť induces an endomorphism of the space hV, W i which we denote as ϕV,W . 41 3.2.2 Theorem. ϕV,W = id. Proof of Theorem 3.2.2: As follows from Proposition 3.1.3 c) it is sufficient to consider the case when V and W are generalized Verma modules, V = Mz , W = N−w where M and N are nil-modules. It is clear that hV, W i = {0} if z 6= w. We start with the following special case. 3.2.3 Proposition. Theorem 3.2.2 is true in the case when V and W are Verma modules. Proof of Proposition 3.2.3: Let V = Vaz , W = ω (Vbz ). As follows from Proposition 3.1.16 the map i : hVa ⊗ ω (Vb )iU0 −→ hV, W iU is an isomorphism. On the other hand, it follows from Proposition 2.2.9 that the operator T ⊗ Ť = T a ⊗ Ť a preserves the subspace Ca ⊗ ω (Cb ) ⊂ M ⊗ N and acts trivially on this subspace. Proposition 3.2.3 is proved. 3.2.4. Consider now the case when V and W are arbitrary generalized Verma modules. Let M, N be nil-modules, z0 ∈ C∗ and A M̌, A Ň be modules as in 3.1.12. As follows from Proposition 2.2.8, the operator T ⊗ Ť defines an endomorphism of the A-module def hM, N −i = hA M̌ ⊗A A Ň − i. We denote this endomorphism of hM, N i by Φ. For any (λ, z) ∈ ΛC∗ × C∗ we denote by Φλ,z the natural morphism Φλ,z : hM, N −i → h(λ,z) M, (λ,z) N − i. 3.2.5 Lemma. a) The A-module hM, N − i is free as an A-module, b) Φλ,z is surjective for all λ ∈ ΛC∗ , z ∈ C∗ , c) the kernel of Φλ,z is equal to mλ,z hM, N − i. Proof: Follows from Propositions 3.1.13 and 3.1.16. Now we can finish the proof of Theorem 3.2.2 in the case when V and W are generalized Verma modules. Really since the morphism Φ0,1 is surjective it is sufficient to prove that Φ = Id. On the other hand, since the A-module hM, N −i is free, it is sufficient to show that the induced endomorphism Φλ,z on h(λ,z) M, (λ,z) N i is equal to Id for generic (λ, z) . But this follows from Propositions 3.1.13 c) and 3.2.3. Theorem 3.2.2 is proved. 3.2.6 Proposition. Theorem 2.5.7 is true in the case when V and W are Verma modules. 42 Proof: We have V = Vaz , W = ω (Vbz ) for some a, b ∈ ′ Λ. As follows from Theorem 2.4.7 the natural imbedding Va ⊗ X ⊗ ω (Vb ) ֒→ (V ⊗ X)[t] ⊗ W defines an isomorphism ∼ hVa ⊗ X ⊗ ω (Vb )iU0 ⊗C An −→ hn ((V ⊗ X)[t] ⊗ W )i (cf. 2.5.2). So it is sufficient to show that for any x ∈ X b−a we have ϕV,X,W (1a ⊗ x ⊗ 1b ) = 1a ⊗ x ⊗ 1b , where 1a , 1b are generators of 1-dimensional spaces Va and Vb and we identify (Va ⊗ X ⊗ ω (Vb ))U0 with its image in hVa ⊗ X ⊗ ω (Vb )i. But this follows immediately from the definitions. Proposition 3.2.6 is proved. 3.2.7. The same arguments as in 3.2.4-3.2.5 show that Theorem 2.5.7 follows from Proposition 3.2.6. Theorem 2.5.7 is proved. 3.3 Completions. 3.3.1. We assume until the end of the section all our infinite-dimensional modules are in [0] C. It is easy to see that for any N in tensor product N ⊗ V lies in [0] [0] C and any finite dimensional U-module V the C. ∨ 3.3.2. We fix until the end of this section a number z such that zq h ∈ C∗ − Q≥0 . For any M = (ρ, M) in Cz we define the spaces Mn and M (n) as in 2.2.10. 3.3.3 Proposition. If N z is a generalized Verma module, V is a finite-dimensional representation of U and M = N z ⊗ V then the natural morphism N ⊗ V −→ M1 is an isomorphism. Proof: Follows from 2.4.15. 3.3.4. We denote by Ez ⊂ Cz be the full subcategory of modules M such that dim M1 < ∞. 3.3.5 Proposition. a) For any M in Ez and any n ∈ N we have dim Mn < ∞. b) For any N in Oz+ and any V in D the tensor product N ⊗ V lies in Ez . Proof: a) The proof is completely analogous to one in §§7.6-7.7 of [KL]. b) Follows from Proposition 3.3.3. 43 3.3.6. For n ∈ N we denote by Ln ⊂ C∗ the set of eigenvalues of Tn on M (n−1) , where M (n−1) ⊂ Mn is the image M(n−1) . Proposition. For any M in Ez and n ∈ N we have Ln+1 ⊂ [ q1k L1 , n≤k≤3n def ∨ where q1 = (ze q h )2d = (z d q)2 . 3.3.7 Proof: We prove the result by induction in n. If n = 0 then there is nothing to prove. Assume that we know the Proposition for n = n0 and we prove it for n = n0 + 1. Any element in M(n−1) is a linear combination of elements of the form Fi m, i ∈ I, m ∈ M(n−2) . Let m be the image of m in M (n−1) . We may assume that m is a generalized eigenvector of Tn−1 on M (n−2) with an eigenvalue equal to λ. It follows then from Proposition 2.2.7 that the image of Fi m in M (n−1) is a generalized eigenvector for S (ii)/2 (ii)/2 Tn with the eigenvalue λ · q1 . The inclusion λ · q1 ∈ n≤k≤3n q1k L1 follows now from the inductive assumption and the inclusion (ii)/2 ∈ {1, 2, 3}. Proposition 3.3.7 is proved. def c def c) and denote by 3.3.8. For any M in Ez we define M = limMn , Tb = limTn ⊂ End(M ← ← c → Mn the natural projection (see 2.2.10). Given M in Ez we define (as in [KL], π bn : M §29) for any ℓ ∈ C, and n ∈ N the number dn (ℓ) to be the dimension of the space ℓ Mn , def where ℓ Mn = ∪m ker(Tn − ℓ)m . It is clear that for any ℓ ∈ C, we have d1 (ℓ) ≤ d2 (ℓ) ≤ · · · ≤ dn (ℓ) ≤ · · · . We define d(ℓ) = lim dn (ℓ). n→∞ 3.3.9 Proposition. For any ℓ ∈ C, we have d(ℓ) < ∞. Proof: Since all operators Tn are invertible we may assume that ℓ ∈ C∗ . It follows from Proposition 3.3.7 that there exists n0 ∈ N such that for all n ≥ n0 , ℓ is not an eigenvalue of the restriction of Tn on M (n−1) . Therefore dn (ℓ) = dn0 (ℓ) for all n ≥ n0 . Proposition 3.3.9 is proved. 44 Remark: We can rephrase the statement of Proposition 3.3.9 by saying that Tb induces c (see §29 in [KL]). an admissible automorphism of M c(∞) ⊂ M c as in 2.2.1 and define a 3.3.10. For any M in Ez we define a submodule M M [0] + c c c c∞ def subspace M = C one ℓ M ⊂ M as in Proposition 29.5 of [KL]. Since M (∞) lies in ℓ∈C∗ c(∞). can define the Sugawara operator T ∈ End M c∞ ⊂ M c(∞). Proposition. a) M c∞ ) = Mn . b) For any n ∈ N we have πn (M Proof: Part a) follows immediately from Propositions 3.3.9, 3.3.6. Part b) follows from Proposition 29.1 in [KL]. 3.3.11. For any M in Ez the projective system πn : Mn → Mn−1 defines an inductive M def ∗ Hom(Mn , C)λ (see 2.1.5). We denote by ֒→ Mn∗ , where Mn∗ = system πn∗ : Mn−1 λ∈ΛC∗ M ∗ the inductive limit M ∗ = limMn∗ . It is easy to see that M ∗ has a natural structure of → U-module. Moreover, M ∗ lies in [0] − C and therefore we can define the Sugawara operator Ť ∈ End(M ∗ ). 3.3.12 Proposition. For any ζ ∈ Mn∗ ⊂ M ∗ and m ∈ M we have ζ(Tn (πn (m)) = (Ť −1 ζ)(m) where we consider ζ as a linear functional on Mn and Ť −1 ζ ∈ M ∗ as a linear functional on M . Proof: Follows from Theorem 3.2.1. 3.3.13 Proposition. Let M be an object in Ez . c(∞) and any n ∈ N we have πn (T m) = Tn πn (m). a) For any m ∈ M c(∞) = M c∞ . b) M c(∞) belongs to O+ . c) The U-module M z c(∞))n → (M c)n = Mn is an isomorphism for all n ∈ N. d) The natural map (M c belongs to Ez . e) M Proof: a) Follows from 2.2.12. 45 c∞ ⊂ M c(∞) follows from Proposition 3.3.6 To prove b) we observe that the inclusion M c(∞) ⊂ M c∞ follows from Proposition 3.1.8 (see the proof of Lemma and the inclusion M 26.4 in [KL]). Part c) follows from Proposition 3.1.8, part d) follows from Proposition 3.3.10 b), and part e) is clear. Proposition 3.3.13 is proved. 3.3.14 Corollary. a) For any module M in Ez the morphisms c ←− M c∞ M −→ M induce isomorphisms c)∗ −→ (M c∞ )∗ . M ∗ ←− (M b) For any W in Oz− and any M in Ez the morphisms c)∨ ) −→ HomU (W, (M c∞ )∨ ) HomU (W, M ∨ ) ←− HomU (W, (M def are isomorphisms, where as always M ∨ = Hom(M, C). Proof: Part a) is equivalent to part d) of Theorem 3.3.13. To prove b) we observe that it follows from the definitions the natural map Hom(W, M ∗ ) → Hom(W, M ∨ ) is an isomorphism. Therefore part b) is a restatement of part a). 3.3.15. In 3.3.15 - 3.3.18 hM i denotes coinvariants of M with respect to U. For any two U-modules W and M in C we define the map hM ⊗ W i∨ → HomU (W, M ∨ ), r → r ∨ , where for any r ∈ hM ⊗ W i∨ ⊂ Hom(M ⊗ W, C) def and any w ∈ W we define r ∨ (w) ∈ M ∨ by the rule r ∨ (w)(m) = r(m ⊗ w). Proposition. The map hM ⊗ W i → HomU (W, M ∨ ) is an isomorphism. Proof: Clear. 3.3.16 Corollary. For any M in Ez and W in Oz− the morphisms c ⊗ W i ←− hM c∞ ⊗ W i hM ⊗ W i −→ hM 46 are isomorphisms. ˙ 3.3.17. For any finite dimensional U-module X we denote by ⊗X the functor from Oz+ to itself such that ˙ def b ∞ (= (V ⊗X)(∞)) b V ⊗X = (V ⊗X) for all V in Oz+ . def ω b b ωW ) For any finite dimensional U-module Y and W in Oz− we define Y ⊗W = (Y ⊗ def ω ˙ and Y ⊗W = ˙ ω W ). (Y ⊗ 3.3.18 Proposition. For any V in Oz+ , W in Oz− and a finite dimensional U-module X the maps are isomorphisms. b ˙ hV ⊗ X ⊗ W i ←− h(V ⊗X) ⊗ W i −→ h(V ⊗X) ⊗ Wi Proof: Follows from Corollary 3.3.16 in the case when M = V ⊗ X. 3.3.19. We say that a U-module V is locally U-finite if for any v ∈ V , Uv is a finitedimensional subspace of V . Proposition. Let M be a U-module in Ez which is locally U-finite. Then the module c∞ is also locally U-finite. M 3.3.20 Proof: The proposition is an immediate consequence of the following general and easy result. def Claim: Let V = (ρ, V ) be a locally finite representation of U, V ∨∨ = (ρ, V ∨∨ ) be the full second dual to V and V ∗∗ ⊂ V ∨∨ be the subspace of vectors which are U+ -finite. Then V ∗∗ is a locally finite representation of U. We will not give a proof of this claim since we will never use Proposition 3.3.19. 3.3.21 Corollary. Let Oz0 ⊂ Oz+ be the subcategory of locally U-finite modules. For ˙ any V in Oz0 and any finite-dimensional representation X of U the module V ⊗X lies in Oz0 . 3.3.22. As in [KL] §27 we denote by Az ⊂ Oz+ the full subcategory of objects V which def admit a filtration 0 = V0 ⊂ V1 ⊂ · · · ⊂ VN = V such that each quotient V n = Vn /Vn−1 , 47 1 ≤ n ≤ N , is isomorphic to a Verma module Vazi , ai ∈ ΛC∗ . We denote by [V ] the element N X def of the group ring C[ΛC∗ ] defined as a sum [V ] = ai . As follows from the Jordan-Hölder n=1 theorem, the element [V ] does not depend on a choice of filtration. 3.3.23. For any finite-dimensional representation X = (ρ, X) of U we define def {X} = X dim(Xλ ) · λ ∈ C[ΛC∗ ], λ∈ΛC∗ where the subspace Xλ of X is defined as in 2.1.5. Proposition. For any V in Az and any U-module X in D ˙ ∈ Az and a) V ⊗X ˙ b) [V ⊗X] = [V ] · {X}. Proof: The proof of a) is completely analogous to the proof of Proposition 28.1 in [KL] and the proof of b) is completely analogous to the proof of Theorem 28.1 in [KL]. 3.4 Comparison of coinvariants. 3.4.1. Given V ∈ Oz+ , X, Y ∈ D and W ∈ Oz− we consider def Γ-modules def b b ) and Pb(V, X, Y, W ) = (V ⊗X)[t] ⊗ (Y ⊗W P (V, X, Y, W ) = (V ⊗ X)[t] ⊗ Y ⊗ W , def ˙ ˙ ). We will often write P , Pb and Q instead of P (V, X, Y, W ), Q(V, X, Y, W ) = (V ⊗X)[t]⊗(Y ⊗W def def Pb(V, X, Y, W ) and Q(V, X, Y, W ). For any n ∈ N we define n P = P ⊗A An , n Pb = Pb ⊗A An , def and n Q = Q ⊗A An . We denote by hn P i, hn Pbi and hn Qi the corresponding An -modules of coinvariants. b ←֓ V ⊗X ˙ ˙ ˙ The natural imbeddings V ⊗ X ֒→ V ⊗X and Y ⊗ W ֒→ Y ⊗W ←֓ Y ⊗W induce the imbeddings coinvariants n P ֒→ n Pb ←֓ n n Q and the corresponding An -morphism of Γn σ η hn P i −→ hn Pb i ←− hn Qi. Theorem. The morphisms n σ and n η are isomorphisms for all V ∈ Oz+ , X, Y ∈ D and W ∈ Oz− . 48 3.4.2 Proof: Let n P0 ⊂ n P be the An -submodule generated by vectors of the form γp, where γ ∈ Γ0 and p ∈ n P . In other words, n P0 is the kernel of the natural projection n P → hn P i. We start with the following result. 3.4.3 Lemma. a) (V ⊗ X)(n) ⊗ (Y ⊗ W ) ⊂ n P0 b) (V ⊗ X) ⊗ (Y ⊗ W )(n) ⊂ n P0 . Proof of Lemma: We prove part a). The proof of part b) is completely analogous. By the definition of the subspace (V ⊗ X)(n) it is sufficient to show that for any ve ∈ V ⊗ X, w e ∈ Y ⊗ W and x ∈ fν , |ν| ≥ n we have x− ve ⊗ w e ∈ n P0 . But this follows from Proposition 2.4.16. 3.4.4 Corollary. The map n σ : hn P i → hn Pbi is an isomorphism. 3.4.5. Consider the map n e def ξ = (n σ)−1 ◦ n η : hn Qi → hn P i. Lemma. The map n ξe is surjective. Proof: Follows from Proposition 3.3.13 d). 3.4.6 Proposition. Theorem 3.4.1 is true in the case when V and W are generalized Verma modules. Proof: In this case we have V = M and W = N−z where M and N are nil-modules and it follows from Proposition 2.5.2 that the An -module n P is free of rank equal to dimhM ⊗ X ⊗ Y ⊗ ω N iU0 . Therefore dimC hn P i = n · dimhM ⊗ X ⊗ Y ⊗ ω N iU0 . On the other hand, it follows from Proposition 3.3.23 and the right exactness of the functor h , i that dimC hn Qi ≤ n · dimhM ⊗ X ⊗ Y ⊗ ω N iU0 . Therefore Proposition 3.4.6 follows from Lemma 3.4.5. 3.4.7. We can now finish the proof of Theorem 3.4.1. Really for any V ∈ Oz+ , W ∈ Oz− we can find exact sequences M1 → M0 → V → 0 and N1 → N0 → W → 0 such that 49 M0 , M1 are generalized Verma modules in Oz+ and N0 , N1 are generalized Verma modules in Oz− . Therefore it follows from Proposition 3.4.6, the right exactness of the functor h , i and the five-homomorphisms lemma that n ξe : hn Qi → hn P i is an isomorphism. Theorem 3.4.1 is proved. 3.4.8 Theorem. Let T = T(ze . The diagram ∨ q h )2 n h Q(V, X, Y, W )i  n  ξe y hn P (V, X, Y, W )i T −1˙ V ⊗X ⊗id −→ (TV−1 ⊗id)·n δX,V,Y ⊗W −→ hn Q(T (V ), T (X), Y, W )i  n  ξe y hn P (T (V ), T (X), Y, W )i is commutative. Proof: We decompose the diagram into a sequence of simpler diagrams whose commutativity was proven already. 50 n h Q(V, X, Y, W )i ˙ ˙ )i h (V ⊗X)[t] ⊗ (Y ⊗W n T −1˙ ⊗id T −1˙ ⊗id V ⊗X V ⊗X hn Q(T (V ), T (X), Y, W )i −→ ˙ (X))[t]⊗Y ⊗W ˙ i h(T (V )⊗T −→ (1) ˙ ˙ )i hn ((V ⊗X)[t] ⊗ (Y ⊗W    y b ˙ )i hn ((V ⊗X)[t] ⊗ (Y ⊗W x    ˙ ˙ )i hn ((V ⊗X)[t] ⊗ (Y ⊗W id⊗ŤY ⊗W ˙ ˙ ˙ )i h(V ⊗X)[t] ⊗ T −1 (Y ⊗W    y −→ (2) id⊗TY ⊗W ˙ b ˙ )i hn (V ⊗X)[t] ⊗ T −1 (Y ⊗W x    −→ (3) id⊗TY ⊗W ˙ ˙ )i h(V ⊗ X)[t] ⊗ T −1 (Y ⊗W −→ (4) ˙ )i hn ((V ⊗X)[t] ⊗ (Y ⊗W    y b )i h (V ⊗X)[t] ⊗ (Y ⊗W x    n n h ((V ⊗X)[t] ⊗ (Y ⊗W )i (TV−1 ⊗id)◦n δX,V,Y ⊗W ˙ −→ (5) (TV−1 ⊗id)◦n δ b X,V,Y ⊗W −→ (6) (TV−1 ⊗id)◦n δX,V,Y ⊗W −→ ˙ )i hn ((V ⊗ X)[t] ⊗ T −1 (Y ⊗W    y b ))i hn ((V ⊗ X)[t] ⊗ T −1 (Y ⊗W x    hn ((V ⊗ X)[t] ⊗ T −1 (Y ⊗ W ))i (7) n (P (V, X, Y, W ) (TV−1 ⊗id)◦n δX,V,Y ⊗W −→ hn P (T (V ), T (X), Y, W )i where the commutativity of (1) follows from Theorem 3.2.2, the commutativity of (4) from Theorem 2.5.7, the vertical map in (2), (3), (5) and (6) are isomorphisms coming from natural imbeddings (see Theorem 3.4.1) and the right vertical isomorphism in (7) comes ∼ from the natural isomorphism hM i −→ hT (M )i as in 1.1.13. Theorem 3.4.5 is proved. 51 def ∨ 3.4.9. Fix z as in 3.1.7, V in Oz+ , W in Oz− , X, Y in D and define u = (ze q h )−2 . For def any n ∈ N we define n R = HomAn (hn Q(V, X, Y, W )i, hnP (V, X, Y, W )i and n def R′ = HomAn (hn Q(T (V ), T (X), Y, W )i, hn P (T (V ), T (X), Y, W )i) and denote by n ∇′ the ⊗id). An -linear map from n R to n R′ such that n ∇′ (a) = (TV−1 ⊗id) ◦ n δV,X,Y ⊗W ◦ (a · TV ⊗X ˙ 3.4.10. The u e-linear isomorphism u b which we constructed in 2.4.22 define an u e-linear def isomorphism ǔ = (b u)−1 such that ǔ : n R′ → n R and we define n ∇ = ǔ ◦ n ∇′ . Then n ∇: n R→ Theorem. n n R is a u e-linear automorphism. We can restate Theorem 3.4.8 as follows. e = ∇(n ξ) ne ξ. 52 §4. Finite-dimensional representations 4.1 The category D. 4.1.1 As before, we denote by D the category of unital finite-dimensional U-modules M such that Z acts on M as identity. e is the quotient of U by the ideal generated by Z − 1 we can identify D Remark: If U e with the category of finite dimensional unital U-modules. It is clear that D has a natural structure of a strict monoidal rigid category (see §1). e admits “loop-like” generators x± , hik , i ∈ I, k ∈ Z such that 4.1.2. The algebra U ik hi0 = Ki , x± i0 = Ei , i ∈ I; and the generators are subject to certain commutation relations explained in [D2] (see also [Be1]). We will use only the following relations: (α) [hik , hjℓ ] = 0 (β) ± 1 [hik , x± jℓ ] = ± k [kaij ] · xj,k+ℓ (γ) − [x+ ik , xjℓ ] = where [m] = q m −q −m q−q −1 δij (θi,k+ℓ −ϕi,k+ℓ ) , q−q −1 and the elements θi,p and ϕi,p , p > 0, are defined from the relations: exp((q − q −1 ) X hip τ p ) = 1 + p>0 exp((q − q −1 ) X X Ki−1 θip τ p X Ki ϕip τ p , p>0 p hip τ ) = 1 + p<0 p<0 where we consider τ as a formal parameter. e be the subalgebra with unity generated by x± , hik , i ∈ I, 4.1.3. Let A+ (resp. A− ) ⊂ U ik where k > 0 (k < 0 resp.). e Proposition. For any X = (ρ, X) in D we have ρ(A+ ) = ρ(A− ) = ρ(U). e e is Proof: We prove that ρ(A+ ) = ρ(U). The proof of the equality ρ(A− ) = ρ(U) completely analogous. e be the span of hik , k ≥ n, 1 ≤ i ≤ r. Put H = ∩n∈N ρ(H(n)). Since Let H(n) ⊂ U dimC End X < ∞ there exists N > 0 such that H = ρ(H(N + p)) for any p ≥ 0. 53 + Let us prove that for any ℓ < 0, ρ(x+ jℓ ) ∈ ρ(A ). Fix a vertex j0 ∈ I such that aj0 j 6= 0. Then [ρ(hj0 N ), ρ(x+ jm )] = 1 [kaj0 j ]ρ(x+ jℓ ), k where m = ℓ − N . We have ρ(hj0 N ) ∈ H = ρ(H(N + p)), for all p ≥ 0 and therefore ρ(hj0 N ) lies in ρ(H(N − ℓ + 1)). Since ρ(hj0 N ) lies in ρ(H(N − ℓ + 1) we can write ρ(hj0 N ) as a linear combination of operators of the form ρ(his ), i ∈ I, s > N − ℓ. Therefore 1 [kaj,ℓ ]ρ(x+ jℓ ) k is a linear combination of operators of the form [ρ(his ), ρ(x+ jm )] = Therefore we see that we have 1 k [kaj0 j ] 1 [kaij ]x+ j,m+s , k + 1 k [kaj0 j ]xjℓ s > N − ℓ, m = ℓ − N. lies in ρ(A+ ). Since the number q is not a root of 1 + 6= 0, and therefore ρ(x+ jℓ ) ∈ ρ(A ). In a similar way, one shows that + + ρ(x− jℓ ) ∈ ρ(A ) for any ℓ ∈ Z. ρ(hik ) ∈ ρ(A ) for k ≥ 0 and for any i ∈ I, we have: ρ(θik ) ∈ ρ(A+ ) for k ≥ 0. The relation (γ) in 4.1.2 implies that ρ(ϕik ) ∈ ρ(A+ ) for k ≥ 0 and all i ∈ I. Since ϕik = θik = 0 if k ≤ 0 we see that the images of all Drinfeld’s e belong to ρ(A+ ). Proposition 4.1.3 is proved. generators of U 4.2 Endomorphisms of tensor products. 4.2.1. Let F = C[t]) be the field of fractions of A(= C[t]), A = C[[t]] be the completion of A and F be the field of fractions of A. It is clear that X(t) = X[t] ⊗A F and X((t)) = X[t] ⊗A F carry the structures of UF and UF -modules respectively. We will denote them by X(t) = (ρX(t) , X(t)), X((t)) = (ρX((t)) , X((t))). For any X = (X, ρX ), Y = (Y , ρY ) in D, the Γ-module structure ρX[t] ⊗ ρY on X[t] ⊗C Y defines a Γ-module structure on finite dimensional, respectively, F and F vector spaces def X(t) ⊗ Y = (X[t] ⊗ Y ) ⊗A F def and X((t)) ⊗ Y = (X[t] ⊗ Y ) ⊗A F . We denote the corresponding Γ-modules by X(t) ⊗ Y = (ρX(t)⊗Y , X(t) ⊗ Y ) and 54 X((t)) ⊗ Y = (ρX((t))⊗Y , X((t)) ⊗ Y ). We denote by ρX((t))⊗Y (UF ) ⊂ End(X((t)) ⊗ Y ) the F -space of the image of ρX((t))⊗Y . 4.2.2. For any X, Y in D we denote (as in 1.1.10) the ring of endomorphisms of X and Y def to be EX , EY and denote by E the tensor product E = EX,Y = EX ⊗C EY . Let def def E = EndUF (X((t)) ⊗C Y ), E = EndUF (X(t) ⊗C Y ). Then E (resp. E) has a natural structure of an F (resp. F ) module and we have a natural imbedding E ֒→ E ֒→ E. 4.2.3 Theorem. The natural morphism E ⊗C F → E is an isomorphism. Proof: We start with the following obvious result. 4.2.4. Let X, Y be finite-dimensional F -vector spaces, BX ⊂ EndF (X), BY ⊂ EndF (Y ) be F -subalgebras containing idX and idY , EX and EY be centralizers of BX and BY in EndF (X) and EndF (Y ) respectively. Lemma. The centralizer of BX ⊗ BY in EndF (X ⊗ Y ) = EndF (X) ⊗F EndF (Y ) is equal to EX ⊗ EY . Proof: Well known. Since ρX((t))⊗Y (UF ) ⊂ ρX((t)) (UF )⊗F ρY (UF ), Lemma 4.2.4 shows that the following result implies the validity of Theorem 4.2.3. 4.2.5 Proposition. For any X = (ρX , X), Y = (ρY , Y ) in D ρX((t))⊗Y (Γ) ⊃ ρX((t)) (UF ) ⊗F ρY (UF ). def 4.2.6 Proof of Proposition 4.2.5: Let BX = ρX (U) ⊂ EndC (X), def BY = ρY (U) ⊂ EndC (Y ), p : EndA (X[t]C ⊗C Y ) → EndC (X ⊗ Y ) be the reduction def mod m, where m = tA ⊂ A and ρ be the composition ρ = p ◦ ρX[[t]]⊗Y : Γ −→ EndC (X ⊗ Y ) = EndC (X) ⊗C EndC (Y ). Nakayama’s lemma shows that Theorem 4.2.3 is a consequence of the following result. 55 Claim. ρ(Γ) ⊃ BX ⊗ BY . 4.2.7 Proof of Claim: It follows from [Be], Th. 4.7 and Prop. 5.3 that for any k > 0, ± i ∈ I we have hik , x± ik ∈ Γ and moreover for any α ∈ {hik , xik }, i ∈ I, k > 0 we have (id ⊗ ψ)∆(α) − α ⊗ 1 ∈ m(UA ⊗A UA ). Here ψ means the automorphism defined in 2.4.2. Therefore ρ(α) = ρX (α) ⊗ idY and we see that ρ(Γ) ⊃ ρX (A+ ) ⊗ idY . As follows from Proposition 4.1.3, ρ(Γ) ⊃ BX ⊗ id. −k α∈Γ Analogously we see that for any i ∈ I, k < 0 and x ∈ {hik , x± ik } we have t and (id ⊗ ψ)∆(t−k α) − 1 ⊗ α ∈ m(UA ⊗ UA ). So ρ(t−k α) = 1 ⊗ α and we have ρ(Γ) ⊃ idX ⊗ ρY (A− ). As follows from Proposition 4.1.3 we have ρ(Γ) ⊃ idX ⊗ BY . This finishes the proof of the Claim. Theorem 4.2.3 is proved. 4.2.8 Corollary. The natural morphism E ⊗C F → E is an isomorphism. Proof: We can consider E and EndF (X((t)) ⊗ Y ) as subspaces in EndF (X((t)) ⊗ Y ). Then E = E ∩ EndF (X(t) ⊗ Y ) and the Corollary follows immediately from Theorem 4.2.3. 4.2.9. Let G be the algebraic C-group such that G(C) is the group of invertible elements in EX ⊗ EY . We can restate Corollary 4.2.7 in the following way. Corollary. The natural morphisms G(F ) → Aut(X(t)⊗Y ) and G(F ) → Aut(X(t) ⊗ Y ) are isomorphisms. 4.2.10. Since the group G is defined over C any automorphism η of the field F preserving the subfield F ⊂ F defines group automorphisms of groups Aut(X(t)⊗Y ) and Aut(X((t))⊗ Y ) which we denote by ηb. 4.3 Intertwiners from X(t) ⊗ Y to Y ⊗ X(t). 4.3.1. For any X, Y in D we denote by J ⊂ HomUF (X((t)) ⊗ Y, Y ⊗ X((t))), J ⊂ HomUF (X(t) ⊗ Y, Y ⊗ X(t)) 56 the subsets of morphisms which are isomorphisms. The group G(F ) acts naturally on J : (a, f ) → af , a ∈ J , f ∈ G(F ). Let Hom(X(t) ⊗ Y, Y ⊗ X(t)) be the F -linear space HomUF (X(t) ⊗ Y, Y ⊗ X(t)) considered as an algebraic variety. 4.3.2 Lemma. a) J is a set of F -points of a Zariski open subset J in a linear subspace of Hom(X(t) ⊗ Y, Y ⊗ X(t)), J = J (F ) and the action of G(F ) on J comes from an algebraic action of G on J . b) Either J is empty or G acts simply transitively on J . Proof: Clear. 4.3.3 Proposition. For any s ∈ J there exists an element f ∈ G(F ) and such that s = s0 f , where s0 ∈ J . Proof: Since the algebraic C-group G is a group of invertible elements in an algebra E (see 4.2.2) it is isomorphic to a semidirect product L ⋉ N , where L is a direct product of a number of general linear groups and N is a unipotent group. Let G(F ) be the F group obtained from G by the extension of scalars from C to F . Then G(F ) is also a semidirect product of L∗F and N ∗F , where L∗F is a direct product of general linear groups and N ∗F is unipotent. As follows from Corollary 4.2.8 and Lemma 4.3.2 b) J is a principal homogeneous G-space. It follows from Proposition 1.33, Propositions 3.1 and 3.6 in [S] that J = 6 φ and moreover J = G(F )J . Proposition 4.3.3 is proved. 4.4 The functional equation. 4.4.1. For any X = (ρX , X), Y = (ρY , Y ) in D we denote by sX[[t]],Y ∈ J the inverse limit of elements n sX[t],Y , where n sX[t],Y are defined in 2.4.19. It can be treated as a linear map X[[t]] ⊗ Y → Y ⊗ X[[t]], where X[[t]] = X ⊗ A. Lemma. There exists an element f ∈ G(F ) such that sX[[t]],Y ·f−1 ∈ J and such an element is unique up to a left multiplication by an element in G(F ). Proof: Follows from Proposition 4.3.3. 57 We denote by LX,Y the subset of all elements f ∈ G(F ) such that sX[[t]],Y · f−1 ∈ J . Corollary. LX,Y is a left G(F ) coset in G(F ). ∨ 4.4.2. Let η be the continuous automorphism of the field F over C such that η(t) = qe2h ·t. Let ηb be the corresponding automorphisms of the groups G(F ) and G(F ) (see 4.2.10). Theorem. ηb(LX,Y ) = LX,Y . 4.4.3. We start the proof of Theorem 4.4.2 with the following observation. Let J ∗ ⊂ HomUF (X(t) ⊗ Y ∗ , Y ∗ ⊗ X(t)), ∗ J ⊂ HomUF (X((t)) ⊗ Y ∗ , Y ∗ ⊗ X((t))), J ∗∗ ⊂ HomUF (X(t) ⊗ Y ∗∗ , Y ∗∗ ⊗ X(t)) J ∗∗ ⊂ HomUF (X((t)) ⊗ Y ∗∗ , Y ∗∗ ⊗ X((t)) be the subsets of isomorphisms as in 4.3.1 and we denote by J ∗ the algebraic subvariety of the linear space Hom(X(t) ⊗ Y ∗ , Y ∗ ⊗ X(t)) corresponding to J ∗ . There exists an action op ⊗ EYop ) × HomUF (X(t) ⊗ Y ∗ , Y ∗ ⊗ X(t)) → HomUF (X(t) ⊗ Y ∗ , Y ∗ ⊗ X(t)) φ : (EX op ⊗ EYop on the space HomUF (X(t) ⊗ Y ∗ , Y ∗ ⊗ X(t)) such that of the algebra EX φ(a, fX ⊗ fY ) = (id ⊗ fX ) ◦ a ◦ (id ⊗ f∗Y ) for fX ∈ EX , fY ∈ EY , a ∈ HomUF (X(t) ⊗ Y ∗ , Y ∗ ⊗ X(t)). The action φ of the algebra (EX ⊗ EY )op on Hom(X(t) ⊗ Y ∗ , Y ∗ ⊗ X(t)) defines an algebraic action (a, g) 7−→ a ∗ g of def G(F ) on J ∗ such that a ∗ g = φ(a, g −1 ) for a ∈ J ∗ , g ∈ G(F ). ∗ 4.4.4. We consider sX[[t]],Y ∗ as an element in J and denote by MX,Y ∗ ⊂ G(F ) the set of all g ∈ G(F ) such that (sX[[t]],Y ∗ ) ∗ (g −1 ) ∈ J ∗ . Lemma. MX,Y ∗ is a left G(F ) coset in G(F ). Proof: Clear. 4.4.5 Proposition. MX,Y ∗ = LX,Y . Proof of Proposition: Consider the action ψ of the algebra EX ⊗ EY on the space HomUF (Y ∗ ⊗ X(t), X(t) ⊗ Y ∗ ) such that ψ(b, fX ⊗ fY ) = (id ⊗ f∗Y ) ◦ b ◦ (id ⊗ fX ) for fX ∈ EX , fY ∈ EY , b ∈ HomUF (Y ∗ ⊗ X(t), X(t) ⊗ Y ∗ ). 58 Lemma. For any a ∈ J ∗ , g ∈ G we have (a ∗ g)−1 = ψ(a−1 , g). Proof: Follows from the definitions. 4.4.6. The action ψ defines an algebraic action (b, g) 7−→ b♦g of G(F ) on HomUF (Y ∗ ⊗ X(t), X(t) ⊗ Y ∗ ) such that b♦g = ψ(b, g) for b ∈ HomUF (Y ∗ ⊗ X(t), X(t) ⊗ Y ∗ ), g ∈ G(F ). As follows from Lemma 4.4.5 we have −1 ∈ Hom(Y ∗ ⊗ X(t), X(t) ⊗ Y ∗ }. MX,Y ∗ = {g ∈ G(F )|s−1 X[[t]],Y ∗ ♦g 4.4.7. Now we can prove Proposition 4.4.5. Choose any fX,Y ∈ LX,Y . Then we have sX,Y = a·fX,Y , where a ∈ J . As follows from Proposition 1.1.18 we have ϕY,Y X[[t]] (sX[[t]],Y ) = Y,Y −1 s−1 X[[t]],Y ∗ . So sX[[t]],Y ∗ = ϕX[[t]] (a · fX,Y ). By Proposition 1.1.10 the right side is equal to Y,Y ϕY,Y X[[t]] (a)♦fX,Y . Since ϕX[[t]] maps HomUF (X(t) ⊗ Y, Y ⊗ X(t)) into HomUF (Y ∗ ⊗ X(t), X(t) ⊗ Y ∗ ) we see that fX,Y ∈ MX,Y ∗ . Proposition 4.4.5 is proved. 4.4.8 Proposition. LX,Y ∗ = MX,Y for all X, Y in D. Proof: Completely analogous to the proof of Proposition 4.4.5. Corollary. LX,Y ∗∗ = LX,Y . 4.4.9. For u ∈ C∗ we denote by θu the automorphism of the field F such that θu (t) = ut. We also denote by θu the continuous extension of θu on F . The automorphism θu define automorphisms of the groups G(F ) and G(F ) which we denote by θbu . Lemma. For any X, Y in D and u ∈ C∗ we have LTu (X),Y = θbu (LX,Y ) and LX,Tu (Y ) = θbu−1 (LX,Y ). Proof: Follows immediately from the definitions. 4.4.10. Now we can finish the proof of Theorem 4.4.2. Really it follows from Corollary 4.4.8, Lemma 4.4.9 and Proposition 2.1.15. Theorem 4.4.2 is proved. Corollary. For any fX,Y ∈ LX,Y we have (*) ηb(fX,Y )f−1 X,Y ∈ G(F ). 59 4.4.11 Lemma. There exists fX,Y ∈ LX,Y such that fX,Y and X[[t]] ⊗ Y preserve X[[t]] ⊗ Y ⊂ X((t)) ⊗ Y and fX,Y (0) = id. Proof: The standard (profinite) topology on F defines a topology on the group G(F ). Since the group G is a semidirect product L ⋉ N , where L is the direct product of a number of copies of GLn and N is a unipotent group, the subgroup G(F ) is dense in G(F ). Lemma 4.4.11 follows now immediately from the equality sX[[t]],Y ≡ σ (mod m), where σ : X ⊗ Y → Y ⊗ X is permutation as in 2.3.3. 4.4.12. Let Amer be the subring of the power series which converges in a neighborhood of zero and define a meromorphic function in C. Proposition. For any X, Y in D there exists f0 ∈ G(Amer ) which satisfies the condition of Lemma 4.4.11. def Proof: Choose any fX,Y ∈ LX,Y and define r = ηb(fX,Y )f−1 X,Y . As follows from Theorem 4.4.2 we have r ∈ G(C(t) ∩ A) and r(0) = Id. Consider the equation (**) ηb(f)f−1 = r on f ∈ G(A). It is easy to see that there exists a unique solution f0 of (**) in G(A) such that f0 (0) = Id and moreover any solution of (**) has a form f = f0 · c for some c ∈ G(C). ∞ Y ∨ r −1 (e q 2h n t). It is One can write f0 ∈ G(A) as an infinite convenient product f0 (t) = n=0 easy to see that this product is also convergent in G(Amer ). Proposition 4.4.12 is proved. Corollary. The map sX[[t]],Y from X ⊗C Y ⊗C A to Y ⊗ X ⊗ A maps the subspace X ⊗C Y ⊗C Amer to Y ⊗C X ⊗C Amer . 4.4.13. As follows from the previous Corollary we can consider s = sX[[t]],Y as a mermorphic function in t with values in the space HomC (X ⊗ Y , Y ⊗ X). The union of the set of poles of s and the points t ∈ C such that s(t) is not an isomorphism is denoted by ΛX,Y . Lemma. ΛX,Y is a discrete subset of C. Proof: It is clear that either ΛX,Y = C or it is discrete. Since s(0) is invertible, 60 ΛX,Y 6= C. The Lemma is proved. e (2) be the full subcategory of the pair of objects in (X, Y ) in D such that 4.4.14. Let D 1∈ / ΛX,Y . As follows from Lemma 4.4.13 for all pairs (X, Y ) in D we have e (2) for all u in a dense open set C∗ − ΛX,Y . We can consider (D e (2) , s) as a (Tu (X), Y ) ∈ D weak braiding on D (see 1.2.1). e (2) , s) is a rigid KZ-data for all z such that Proposition. The data (C, D, Oz± , s± , T± , D ∨ def ze q h ∈ C∗ − Q≥0 , where T± = Tz±1 , s±(V,X) = (sV,X )±1 , and sV,X is as in 2.3.3. Proposition: Follows immediately from the definitions and 4.4.14, 2.3.8. 4.4.15 Remark.: Similarly to 4.4.3 one can prove that “the square of weak braiding” sX,Y (t)sY,X (t−1 ) is an elliptic function on the curve C∗ /e q 2h Y , Y ⊗ X). 61 ∨ Z with values in HomC (X ⊗ §5. The quasi-associativity morphism. 5.1 The definition. ∨ 5.1.1. Fix z such that ze q h ∈ C∗ − Q≥0 and U-modules V in Oz+ , W in Oz− , X, Y in D. ∨ We assume that (ze q h )2m ∈ / ΛX,Y for m ∈ Z. Let P = P (V, X, Y, W ) be the Γ-module as in 3.4.1 and hP i the corresponding A-module of coinvariants. Proposition. hP i is a finitely generated A-module. Proof: As follows from Proposition 3.1.8, there exist exact sequences M1′ → M0′ → V → 0 and N1′ → N0′ → W → 0 where Mk′ and Nk′ are generalized Verma modules in Oz+ and Oz− , respectively, k = 0, 1. Therefore Proposition 5.1.1 follows from Lemma 2.5.5 and the right exactness of the functor h i. 5.1.2. Let Q, n P, n Q be Γ-modules as in 3.4.1 and hQi, hnP i and hn Qi be the corresponding modules of coinvariants. Proposition. a) hQi is a finitely generated A-module. b) The natural An -module morphisms n πP : hP i ⊗A An → hn P i and n πQ : hQi ⊗A An → hn Qi are isomorphisms for all n ∈ N. Proof: a) Follows from Proposition 5.1.1 and Theorem 3.4.1 and b) follows from Proposition 2.5.4. 5.1.3. For any n ∈ N we denote by n ξ = (n πP )−1 ◦ n ξe ◦ n πQ where n ξ : hP i ⊗A An → hQi ⊗A An the composition ne ξ is the isomorphism from hn Qi to hn P i as in 3.4.5. It follows from Theorem 3.4.1 and Proposition 5.1.2 that n ξ is an isomorphism for all n ∈ N. It is clear that the isomorphisms n ξ are compatible with the natural projections hP i ⊗A An+1 → hP i ⊗A An and hQi ⊗A An+1 → hQi ⊗A An . Therefore the family n def ξ defines an isomorphism ξ : hQi → hP i of A-modules, where hQi = limhQi ⊗A An , ← def hP i = limhP i ⊗A An and A = limAn . ← ← Since hP i, hQi are finitely generated A-modules we can identify the A-modules hP i, hQi with the tensor products hP i ⊗A A and hQi ⊗A A correspondingly. Let Amer be the ring as in Section 4. 62 def def 5.1.4. Let hP imer = hP i ⊗A Amer , hQimer = hQi ⊗A Amer . We can identify hP i with hP imer ⊗Amer A and hQi with hQimer ⊗Amer A. ∼ Theorem. There exists an isomorphism ξmer : hQimer −→ hP imer such that ξ = ξmer ⊗ 1. Proof: For any n ∈ N we denote by n R the An -module HomAn (hn Qi, hn P i) and by n ∇ the u e-linear automorphism of n R as in 3.4.10. def Let R = lim n R. We can identify R with the A-module ← HomA (hQi, hP i). Let def Rmer = HomAmer (hQimer , hPmer i). One can consider Rmer as an Amer -submodule of R and R = Rmer ⊗Amer A. We denote by ∇ the u e-linear automorphism of the module def R = lim n R which is the projective limit of n ∇. ← 5.1.5 Lemma. ∇(Rmer ) ⊂ Rmer . Proof: Follows from Corollary 4.4.12, Proposition 2.4.22 and the definition of ∇. 5.1.6. Our proof of Theorem 5.1.4 is based on the following result. Proposition. Let ϕ : DR → Mn (C) be a holomorphic function in a disc |t| ⊂ DR = {t ∈ C |t| < R} such that det ϕ(0) 6= 0, ψ : C → GLn (C) be a polynomial function, p a complex number such that 0 < |p| < 1 and F (t) ∈ A ⊗C Mn (C) be a formal power series solution of the difference equation (*) F (pt)ψ(t) = ϕ(t)F (t) such that F (0) = Id. Then the series F (t) is convergent in the disc |t| < R . kϕ(0)k+1 Moreover if ϕ(t) has a meromorphic continuation to C then F (t) has a meromorphic continuation to C. Proof: As follows from (*) we have ϕ(0) = ψ(0). Let us write the expansions for ϕ, ψ and F ϕ(t) = ∞ X j ϕj t , ψ(t) = j=0 N X j ψj t , F (t) = j=0 ∞ X f j tj . j=0 Then we can rewrite (*) in the form (**) j p fj ϕ(0) − ϕ(0)fj = j X k=1 63 ϕk fj−k − N X k=1 fj−k ψk . Since ϕ is convergent in DR we have kϕj k ≤ r −j for all r < R. Let Kj be an endomorphism def of Mn (C) such that Kj M = pj M ϕ(0) − ϕ(0)M for M ⊂ Mn (C) and K > 0 be a constant such that kKj−1 k < K for j ≫ 0. Let us prove that kfj k < C(r/K ′ )−j suitable constants K ′ and C. This will imply the validity of the first part of Proposition 5.1.6. 5.1.7. Applying the norm to both sides of (**) we get an inequality (for j ≫ 0)) ′ kfj k ≤ K ( j X r −k kfj−k k). k=1 This implies that the sequence kfj k is dominated by the sequence gj which satisfies the j X equalities gj = K ′ ( r −k gj−k ) for all j ≫ 0. The sequence gj satisfies the equation k=1 rgj+1 /K ′ − gj /K ′ = gj and therefore gj is a geometric progression. So fj is dominated by a geometric progression and therefore F (t) is convergent in a neighborhood of 0. If ϕ has a meromorphic continuation to C, it follows then from (*) that F has a meromorphic continuation to C. Proposition 5.1.6 is proved. 5.1.8 Corollary. Let L, M be finitely generated A-modules, ϕ ∈ (End L) ⊗A Amer , ψ ∈ (End M ) and ξ ∈ Hom(L, M ) ⊗A A be such that ϕ(0) ∈ Aut L/tL, ψ(0) ∈ Aut M/tM , ξ(0) ∈ Isom(L/tL, M/tM ) and the formal power series ξ satisfy the equation ξ(pt) = ϕ(t)ξ(t)ψ(t). then ξ ∈ Hom(L, M ) ⊗A Amer . ∨ 5.1.9. We can now prove Theorem 5.1.4. We put p = (ze q h )−1 . As follows from Proposition 3.1.8 it is sufficient to prove Theorem 5.1.4 in the case when V and W are generalized Verma modules. In this case it follows from Proposition 2.5.2 that P and Q are free Amodules and the result follows from Theorem 3.4.8 and Corollary 5.1.8. Theorem 5.1.4 is proved. 64 5.2.1. Let z, V, W, X and Y satisfy the assumptions of 5.1.1. It follows from the Theorem 5.1.4 that there exists a small disc |t| < ǫ such that ξmer defines an isomorphism of hQimer and hP imer at any point of this disc. Since |ze q h | > 1 we can find an even integer m < 0 such ∨ ∨ that |(ze q h )m | < ǫ so we get an isomorphism of hQi and hP i at t0 = (ze q h )m . Iterating u e-linear isomorphism ∇ from Lemma 5.1.5 and using Corollary 2.4.22 and the condition ∨ (ze q h )n ∈ / ΛX,Y , for n ∈ Z we obtain the isomorphism of hQi and hP i at t1 = 1. 5.2.2 Proposition. We have constructed a functorial isomorphism ˙ ˙ iU ≃ hV, X, Y, W iU ∇ : hV ⊗X, Y ⊗W where h· iU denotes the vector space of U-coinvariants. 5.2.3. Corollary. Let z, V, X and Y be as in 5.1.1. Then the isomorphism ∇ from 5.2.2 gives rise to the functorial quasi-associativity constraint ˙ ˙ ≃ V ⊗(X ˙ aV,X,Y : (V ⊗X) ⊗Y ⊗ Y ). It satisfies the pentagon axiom (see [M]) with respect to X and Y . Proof: Follows from 5.2.2 in the same way as the isomorphism 18.2 (b) follows from the theorem 17.29 in [KL]. 5.2.4 Remark: The following is the tautological reformulation of 5.2.3. We say that X ∨ and Y in D are in generic position with respect to the elliptic curve E = C∗ /(e q h z)2Z if the image of ΛX,Y under the natural projection C∗ → E does not contain the unity of the group E. If X and Y are in generic position with respect to the elliptic curve E then 5.2.3 holds. 65 References [Be] Beck, J.: Braid group action and quantum affine algebras. Commun. Math. Phys., 165: 3 (1994), 555-568. [D1] Drinfeld, V.G.: On quasitriangular quasi-Hopf algebras closely related to Gal(Q/Q), Algebra i Anal. 2 (1990), 149-181. [D2] Drinfeld, V.: A new realization of Yangians and quantized affine algebras. Preprint of the Phys.-Tech. Inst. of Low Temperatures, 1986. [D3] Drinfeld, V.: Quantum groups. Proc. ICM (Berkeley, vol. 2, pp. 789-819. [DM] Deligne, P., Milne, J.S.: Tannakian Categories, Lecture Notes in Math. 900, SpringerVerlag, 1981, 104-228. [FR] Frenkel, I., Reshetikhin, N.: Quantum affine algebras and holonomic difference equations. Commun. Math. Phys., 146 (1992), 1-60. [K] Kac, V.: Infinite-dimensional Lie algebras. Cambridge University Press (1991). [KL] Kazhdan, D., Lusztig, G.: Tensor structures arising from affine Lie algebras, I-IV. American J. Math., 1993-1994. [L] Lusztig, G.: Introduction to quantum groups. Birkhäuser, 1993. [M] Mac Lane, S.: Categories for the working mathematician. Springer-Verlag (1971). [S] Serre, J.-P.: Cohomologie Galoisienne, Collège de France, 1963. [TUY] Tsuchiya, A., Ueno, K., and Yamada, Y., Conformal field theory on universal family of stable courses with gauge symmetries, Advanced Stud. Pure Math., vol. 19, Academic Press, Boston, MA, 1989, pp. 459-565. 66