QUASI{COXETER CATEGORIES AND A RELATIVE …mathserver.neu.edu/.../Preliminary_files/ATL1.pdf ·...

$: QUASI{COXETER CATEGORIES AND A RELATIVE …mathserver.neu.edu/.../Preliminary_files/ATL1.pdf · 2013-06-02 · QUASI{COXETER CATEGORIES AND A RELATIVE EK FUNCTOR 3 k[[~]]{modules.$
QUASI–COXETER CATEGORIES AND A RELATIVE

ETINGOF–KAZHDAN QUANTIZATION FUNCTOR

ANDREA APPEL AND VALERIO TOLEDANO LAREDO

Abstract. Let g be a symmetrizable Kac–Moody algebra and U~g itsquantized enveloping algebra. The quantum Weyl group operators ofU~g and the universal R–matrices of its Levi subalgebras endow U~g witha natural quasi–Coxeter quasitriangular quasibialgebra structure whichunderlies the action of the braid group of g and Artin’s braid groups onthe tensor product of integrable, category O modules. We show that thisstructure can be transferred to the universal enveloping algebra Ug[[~]].The proof relies on a modification of the Etingof–Kazhdan quantizationfunctor, and yields an isomorphism between (appropriate completionsof) U~g and Ug[[~]] preserving a given chain of Levi subalgebras. Wecarry it out in the more general context of chains of Manin triples, andobtain in particular a relative version of the Etingof–Kazhdan functorwith input a split pair of Lie bialgebras. Along the way, we developthe notion of quasi–Coxeter categories, which are to generalised braidgroups what braided tensor categories are to Artin’s braid groups. Thisleads to their succint description as a 2–functor from a 2–category whosemorphisms are De Concini–Procesi associahedra. These results will beused in the sequel to this paper to give a monodromic description of thequantum Weyl group operators of an affine Kac–Moody algebra, extend-ing that obtained by the second author for a semisimple Lie algebra.

Contents

1. Introduction 22. Quasi–Coxeter categories 53. Etingof-Kazhdan Quantization 174. Generalized Etingof–Kazhdan functor 285. Quantization of Verma modules 416. Universal relative Verma modules 447. Chains of Manin triples 508. An equivalence of quasi–Coxeter categories 52References 57

Date: December 30, 2012.Both authors were supported in part through the NSF grant DMS–0854792.

1

2 A. APPEL AND V. TOLEDANO LAREDO

1. Introduction

1.1. This is the first of a series of three papers the aim of which is toextend the description of the monodromy of the rational Casimir connectionin terms of quantum Weyl group operators given in [TL3, TL4, TL5] to thecase of an affine Kac–Moody algebra g.

The method we follow is close to that of [TL4], and relies on the notion of aquasi–Coxeter quasitriangular quasibialgebra (qCqtqba), which is informallya bialgebra carrying actions of a given generalised braid group and Artin’sbraid groups on the tensor products of its modules. A cohomological rigidityresult, proved in the second paper of this series [ATL1], shows that thereis at most one such structure with prescribed local monodromies on theclassical enveloping algebra Ug[[~]]. It follows that the generalised braidgroup actions arising from quantum Weyl groups and the monodromy ofthe Casimir connection [ATL2] are equivalent, provided the quasi–Coxeterquasitriangular quasibialgebra structure responsible for the former can betransferred from U~g to Ug[[~]]. This result is the purpose of the presentarticle.

1.2. Its proof differs substantially from that given in [TL4] in that, for asemisimple Lie algebra g, the transfer of structure ultimately rests on thevanishing of the first and second Hochschild cohomology groups of Ug[[~]],and in particular on the fact that U~g and Ug[[~]] are isomorphic as algebras,a fact which does not hold for affine Kac–Moody algebras. Rather thanthe cohomological methods of [TL4], we use instead the Etingof–Kazhdan(EK) quantization functor [EK96, EK98, EK08], which yields in particulara canonical isomorphism

ΨEK : U~g→ Ug[[~]]

between the completions of U~g and Ug[[~]] with respect to category O.Surprisingly perhaps, and despite its functorial nature, the isomorphism

ΨEK does not preserve the inclusions of Levi subalgebras

U~gD ⊆ U~g UgD[[~]] ⊆ Ug[[~]]

determined by a subdiagram D of the Dynkin diagram of g, something whichis required by the transfer of structure. The bulk of this paper is thereforedevoted to modifying ΨEK so that it is compatible with such inclusions.

1.3. To outline our construction, which works more generally for an in-clusion (gD, gD,−, gD,+) ⊂ (g, g−, g+) of Manin triples over a field k ofcharacteristic zero, recall that the main step in the EK quantization ofg is the construction of a tensor functor F EK from the Drinfeld categoryDΦ(g) of (equicontinuous) g–modules with associativity constraints givenby a fixed Lie associator Φ, to the category Vectk[[~]] of topologically free

QUASI–COXETER CATEGORIES AND A RELATIVE EK FUNCTOR 3

k[[~]]–modules. The quantization U~g is then essentially defined as the Hopfalgebra of endomorphisms of the fibre functor F EK.1

To construct an isomorphism compatible with the inclusion gD ⊂ g, wefirst construct a relative fiber functor, that is a tensor functor ΓD on DΦ(g)whose target category is DΦ(gD) rather than Vectk[[~]]. Denoting by F EK

Dthe Etingof–Kazhdan fiber functor for gD, we then show the existence of anatural transformation between the composition F EK

D ΓD and F EK. Thisyields the sought for inclusion

U~gD = End(F EKD ) → End(F EK

D ΓD) ∼= End(F EK) = U~g

1.4. The construction of the tensor restriction functor Γ is very much in-spired by [EK96]. The principle adopted by Etingof and Kazhdan is thefollowing. In a k-linear monoidal category C, a coalgebra structure on anobject C ∈ Obj(C) induces a tensor structure on the Yoneda functor

hC = HomC(C,−) : C → Vectk

If C is braided and C1, C2 are coalgebra objects in C, then so is C1⊗C2, andthere is therefore a canonical tensor structure on hC1⊗C2 .

If g is finite–dimensional, the polarization Ug ' M− ⊗M+, where M±are the Verma modules indg

g∓ C, realizes Ug as the tensor product of two

coalgebra objects in RepΦ(Ug[[~]]). This yields the tensor structure on theforgetful functor

hUg : RepΦ(Ug)[[~]]→ Vectk[[~]]

The main idea is to apply the same principle to the (abelian) restrictionfunctor i∗D : RepΦ(Ug[[~]])→ RepΦ(UgD[[~]]). We therefore factorize Ug asa tensor product of two coalgebra objects L−, N+ in the braided monoidalcategory of (g, gD)–bimodules, with associator (Φ ·Φ−1

D ), where Φ−1D acts on

the right. Just as the modules M−,M+ are related to the decompositiong = g− ⊕ g+, L− and N+ are related to the asymmetric decomposition

g = m− ⊕ p+

where m− = g− ∩ g⊥D,+ and p+ = gD ⊕ m+. This factorization induces atensor structure on the functor Γ = hL−⊗N+ , canonically isomorphic to therestriction functor i∗D through the right gD–action on L−⊗N+. As in [EK96,Part II], this tensor structure can also be defined in the infinite–dimensionalcase.

1if g is infinite–dimensional, End(FEK) is only a topological bialgebra. To remedy this,Etingof–Kazhdan first obtain a quantization of Ug−[[~]] inside End(FEK), and then defineU~g as the quantum double of U~g−. We shall overlook this point in the Introduction andargue as if g were finite–dimensional.


1.5. To construct an isomorphism F EKD ΓD ∼= F EK, we note that the func-

tors F EK, F EKD factor through the categories Rep(U~g) and Rep(U~gD) re-

spectively, thus yielding a diagram

DΦ(g)FEK

//

Γ

FEK%%

Rep(U~g)

xx

Vectk[[~]]oo

KS

DΦ(gD)FEKD

//

FEKD

99

Rep(U~gD)

ff

where the diagonal arrows on the right–hand side are the forgetful functors toVectk[[~]], and the dotted horizontal arrow on the left–hand side is the soughtfor natural transformation. Since the EK quantization is functorial withrespect to morphisms of Manin triples such as the inclusion iD : gD → g,the rightmost triangle can be filled in by adding the quantized pull–backfunctor (iD)∗~ : Rep(U~g)→ Rep(U~gD), and we are reduced to constructinga natural transformation making the following diagram commute

DΦ(g)FEK

//

Γ

Rep(U~g)

v~

(iD)∗~

DΦ(gD)FEKD

// Rep(U~gD)

To then remark, as suggested to us by P. Etingof, that a quantum analogueΓ~ of Γ can be similarly defined using a quantum version L~−, N

~+ of the

modules L−, N+. The functor Γ~ = HomUEK~ g(L

~− ⊗ N~

+,−) is naturally

isomorphic to (iD)∗~ as tensor functor, since there is no associator involvedon this side and their structures are trivial. Moreover, an identification

FEKD Γ ' Γ~ FEK

is readily obtained, provided one establishes isomorphisms of (UEK~ g, UEK

~ gD)–bimodules

FEKD FEK(L−) ' L~− and FEKD FEK(N+) ' N~+

1.6. While forM± it is easy to construct an isomorphism between FEK(M±)and the quantum counterparts of M±, the proof for L−, N+ is more involved.It relies on a description of the quantization functor FEK in terms of Propcategories (cf. [EK98, EG]) and the realization of L−, N+ as universal ob-jects in the colored Prop describing the inclusion of bialgebras gD ⊂ g.


This yields in particular a relative extension of the EK functor with inputa pair of Lie bialgebras a, b which is split, i.e., endowed with maps a i

p bsuch that p i = id.

1.7. Outline of the paper. We begin in Section 2 by extending the defi-nition of quasi–Coxeter quasitriangular quasibialgebra given in [TL4] to thecompletion of such an algebra with respect to a given category of modulesand a fiber functor on this category. We do so by translating the definitionin categorical terms, and obtain in particular a concise definition of a quasi–Coxeter category as a 2–functor from a 2–category whose morphisms areDe Concini–Procesi associahedra. In Section 3, we review the constructionof the Etingof–Kazhdan quantization functor and the isomorphism ΨEK

in [EK08]. In Section 4, we modify this construction by using generalizedVerma modules, and obtain a relative fiber functor Γ : DΦ(g) → DΦ(gD).In Section 5 we define the quantum analogues of (generalized) Verma mod-ules. Using suitably defined Props we then show, in Section 6 that theseVerma modules are isomorphic to the quantization of their classical coun-terparts. In Section 7, we use these results to show that, for any given chainof Manin triples ending in a given g, there exists a quantization of Ug whichis compatible with each inclusion and independent, up to isomorphism, ofthe choice of the given chain. Finally, in Section 8, we apply these resultsto the case of a Kac–Moody algebra g and obtain the desired tranport of itsquasi–Coxeter quasitriangular quasibialgebra structure to the completion ofUg[[~]] with respect to category O, integrable modules.

1.8. Acknowledgments. We are very grateful to Pavel Etingof for his in-terest in the present work and for many enlightening discussions.

2. Quasi–Coxeter categories

The goal of this section is to adapt the definition of quasi–Coxeter quasi-triangular quasibialgebras, introduced in [TL4, Section 3], to completion ofD-bialgebras with respect to a fixed family of subcategories (e.g., categoryO), compatible with restriction. This description amounts to the definitionof a categorical notion of quasi–Coxeter quasitriangular quasibialgebras.

2.1. Algebras arising from fiber functors. We shall repeatedly need thefollowing elementary

Lemma. Consider the following situation

C

F ++

H // D

G

α

w

Awhere A, C,D are additive k–linear categories, F,G,H functors, and α isan invertible transformation. If H is an equivalence of categories, the map


End(G) −→ End(F ) given by

gW 7→ Ad(α−1V )(gH(V ))

is an algebra isomorphism.

2.2. Diagrams and maximal nested sets. We review some basic defi-nitions from [TL4, Section 2.1]. By a diagram we shall mean a nonemptyundirected graph D with no multiple edges or loops. We denote the set ofvertices of D by V(D) and set |D| = |V(D)|. A subdiagram B ⊆ D is a fullsubgraph of D, that is, a graph consisting of a subset V(B) of vertices ofD, together with all edges of D joining any two elements of V(B). We willoften abusively identify such a B with its set of vertices and write i ∈ B tomean i ∈ V(B). We denote by SD(D) the set of subdiagrams of D.

The union B1 ∪ B2 of two subdiagrams B1, B2 ⊂ D is the subdiagramhaving V(B1) ∪ V(B2) as its set of vertices. Two subdiagrams B1, B2 ⊂ Dare orthogonal if V(B1) ∩ V(B2) = ∅ and no two vertices i ∈ B1, j ∈ B2 arejoined by an edge in D. B1 and B2 are compatible if either one contains theother or they are orthogonal.

Let D be a diagram with connected components D1, . . . , Dr.

Definition. A nested set on D is a collection H of pairwise compatible,connected subdiagrams of D which contains the connected components ofD.

We denote by ND the partially ordered set of nested sets on D, orderedby reverse inclusion. ND has a unique maximal element 1 = Di and itsminimal elements are the maximal nested sets. We denote the set of maximalnested sets on D by Mns(D). Every nested set H on D is uniquely deter-mined by a collection Hiri=1 of nested sets on the connected componentsof D. We therefore obtain canonical identifications

ND =

r∏i=1

NDi and Mns(D) =

r∏i=1

Mns(Di)

The De ConciniProcesi associahedron AD is the regular CW–complexwhose poset of (nonempty) faces is ND. It easily follows from the definitionthat

AD =

r∏i=1

ADi

This can be realized as a convex polytope of dimension |D| − r. For anyH ∈ ND, we denote by dim(H) the dimension of H in AD.

For any F ∈ Mns(D), B ∈ F , we denote by FB ∈ Mns(B) the maximalnested set induced by F on B.


For any nested set H on D and B ∈ H, we set iH(B) =⋃mi=1Bi where

the Bi’s are the maximal elements of H properly contained in B.

Definition. Set αBH = B \ iH(B). We denote by

n(B;H) = |αBH| and n(H) =∑B∈H

(n(B;H)− 1

)An element B ∈ H is called unsaturated if n(B;H) > 1.

We recall the following

Proposition. .

(i) For any nested set H ∈ ND,

n(H) = |D| − |H| = dim(H)

(ii) If H is a maximal nested set if and only if n(B;H) = 1 for anyB ∈ H.

(iii) Any maximal nested set is of cardinality |D|.

For any F ∈ Mns(D), B ∈ F , iF (B) denotes the maximal element in Fproperly contained in B and αBF = B \ iF (B) consists of one vertex, denotedαBF .

2.3. Quotient diagrams. Let B ⊂ D a proper subdiagram with connectedcomponents B1, . . . , Bm.

Definition. The set of vertices of the diagram D/B is V(D) \ V(B). Twovertices i 6= j of D/B are linked by an edge if and only if the following holdsin D:

i 6⊥ j or i, j 6⊥ Bi for some i = 1, . . . ,m

For any connected subdiagram C ⊆ D not contained in B, we denote byC ⊆ D/B the connected subdiagram with vertex set V (C) \ V (B).We havethe following

Lemma. Let C1, C2 * B be two connected subdiagrams of D which arecompatible. Then

(i) C1, C2 are compatible unless C1 ⊥ C2 and C1, C2 ⊥Bi for some i.(ii) If C1 is compatible with every Bi, then C1 and C2 are compatible.

In particular, if F is a nested set on D containing each Bi, then F = C,where C runs over the elements of F such that C * B, is a nested set onD/B.

Let now A be a connected subdiagram of D/B and denote by A ⊆ D theconnected sudbdiagram with vertex set

V (A) = V (A)⋃

i:Bi⊥V (A)

V (Bi)


Clearly, A1 ⊆ A2 or A1 ⊥ A2 imply A1 ⊆ A2 and A1 ⊥ A2 respectively, so

the lifting map A→ A preserves compatibility.

For any connected subdiagrams A ⊆ D/B and C ⊆ D, we have

A = A and C = C⋃

i:Bi⊥CBi

In particular, C = C if, and only if, C is compatible with B1, . . . , Bm and

not contained in B. The applications C → C and A → A therefore yield abijection between the connected subdiagrams of D which are either orthog-onal to or strictly contain each Bi and the connected subdiagrams of D/B.This bijection preserves compatibility and therefore induces an embeddingND/B → ND. This yields an embedding

ND/B ×NB = ND/B ×(NB1 × · · · × NBm

)→ ND

with image the poset of nested sets on D containing each Bi. Similarly, forany B ⊆ B′ ⊆ B′′, we obtain a map

∪ : NB′′/B′ ×NB′/B → NB′′/BThe map ∪ restricts to maximal nested sets. For any B ⊂ B′, we denote byMns(B′, B) the collection of maximal nested sets on B′/B. Therefore, forany B ⊂ B′ ⊂ B′′, we obtain an embedding

∪ : Mns(B′′, B′)×Mns(B′, B)→ Mns(B′′, B)

such that, for any F ∈ Mns(B′′, B′),G ∈ Mns(B′, B),

(F ∪ G)B′/B = G

2.4. Elementary and equivalent pairs.

Definition. An ordered pair (G,F) in Mns(D) is called elementary if Gand F differ by one element. A sequence H1, . . . ,Hm in Mns(D) is calledelementary if |Hi+1 \ Hi| = 1 for any i = 1, 2, . . . ,m− 1.

Definition. The support supp(F ,G) of an elementary pair in Mns(D) is theunique unsaturated element of F∩G. The central support z supp(F ,G) is theunion of the maximal elements of F ∩ G properly contained in supp(F ,G).Thus

z supp(F ,G) = supp(F ,G) \ αsupp(F ,G)F∩G

Definition. Two elementary pairs (F ,G), (F ′,G′) in Mns(D) are equivalentif

supp(F ,G) = supp(F ′,G′)α

supp(F ,G)F = α

supp(F ′,G′)F ′ α

supp(F ,G)G = α

supp(F ′,G′)G′


2.5. D–categories. Recall that, given a diagram D, a D–algebra is a pair(A, ABB∈SD(D)), where A is an associative algebra and ABB∈SD(D) is acollection of subalgebras indexed by SD(D) and satisfying

AB ⊆ AB′ if B ⊆ B′ and [AB, AB′ ] = 0 if B ⊥ B′

The following rephrases the notion of D–algebras in terms of their cate-gory of representations.

Definition. A D–category

C = (CB, FBB′)

is the datum of

• a collection of k–linear additive categories CBB⊆D• for any pair of subdiagrams B ⊆ B′, an additive k–linear functorFBB′ : CB′ → CB1

satisfying the following properties

• For any B ⊆ D, FBB = idCB .• For any B ⊆ B′ ⊆ B′′, FBB′ FB′B′′ = FBB′′ .• Given B =

⊔rj=1Bj , with Bj ∈ SD(D) pairwise orthogonal, the

images in End(FB) of the maps

End(FBj )→ End(FBjFBjB) = End(FB)

pairwise commute.

Remark. It may seem more natural to replace the equality of functorsFBB′ FB′B′′ = FBB′′ by the existence of invertible natural transformationsαB′

BB′′ : FBB′ FB′B′′ ⇒ FBB′′ for any B ⊆ B′ satisfying the associativity

constraints αB′

BB′′′ FBB′(αB′′

B′B′′′) = αB′′

BB′′′ (αB′

BB′′)FB′′B′′′ for any B ⊆ B′ ⊆B′′ ⊆ B′′′. A simple coherence argument shows however that this leads to anotion of D–category which is equivalent to the one given above.

Remark. We will usually think of C∅ as a base category and at the func-tors F as forgetful functors. Then the family of algebras End(FB) defines,through the morphisms α, a structure of D–algebra on End(FD). Conversely,every D–algebra A admits such a description setting CB = RepAB for B 6= ∅and C∅ = Vectk, FBB′ = i∗B′B, where iB′B : AB ⊂ AB′ is the inclusion.

Remark. The above definition may be rephrased as follows. Let I(D) bethe category whose objects are subdiagrams B ⊆ D and morphisms B′ → Bthe inclusions B ⊂ B′. Then a D–category is a functor

C : I(D)→ Cat

1When B = ∅ we will omit the index B.


2.6. Strict morphisms of D–categories. The interpretation ofD–categoriesin terms of I(D) suggests that a morphism of D–categories C, C′ is one ofthe corresponding functors

I(D)

C##

C′

==

Cat

This yields the following definition. For simplicity, we assume that C∅ = C′∅.

Definition. A strict morphism of D–categories C, C′ is the datum of

• for any B ⊆ D, a functor HB : CB → C′B• for any B ⊆ B′, a natural transformation

CB′HB′ //

FBB′

C′B′

F ′BB′

γBB′

zCB

HB

// C′B

(2.1)

such that

• H∅ = id• γBB = idHB• For any B ⊆ B′ ⊆ B′′,

γBB′′ = γBB′ γB′B′′where is the composition of natural transformations defined by

CB′′ //

C′B′′

y

CB′ //

C′B′

y

CB // C′B

(2.2)

The diagram (2.1), withB = ∅, induces an algebra homomorphism End(F ′B′)→End(FB′) which, by (2.2) is compatible with the maps End(FB)→ End(FB′)and End(F ′B) → End(F ′B′) for any B ⊂ B′. As pointed out in [TL4, 3.3],this condition is too restrictive and will be weakened in the next paragraph.

2.7. Morphisms of D–categories.

Definition. A morphism of D–categories C, C′, with C∅ = C′∅, is the datumof

• for any B ⊆ D a functor HB : CB → C′B


• for any B ⊆ B′ and F ∈ Mns(B,B′), a natural transformation

CB′HB′ //

FBB′

C′B′

F ′BB′

γFBB′

zCB

HB

// C′B

such that

• H∅ = id• γFBB = idHB• for any B ⊆ B′ ⊆ B′′, F ∈ Mns(B,B′), G ∈ Mns(B′, B′′),

γFBB′ γGB′B′′ = γF∨GBB′′

Remark. For any F ∈ Mns(B′), the natural transformation γFB′ induces an

algebra homomorphism ΨFB′ : End(F ′B′)→ End(FB′) such that the followingdiagram commutes for any B ∈ F

End(F ′B′)ΨFB′ // End(FB′)

End(F ′B)ΨFBB //

OO

End(FB)

OO

In particular, the collection of homomorphisms ΨFD defines a morphism ofD–algebras End(F ′D)→ End(FD) in the sense of [TL4, 3.4].

Remark. The above definition may be rephrased as follows. Let M(D)be the category with objects the subdiagrams B ⊆ D and morphismsHom(B′, B) = Mns(B′, B), with composition given by union. There isa forgetful functor M(D) → I(D) which is the identity on objects andmaps F ∈ Mns(B′, B) to the inclusion B ⊆ B′. Given two D–categoriesC, C′ : I(D) → Cat a morphism C → C′ as defined above coincides with amorphism of the functors M(D)→ Cat given by the composition

M(D) // I(D)C //

C′// Cat

2.8. Quasi–Coxeter categories.

Definition. A labeling of the diagram D is the assignment of an integermij ∈ 2, 3, . . . ,∞ to any pair i, j of distinct vertices of D such that

mij = mji mij = 2

if and only if i ⊥ j.

Let D be a labeled diagram.


Definition. The Artin braid group BD is the group generated by elementsSi labeled by the vertices i ∈ D with relations

SiSj · · ·︸︷︷︸mij

= SjSi · · ·︸︷︷︸mij

for any i 6= j such that mij < ∞. We shall also refer to BD as the braidgroup corresponding to D.

Definition. A quasi–Coxeter category of type D

C = (CB, FBB′, ΦFG, Si)

is the datum of

• a D–category C = (CB, FBB′)• for any elementary pair (F ,G) in Mns(B,B′), a natural transforma-

tion

ΦFG ∈ Aut(FBB′)

• for any vertex i ∈ V(D), an element

Si ∈ Aut(Fi)

satisfying the following conditions

• Orientation. For any elementary pair (F ,G),

ΦGF = Φ−1FG

• Coherence. For any elementary sequencesH1, . . . ,Hm andK1, . . . ,Klin Mns(B,B′) such that H1 = K1 and Hm = Kl,

ΦHm−1Hm · · ·ΦH1H2 = ΦKl−1Kl · · ·ΦK1K2

• Support. For any elementary pair (F ,G) in Mns(B,B′), let S =supp(F ,G), Z = z supp(F ,G) ⊆ D and

F = F|supp(F ,G)z supp(F ,G) G = G|supp(F ,G)

z supp(F ,G)

Then

ΦFG = idBZ ΦF G idB′S


where the expression above denotes the composition of naturaltransformations

CB′

FBB′

FBB′

ΦGF +3

CB′

FSB′

CS

FZS

FZS

ΦGF +3=

CZFBZ

CB CB

• Forgetfulness. For any equivalent elementary pairs (F ,G), (F ′,G′)in Mns(B,B′)

ΦFG = ΦF ′G′

• Braid relations. For any pairs i, j of distinct vertices of B, suchthat 2 < mij <∞, and elementary pair (F ,G) in Mns(B) such thati ∈ F , j ∈ G, the following relations hold in End(FB)

Ad(ΦGF )(Si) · Sj · · · = Sj · Ad(ΦGF )(Si) · · ·

where, by abuse of notation, we denote by Si its image in End(FB)and the number of factors in each side equals mij .

Remark. To rephrase the above definition, consider the 2–category qC(D)

obtained by adding to M(D) a unique 2–isomorphism ϕBB′

GF : F → G for anypair of 1–morphisms F ,G ∈ Mns(B′, B), with the compositions

ϕBB′

HG ϕBB′

GF = ϕBB′

HF and ϕBB′

F2G2 ϕB′B′′F1G1

= ϕBB′′

F2∪F1 G2∪G1

where F ,G,H ∈ Mns(B′, B), B ⊂ B′ ⊆ B′′ and F1,G1 ∈ Mns(B′′, B′),F2,G2 ∈ Mns(B′, B). There is a unique functor qC(D) → I(D) extendingM(D) → I(D), and a quasi–Coxeter category is the same as a 2–functorqC(D)→ Cat fitting in a diagram

qC(D) //

##

Cat

I(D)

<<

Note that, for any B ⊂ B′, the category HomqC(D)(B′, B) is the 1–

groupoid of the De Concini–Procesi associahedron on B′/B [TL4].


2.9.

Definition. A morphism of quasi–Coxeter categories C, C′ of type D is amorphism (H, γ) of the underlying D–categories such that

• For any i ∈ B, the corresponding morphism Ψi : End(F ′i )→ End(Fi)satisfies

Ψi(S′i) = Si

• For any elementary pair (F ,G) in Mns(B,B′),

HB(ΦFG) γFBB′ (Φ′GF )HB′ = γGBB′

in Nat(F ′BB′ HB′ , HB FBB′), as in the diagram

C′B′

Φ′FG//

55

HB′

CB′

ΦFG +3 γGpx

γF

C′B55

HB

CB

Remark. Note that the above condition can be alternatively stated in termsof morphisms ΨF as the identity

ΨG Ad(ΦGF ) = Ad(Φ′GF ) ΨF

2.10. Strict D–monoidal categories.

Definition. A strict D–monoidal category C = (CB, FBB′, JBB′) isa D–category C = (CB, FBB′) where

• for any B ⊆ D, (CB,⊗B) is a strict monoidal category• for any B ⊆ B′, the functor FBB′ is endowed with a tensor structureJBB′

with the additional condition that, for every B ⊆ B′ ⊆ B′′, JBB′ JB′B′′ =JBB′′ .

Remark. The tensor structure JB induces on End(FB) a coproduct ∆B :End(FB)→ End(F 2

B), where F 2B := ⊗ (FB FB), given by

gV V ∈CB 7→ ∆B(g)VW := Ad(JBVW )(gV⊗W )V,W∈CB


Moreover, for any B ⊆ B′, End(FB) is a subbialgebra of End(FB′), i.e., thefollowing diagram is commutative

End(FB)

∆B // End(F 2B)

End(FB′)∆B′

// End(F 2B′)

Remark. Note that a strict D–monoidal category can be thought of asfunctor

C : I(D)→ Cat⊗0

where Cat⊗0 denotes the 2–category of strict monoidal category, with monoidalfunctors and gauge transformations.

Definition. A morphism of strict D–monoidal categories is a natural trans-formation of the corresponding 2–functors M(D)→ Cat⊗0 , obtained by com-position with M(D)→ I(D).

2.11. D–monoidal categories.

Definition. A D–monoidal category

C = ((CB,⊗B,ΦB), FBB′, JFBB′)

is the datum of

• A D–category ((CB, FBB′) such that each (CB,⊗B,ΦB) is atensor category, with C∅ a strict tensor category, i.e., Φ∅ = id.

• for any pair B ⊆ B′ and F ∈ Mns(B,B′), a tensor structure JBB′

Fon the functor FBB′ : CB′ → CB

with the additional condition that, for any B ⊆ B′ ⊆ B′′, F ∈ Mns(B′′, B′),G ∈ Mns(B′, B),

JGBB′ JFB′B′′ = JF∪GBB′′

Remark. The usual comparison with the algebra of endomorphisms leadsto a collection of bialgebras (End(FB),∆F , ε) endowed with multiple coprod-ucts, indexed by Mns(B).

Remark. A D–monoidal category can be thought of as a functor M(D)→Cat⊗ fitting in a diagram

M(D)

// Cat⊗

wI(D) // Cat


Accordingly, a morphism of monoidal D–categories is one of the correspond-ing functors.

M(D)

C%%

C′

<<

Cat⊗

2.12. Fibered D–monoidal categories. We shall often be concerned withD–monoidal categories such that the underlying categories (CB,⊗B) arestrict, and the functors FBB′ : (CB′ ,⊗B′) → (CB,⊗B) are tensor functors.This may be described in terms of the category M(D) as follows. Let DCat⊗

be the 2–category of Drinfeld categories, that is strict tensor categories (C,⊗)endowed with an additional associativity constraint Φ making (C,⊗,Φ) amonoidal category. There is a canonical forgetful 2–functor DCat⊗ → Cat⊗0 .

We shall say that a D–monoidal category fibers over a strict D–monoidalcategory if the corresponding functor M(D)→ Cat⊗ maps into DCat⊗ andfits in a commutative diagram

M(D)

// DCat⊗

v~I(D) // Cat⊗0

In this case, the coproduct ∆F on a bialgebra End(FB) is the twist of areference coassociative coproduct ∆0 on End(FD) such that ∆0 : End(FB)→End(F 2

B).

2.13. Braided D–monoidal categories.

Definition. A braided D–monoidal category

C = ((CB,⊗B,ΦB, βB), (FBB′ , JFBB′)is the datum of

• a D–monoidal category ((CB,⊗B,ΦB), (FBB′ , JFBB′)• for every B ⊆ D, a commutativity constraint βB in CB, defining a

braiding in (CB,⊗B,ΦB).

Remark. Note that the tensor functors (FBB′ , JFBB′) : CB′ → CB are not

assumed to map the commutativity constraint βB′ to βB.

Definition. A morphism of braided D–monoidal categories from C to C′ isa morphism of the underlying D–monoidal categories such that the functorsHB : CB → C′B are braided tensor functors.

Remark. The fact that HB are braided tensor functors automatically im-plies that

Ψ⊗2F ((RB)JF ) = (R′B)J ′F


in analogy with [TL4], where RB = (12) βB, and we are assuming thatC∅ = C′∅ is a symmetric strict tensor category.

2.14. Quasi–Coxeter braided monoidal categories.

Definition. A quasi–Coxeter braided monoidal category of type D

C = ((CB,⊗B,ΦB, βB), (FBB′ , JFBB′), ΦFG, Si)

is the datum of

• a quasi–Coxeter category of type D,

C = (CB, FBB′, ΦFG, Si)

• a braided D–monoidal category

C = ((CB,⊗B,ΦB, βB), (FBB′ , JBB′

F ))

satisfying the following conditions

• for any B ⊆ B′, and G,F ∈ Mns(B,B′), the natural transformationΦFG ∈ Aut(FBB′) determines an isomorphism of tensor functors

(FBB′ , JGBB′)→ (FBB′ , J

FBB′), that is for any V,W ∈ CB′ ,

(ΦGF )V⊗W (JFBB′)V,W = (JGBB′)V,W ((ΦGF )V ⊗ (ΦGF )W )

• for any i ∈ D, the following holds:

∆i(Si) = (Ri)21Ji · (Si ⊗ Si)

A morphism of quasi–Coxeter braided monoidal categories of type D isa morphism of the underlying quasi–Coxeter categories and braided D–monoidal categories.

Remark. A quasi–Coxeter braided monoidal category of type D determinesa 2–functor qC(D)→ Cat⊗ fitting in a diagram

qC(D)

// Cat⊗

wI(D) // Cat

Note however that this functor does not encode the commutativity con-straints βB and automorphisms Si.

3. Etingof-Kazhdan Quantization

We review in this section the results obtained in [EK96, EK08].


3.1. Lie bialgebras and Manin triples. A Manin triple is the data ofa Lie algebra g with a nondegenerate invariant inner product 〈, 〉, g+, g−isotropic Lie subalgebras, such that

(M1) g = g+ ⊕ g− as vector space(M2) the inner product defines an isomorphism g+ ' g∗−(M3) the commutator of g is continuous with respect to the topology ob-

tained by putting the discrete topology on g− and the week topologyon g+

Under these assumptions, g− is naturally a Lie bialgebra, since the isomor-phism g+ ' g∗− defines a commutator on g∗−. This commutator turns out tobe a 1-cocycle (cf. [Dri86]). Notice that, in absolute generality, g+ is only atopological Lie bialgebra, i.e. δ(g+) ⊂ g+⊗g+.

By Manin subtriple we mean a topological Lie subalgebra gD ⊂ g suchthat the restriction 〈, 〉|gD defines an invariant inner product on gD withisotropic Lie subalgebras gD,± such that (gD, gD,+, gD,−) is a Manin triple.

It is known that, if g− is N-graded of finite type (i.e. with dim(g−)n <∞for all n ∈ N), then we can formulate the definition of graded Manin triplewhere the isomorphism g+ ' g∗− is instead an isomorphism with the gradeddual of g−, i.e. g∗− =

⊕n∈N(g−)∗n. Under this assumption, g+ and then g

have natural structures of Lie bialgebras.

3.2. Glossary of topology. Recall few basic definition from [EK96]. Let Sbe a set, T a topological space and TS the space of maps from S to T . Thisspace has a natural weak topology, which is the weakest topology in whichall the evaluation maps TS → T are continuous. Namely, let B be the basisof the topology on T . For an integer n ≥ 1, elements sii=1...n ⊂ S andopen sets Ui ∈ B, i = 1 . . . n, define

Y (s1, . . . , sn, U1, . . . , Un) := f ∈ TS | f(si) ∈ Ui, i = 1, . . . , n

Let B be the collection of all such sets Y . This is a basis of a topology onTS which is called weak topology.

Let k be a field of characteristic zero with the discrete topology. Let Vbe a topological vector space over k. The topology on V is called linear ifopen subspaces of V form a basis of neighborhood of zero. Notice that inany topological space, an open subspace is also closed.

Let V be a topological vector space over k with linear topology. V is calledseparated if the map V → lim(V/U) is a monomorphism, where U runs overopen subspaces of V . V is called complete if the map V → lim(V/U) is anepimorphism.


In particular, if M is a complete vector space with a countable basis ofneighborhoods of zero, then there exists a filtration M = M0 ⊃ M1 ⊃ . . .such that ∩n≥0Mn = 0 and Mn is a basis of neighborhoods of zero inM . In this case, M = limn→∞M/Mn.

Let M,N be topological spaces over k. We denote by Homk(M,N) thespace of continuous linear operators from M to N , equipped with the weaktopology. A basis of neighborhoods of zero in Homk(M,N) is generated bysets of the form Y (v, U) = A ∈ Homk(M,N) | Av ∈ U, where v ∈M andU ⊂ N is an open set.

In particular, if N = k with the discrete topology, the space Homk(M,k)is the space of all continuous linear functionals on M , which we denote byM∗. Clearly, a basis of neighborhoods of zero in M∗ consists of orthogo-nal complements of finite-dimensional subspaces in M . In particular, if Mis discrete (so the topological dual coincide with the algebraic dual), thenthe canonical embedding M → (M∗)∗ is an isomorphism of linear spaces.However, if M is infinite dimensional (this implies that M∗ is not discrete),this embedding is not an isomorphism of topological vector spaces since thespace (M∗)∗ is not discrete.

Let V,W be complete vector spaces. Consider the space

V ⊗W := lim(V/V ′)⊗ (W/W ′)

where the limit is taken over open subspaces V ′ ⊂ V,W ′ ⊂ W . A basis ofneighborhoods of zero in V ⊗W is the collection of subspaces V ⊗W ′+V ′⊗W ,with V ′,W ′ open subspaces in V,W .

Remark. (i) Any discrete vector space V is complete. Moreover, thetopological vector space M = V [[~]] of formal power series in h withcoefficients in V is a complete vector space.

(ii) Let V be a complete vector space, U ⊂ V an open set. Then U iscomplete and V/U is discrete.

(iii) Let V a discrete space. Then V ⊗k[[~]] = V [[~]].

The additive category of completed vector spaces, in which morphism arecontinuous linear operators, equipped with completed tensor product ⊗, isa symmetric tensor category.

Let V be a complete vector space over k. Then the space V [[~]] = V ⊗k[[~]]of formal power series in h with coefficients in V is also a complete vec-tor space. Moreover, V [[~]] has a natural structure of a topological k[[~]]-module. We call a topological k[[~]]-module complete if it is isomorphic toV [[~]] for some complete V .


Let A be the category of complete k[[~]]-module, where morphisms arecontinuous k[[~]]-linear maps. It is an additive category. We can define atensor product on A to be the quotient of the tensor product V ⊗W by theimage of the operator h⊗1−1⊗h. This tensor product will be still denotedby ⊗. It is clear that for V,W ∈ A V ⊗W ∈ A. The category A equippedwith the bifunctor ⊗ is a symmetric monodical category.

We have an extension of scalar functor from the category of completevector spaces to A, V 7→ V [[~]]. This functor respects the tensor product,i.e. (V ⊗W )[[~]] is naturally isomorphic to V [[~]]⊗W [[~]].

3.3. Equicontinuous modules. Let M be a topological vector space overk, and Ax, x ∈ X be a family of elements of EndM . We say that thefamily Ax is equicontinuous if, for every neighborhood of zero U ⊂ Mthere exists another neighborhood of zero U ′ ⊂ M such that AxU ⊂ U ′ forall x ∈ X. For example, if M is complete and A ∈ EndM is any continuouslinear operator, then λA, λ ∈ k is equicontinuous.Fix now a topological Lie algebra g.

Definition. Let M be a complete vector space. We say that M is anequicontinuous g-module if:

(E1) the map πM : g→ EndM is a continuous homomorphism of topo-logical Lie algebras;

(E2) πM (g)g∈g is an equicontinuous family of linear operators.

Example. If M is a complete vector space with a trivial g-module struc-ture, then M is an equicontinuous g-module.

Let V,W be equicontinuous g-modules. Then V ⊗W has a natural struc-ture of equicontinuous g-module. Moreover, (V ⊗W )⊗U is naturally iden-tified with V ⊗(W ⊗U) for any equicontinuous g-modules V,W,U . The cat-egory of equicontinuous g-modules is then a monoidal category. It is alsosymmetric since V ⊗W and W ⊗V are identified by permutation of compo-nents. We denote this category by Repeq g.

Let (g, g+, g−) be a Manin triple. Let ai, i ∈ I be a basis for g− andbi, i ∈ I ⊂ g+ be the linear functionals on g− defined by 〈bi, aj〉 = δij .

Lemma. Let M be a complete vector space with a continuous homomor-phism g → EndM . Then for all m ∈ M and any neighborhood of zeroU ⊂M , one has bim ∈ U for all but finitely many i ∈ I.

Proof. A basis of neighborhood of zero in g+ is given by sets

Yj = f ∈ g+ | 〈f, aj〉 = 0

Let is ∈ I be any sequence of distinct elements. Since for any Yj thereexists is0 ∈ I such that bis ∈ Yj for s > s0, then lims→∞ b

is = 0. By


continuity of the map g → EndM , lims→∞ bism = 0 for any m ∈ M . This

means that bim ∈ U for almost all i ∈ I.

3.4. Verma modules. In [EK96], Etingof and Kazhdan constructed twomain examples of equicontinuous g-modules in the case when g belongs toa Manin triple (g, g+, g−).Consider the Verma modules

M+ = Indgg− 1 M− = Indg

g+1

where 1 denotes the trivial 1-dimensional representation. The modules M±are freely generated by U(g±) by a vector 1± such that g∓1±. Therefore,they are naturally identified as vector spaces to U(g±) via x1± → x.The modules M− and M∗+, with appropriate topologies, are equicontinuousg-modules.

Lemma. The module M−, equipped with the discrete topology, is an equicon-tinuous g-module.

The topology on M+ comes from the identification of vector spaces M+ 'U(g+) =

⋃n≥0 U(g+)n, where U(g+)n is the set of elements of degree ≤ n.

Furthermore we have a linear isomorphism

ξn :

n⊕j=0

Sjg+ → U(g+)n

where Sjg+ is the j-th symmetric power of g+ and, as usual, S0g+ = g⊗0+ =

k. Since Sjg+ has a natural weak topology coming from the embedding

Sjg+ → (g⊗j− )∗, the isomorphism ξn defines a topology on U(g+)n. More-over, by definition, U(g+)m is a closed subspace in U(g+)n. This allowsus to equip U(g+), and therefore M+, with the topology of the colimit. Bydefinition, a set U ⊂ U(g+) is open in this topology if and only if U∩U(g+)nis open for all n.

Lemma. For any g ∈ g, πM+(g) ∈ EndM+ is a continuous operator.

Consider now the vector space M∗+ of continuous linear functionals onM+. By definition, M∗+ is naturally isomorphic to lim(U(g+)n)∗ as n→∞.

As vector space, (U(g+)n)∗ = (Sjg+)∗ = Sjg−. Therefore, it is natural toput the discrete topology on (U(g+)n)∗. This equip the module M∗+ witha natural structure of a complete vector space. It is also equipped with afiltration by subspaces (M∗+)n satisfying

0→ (M∗+)n →M∗+ → (U(g+)n)∗ → 0

and such that M∗+ = limM∗+/(M∗+)n.

Remark. The topology of the limit on M∗+ does not coincide, in general,with the weak topology of the dual. In fact, it is stronger than the weaktopology.


By previous lemma, the dual of the g-action on M+ defines a g-action onM∗+.

Lemma. M∗+ is an equicontinuous g-module.

Remark. if dim g− = ∞, then M+ is not, in general, an equicontinuousg-module, since the family of operators πM+(g), g ∈ g− may fail to beequicontinuous.

3.5. The Casimir element. Consider now the tensor product a⊗ a∗ andits embedding into End a by (x ⊗ f)(y) = f(y)x, for x, y ∈ a, f ∈ a∗. Thisembedding defines a topology on a⊗ a∗, obtained by restriction of the weaktopology in End a. Since the image of a⊗a∗ is dense in End a, the topologicalcompletion a⊗a∗ is identified with End a.

Lemma. Let V,W ∈ Repeq g. The map πV ⊗ πW : a ⊗ a∗ → End(V ⊗W )extends to a continuous map a⊗a∗ → End(V ⊗W ).

Let r ∈ a⊗a∗ be the element corresponding to the identity operator underthe identification with End a. Let rop ∈ a∗⊗a to be the element obtainedfrom r by permutation of components. We define the Casimir element byΩ := r+rop ∈ a⊗a∗⊕a∗⊗a. Clearly, r =

∑ai⊗bi and Ω =

∑(ai⊗bi+bi⊗ai).

Consider V,W ∈ Repeq g and denote ΩVW := πV ⊗ πW (Ω). This endomor-phism of V ⊗W is well defined and continuous. Moreover, it commutes withthe action of g, so it is an endomorphism of V ⊗W as equicontinuous g-module.

Remark. Although the Casimir operator is defined on the tensor product ofany two equicontinuous g-modules, the Casimir element C =

∑(aib

i + biai)has no meaning as an operator on an equicontinuous g-module.

3.6. Drinfeld category. Let Rep (g)[[~]] denote the category whose objectare deformed equicontinuous g-modules and

HomRep (g)[[~]](U [[~]], V [[~]]) := Homg(U, V )[[~]]

Following [Dri89], we can define a structure of braided monoidal categoryon Rep g[[~]] using a fixed Lie associator Φ, the bifunctor ⊗ and the Casimir

operator Ω. Define the braiding βVW = σ eh2

ΩVW . Let now γ be the func-torial isomorphism defined by γVW = β−1

WV . It is clear that γ is a braidingon Rep g[[~]]. We denote this braided tensor category RepΦ(Ug[[~]]).

It is an easy exercise to prove that the category of equicontinuous g–modules is equivalent to the category of Yetter-Drinfeld module over g−,YDg−. The equivalence holds at the level of tensor structure induced bythe choice of an associator Φ.


3.7. Properties of the Verma modules. The modules M± are identifiedas vector spaces with U(g±). Thus, we can define the maps i± : M± →M±⊗M± given by comultiplication in the enveloping universal algebrasU(g±). These maps are U(g)-intertwiners since they are U(g±)-intertwinersand map the vector 1± to the g∓-invariant vector 1± ⊗ 1±.For any f, g ∈ M∗+, consider the linear functional M+ → k defined byv 7→ (f ⊗ g)(i+(v)). This functional is continuous, so it belongs to M∗+.Define accordingly the map i∗+ : M∗+⊗M∗+ →M∗+. It is clear that i∗+ is con-

tinuous and it extends to a morphism in Rep (g)[[~]], i∗+ : M∗+⊗M∗+ →M∗+.

Let now V ∈ Rep (g). Consider the space Homg(M−,M∗+⊗V ), where as

usual Homg denotes the set of continuous homomorphisms. Equip this spacewith the weak topology.

Lemma. The complete vector space Homg(M−,M∗+⊗V ) is isomorphic to V

as topological vector space. The isomorphism is given by f 7→ (1+⊗1, f(1−)).

Proof. By Frobenius reciprocity, Homg(M−,M∗+⊗V ) is isomorphic, as a

topological vector space, to the space of invariants (M∗+⊗V )g+ , via f 7→f(1−). Consider now the space Homk(M+, V ), equipped with the weaktopology, and the map

φ : (M∗+⊗V )→ Homk(M+, V )

given by φ(f⊗v)(x) = f(x)v, f ∈M∗+, x ∈M+, v ∈ V . Clearly, φ is injectiveand continuous. Moreover, φ restricts to an isomorphism

φ : (M∗+⊗V )g+ → Homg(M+, V )

Indeed, φ(M∗+⊗V )g+ ⊂ Homg(M+, V ). So, we need to show that any con-

tinuous g−-intertwiner g : M+ → V is of the form φ(g′), g′ ∈ (M∗+⊗V )g+ .Notice that, since V is complete, we have V ' limV/U , where the limit istaken over the open sets of V . Therefore, we have a canonical isomorphismHomk(M+, V ) ' lim Homk(M+, V/U).For any map g ∈ Homg+(M+, V ), g−1(U) ∩ U(g+)n is open in U(g+)n, i.e.it is a complement to a finite dimensional subspace. Then composing withthe projection pU : V → V/U , the map gU := pU g sends U(g+)n into afinite dimensional linear subspace in V/U and, taking the limit

(M∗+⊗V )g+ = limn

limU⊂V

((U(g+)n)∗ ⊗ V/U)

Homg+(M+, V ) = limn

limU⊂V

Homg+((U(g+))n, V/U)

it is possible to determine g′ ∈ (M∗+⊗V )p+ such that g = φ(g′).

By Frobenius reciprocity, the space Homcg+

(M+, V ) is isomorphic to V as

a topological vector space via f 7→ f(1+). The lemma is proved.


3.8. The fiber functor and the EK quantization. Let then F : RepΦ(Ug−[[~]])→A be the functor given by

F (V ) = Hom(M−,M∗+ ⊗ V )

There is a natural transformation

J ∈ Nat(⊗ (F F ), F ⊗)

defined, for any v ∈ F (V ), w ∈ F (W ), by

JVW (v ⊗ w) = (i∨+ ⊗ 1⊗ 1)A−1β−123 A(v ⊗ w)i−

where A is defined as a morphism

(V1 ⊗ V2)⊗ (V3 ⊗ V4)→ V1 ⊗ ((V2 ⊗ V3)⊗ V4)

by the action of (1⊗ Φ2,3,4)Φ1,2,34.

Theorem. The natural transformation J is invertible and defines a tensorstructure on the functor F .

The tensor functor (F, J) is called fiber functor. The algebra of endo-morphisms of F is therefore naturally endowed with a topological bialgebrastructure, as described in the previous section.1

The object F (M−) ∈ A has a natural structure of Hopf algebra, definedby the multiplication

m : F (M−)⊗ F (M−)→ F (M−) m(x, y) = (i∨+ ⊗ 1)Φ−1(1⊗ y)x

and the comultiplication

∆ : F (M−)→ F (M−)⊗ F (M−) ∆(x) = J−1(F (i−)(x))

The algebra F (M−) is naturally isomorphic as a vector space with M− 'Ug− and

Theorem. The algebra UEK~ g− = F (M−) is a quantization of the algebra

Ug−.

In [EK98], it is shown that this construction defines a functor

QEK : LBA(k)→ QUE(K)

where LBA(k) denotes the category of Lie bialgebras over k and QUE(K)denotes the category of quantum universal enveloping algebras over K =k[[~]]. Another important result in [EK98] states the invertibility of thefunctor QEK , obtained using the action of the Grothendieck–Teichmullergroup.

1By topological bialgebra we do not mean topological over k[[~]]. We are insteadreferring to the fact that the algebra End(F ) has a natural comultiplication ∆ : End(F ) →End(F 2), where End(F 2) can be interpreted as an appropriate completion of End(F )⊗2.


The map

m− : UEK~ g− → End(F ) m−(x)V (v) = (i∨+ ⊗ 1)Φ−1(1⊗ v)x

where V ∈ YDΦ(Ug−[[~]]) and v ∈ F (V ), is, indeed, an inclusion of Hopfalgebras. The map m− defines an action of UEK

~ g− on F (V ). Moreover, themap

F (V )→ F (M−)⊗ F (V ) v 7→ RJ(1⊗ v)

where RJ denotes the twisted R–matrix, defines a coaction of UEK~ g− on

F (V ) compatible with the action, therefore

Theorem. The fiber functor F : YDΦ(Ug−[[~]])→ A lifts to an equivalenceof braided tensor categories

FEK : YDΦ(Ug−[[~]])→ YD(UEK~ g−)

The fact that FEK is an equivalence of categories is a consequence of theinvertibility of QEK .

3.9. Generalized Kac-Moody algebras. Denote by k a field of charac-teristic zero. We recall definitions from [Kac] and [EK08]. Let A = (aij)i,j∈Ibe an n×n symmetrizable matrix with entries in k, i.e. there exists a (fixed)collection of nonzero numbers dii∈I such that diaij = djaji for all i, j ∈ I.Let (h,Π,Π∨) be a realization of A. It means that h is a vector space ofdimension 2n − rk(A), Π = α1, . . . , αn ⊂ h∗ and Π∨ = h1, . . . , hn ⊂ hare linerly independent and (αi, hj) = aji.

Definition. The Lie algebra g = g(A) is generated by h, ei, fii∈I withdefining relations

[h, h′] = 0 h, h′ ∈ h; [h, ei] = (αi, h)ei

[h, fi] = −(αi, h)fi; [ei, fj ] = δijhi

Assume that A has been fixed. There exists a unique maximal ideal r ing that intersect h trivially. Let g := g/r. The algebra g is called general-ized Kac-Moody algebra. The Lie algebra g is graded by principal gradationdeg(ei) = 1, deg(fi) = −1,deg(h) = 0, and the homogenous component arefinite-dimentional.

Let us now choose a non degenerate bilinear symmetric form on h suchthat 〈h, hi〉 = d−1

i (αi, h). Following [Kac], there exists a unique extensionof the form 〈, 〉 to an invariant symmetric bilinear form on g. For this ex-tension, one gets 〈ei, fj〉 = δijd

−1i . The kernel of this form is r and thus the

form descends to a non degenerate form on g.

Let n±, b± be the nilpotent and the Borel subalgebras of g, i.e. n± aregenerated by ei, fi, respectively, and b± := n± ⊕ h. Since [n±, h] ⊂ n±,


we get Lie algebra maps ¯ : b± → h and we can consider the embeddings ofLie subalgebras η± : b± → g⊕ h given by

η±(x) = (x,±x)

Define the inner product on g⊕ h by 〈, 〉g⊕h = 〈, 〉g − 〈, 〉h.

Proposition. The triple (g⊕ h, b+, b−) with inner product 〈, 〉g⊕h and em-beddings η± is a graded Manin triple.

Under the embeddings η±, the Lie subalgebras b± are isotropic with re-spect to 〈, 〉g⊕h. Since 〈, 〉g and 〈, 〉h are invariant symmetric non degeneratebilinear form, so is 〈, 〉g⊕h.

The proposition implies that g ⊕ h, b+, b− are Lie bialgebras. Moreover,b+ ' bcop− as Lie bialgebras (where ∗ denotes the restricted dual space andby cop we mean the opposite cocommutator). The cocommutator δ on thesealgebras is easily computed:

δ(h) = 0, h ∈ h ⊂ b±;

δ(ei) =di2

(ei ⊗ hi − hi ⊗ ei) =di2ei ∧ hi; δ(fi) =

di2fi ∧ hi

The Lie subalgebra (0, h) | h ∈ h is therefore an ideal and a coideal ing ⊕ h, and so the quotient g = (g ⊕ h)/h is also a Lie bialgebra with Liesub-bialgebras b± and the same cocommutator formulas.

3.10. Quantization of Kac–Moody algebras and category O. In [EK08],Etingof and Kazhdan proved that, for any symmetrizable irreducible Kac-Moody algebra g, the quantization UEK

~ g is isomorphic with the Drinfeld–Jimbo quantum group U~g.

In particular, they construct an isomorphism of Hopf algebras U~b+ 'UEK~ b+, inducing the identity on Uh[[~]], where b+ is the Borel subalgebra

and h is the Cartan subalgebra of g. Thanks to the compatibility with thedoubling operations

DUEK~ b+ ' UEK

~ Db+

proved by Enriquez and Geer in [EG], the isomorphism for the Borel subal-gebra induces an isomorphism U~g ' UEK

~ g.

Recall that the category O for g, denoted Og is defined to be the categoryof all h–diagonalizable g–modules V , whose set of weights P(V ) belong to aunion of finitely many cones

D(λs) = λs +∑i

Z≥0αi λs ∈ h∗, s = 1, ..., r

and the weight subspaces are finite–dimensional. We denote by Og[[~]] thecategory of trivially deformed g–representations,i.e., representations of g ontopologically free k[[~]]–modules with the above properties (with weights in


h∗[[~]]).

In a similar way, one defines the category OU~g: it is the category ofU~g–modules which are topologically free over k[[~]] and satisfy the sameconditions as in the classical case.

The morphism of Lie bialgebras

Db+ → g ' Db+/(h ' h∗)

gives rise to a pullback functor

Og → YD(Ub+) =⇒ Og,Φ[[~]]→ YDΦ(Ub+[[~]])where Og,Φ denotes the category Og with the tensor structure of the Drinfeldcategory. Similarly, the morphism of Hopf algebras

DUEK~ b+ → UEK

~ g ' U~ggives rise to a pullback functor

OU~g → YD(UEK~ b+)

Theorem. The equivalence FEK reduces to an equivalence of braided tensorcategories

FEKO : Og,Φ[[~]]→ OU~g

which is isomorphic to the identity functor at the level of h–graded k[[~]]–modules.

3.11. The isomorphism ΨEK . In [EK08], Etingof–Kazhdan showed thatthe equivalence FEK induces an isomorphism of algebras

ΨEK : Ug[[~]]→ U~g

whereUg = limUβ Uβ = Ug/Iβ, β ∈ NI

Iβ being the left ideal generated by elements of weight less or equal β (anal-

ogously for U~g, cf. [EK08, Sec. 4]).

Proposition. The isomorphism ΨEK coincides with the isomorphism in-duced by the equivalence FEKO , as explained in Section 2.1.

Proof. The identification of the two isomorphism is constructed in thefollowing way:

(a) First, we show that there is a canonical map

End(FO)→ CEnd(U)

(Endg(U))

(b) There is a canonical multiplication in U , so that(i) There is a canonical map

CEnd(U)

(Endg(U))→ U


(ii) For every V ∈ O the action of Ug lifts to an action of U

Ug //

End(V )

U

;;

(c) It defines a map U → End(FO) and we have an isomorphism ofalgebras

U ' End(FO)

If g is a semisimple Lie algebra, the equivalence of categories FEK leadsto an isomorphism of algebras

U(Db+)[[~]] ' DUEK~ b+ =⇒ Ug[[~]] ' U~g

which is the identity modulo h. Toledano Laredo proved in [TL4, Prop. 3.5]that such an isomorphism cannot be compatible with all the isomorphisms

Uslαi2 [[~]] ' U~slαi2 ∀i

where αi are the simple roots of g. This amounts to a simple proof thatthe isomorphism ΨEK cannot be, in general, an isomorphism of D–algebras.

4. Generalized Etingof–Kazhdan functor

4.1. In this section, we define parabolic subalgebras induced by a fixed sub–Manin triple gD ⊂ g. We use these subalgebras to decompose g with respectto gD and construct a relative version of the Verma modules M+,M−. Wethen construct a tensor structure on the restriction functor, proving thefollowing

Theorem. For any sub–Manin triple iD : (gD, gD,+, gD,−) → (g, g+, g−),there is a tensor functor

Γ : RepΦ(Ug[[~]])→ RepΦD(UgD[[~]])

canonically isomorphic to the restriction functor i∗D.

4.2. Parabolic Lie subalgebras. Given an inclusion of Manin triples iD :gD ⊂ g, we denote by m± the subspaces

m± = g± ∩ (gD,∓)⊥

Proposition.

(i) The subspaces m± ⊆ g± are ideals in g±, so that

g± = m± o gD,±


(ii) [m±, gD] ⊂ m± so that the subspaces

p± = m± o gD

are Lie subalgebras of g.(iii) If the inclusion iD is compatible with a finite N–grading, then the

Lie subalgebras m± ⊆ g± are coideals, i.e.,

δ(m±) ⊂ m± ⊗ gD,± + gD,± ⊗m±

and p± are Lie subbialgebras with homomorphisms of Lie bialgebras

p± ⊂ g p± → gD

Proof. For any x ∈ m+ and y1, y2 ∈ gD,

〈[x, y1], y2〉 = 〈x, [y1, y2]〉 = 0

and therefore [m+, gD] ⊂ m+. For any x1, x2 ∈ m+ and y ∈ gD,−, we have,since gD,− is a Lie sub-bialgebra of g−,

〈[x1, x2], y〉 = 〈x1 ⊗ x2, δ(y)〉 = 0 or 〈[x1, x2], y〉 = 〈x1, [x2, y]〉 = 0

since [x2, y] ∈ m+ ⊂ g+ that is isotropic. Therefore m+ is a Lie subalgebraof g+. On the other hand, for any x ∈ m+ and y1, y2 ∈ gD,

〈δ(x), y1 ⊗ y2〉 = 〈x, [y1, y2]〉 = 0

and δ(m+) ⊂ m+ ⊗ g+ + g+ ⊗ m+, but m+ is not in general a Lie sub-bialgebra. Analogously for m−.We then get a vector space decomposition g± = m± ⊕ gD,± and a Lie bial-gebra map g± → gD,±. It is also possible to define the Lie subalgebras

p± = m± ⊕ gD ⊂ g

If we assume the existence of a compatible grading on g and gD, as explainedabove, then the natural maps

p± ⊂ g p± → gD

are morphisms of Lie bialgebras. This completes the proof of the proposition.

4.3. Diagrammatic Lie bialgebras. Assume now that g is a Kac-Moodyalgebras of Dynkin type Dg. For any connected Dynkin subdiagram D ⊂Dg, we can identify a Lie sub-bialgebra gD ⊂ g. Moreover, the Manintriple attached to gD, (gD⊕hD, bD,+, bD,−), is a graded Manin sub-triple of(g⊕ h, b+, b−).

4.4. The relative Verma Modules.

Definition. Given a sub-Manin triple gD ⊂ g and the subsequent decom-position g = m− ⊕ p+, we define the relative Verma modules

L− = indgp+

1 N+ = indgm− 1

Lemma. For any sub-Manin triple gD ⊂ g, the g-modules L−, N∗+ are

equicontinuous.


4.4.1. Equicontinuity of L−. As vector spaces,

L− = Indgp+C ' Um−

so it is natural to equip L− with the discrete topology. Consequently, condi-tion (e1) becomes trivial and condition (e2) reduces to verify that, for everyelement v ∈ L− the set

Yv = b ∈ g+| b.v = 0

is a neighborhood of zero in g+. Since Um− embeds naturally in Ug− theproof is identical to [EK96, Lemma 7.2].We proceed by induction on the length of v = ai1 . . . ain1−. If n = 0,then v = 1− and Yv = g+. If n > 1, then assume v = ajw, with w =ai1 . . . ain−11− and Yw open in g+. For every x ∈ g+

x.v = x.(ajw) = [x, aj ].w + (ajx).w

Call Z the subset of g+

Z = x ∈ g+| [x, aj ] ∈ Yw

Z is open in g+, by continuity of bracket [, ], and clearly Z ∩ Yw ⊂ Yv.

4.4.2. Equicontinuity of N∗+. As a vector space,

N+ = Indgm−C ' Up+ ' colimUnp+

where Unp+ is the standard filtration of Up+ and

Unp+ 'n⊕j=0

Sjp+ 'n⊕j=0

j⊕i=0

(Sj−ig+ ⊗ SigD,−

)We turn this isomorphism into an isomorphism of topological vector spaces,taking on Sj−ig+ and SigD,− the topology induced by the embeddings

Sj−ig+ → (g⊗j−i− )∗ SigD,− → (g⊗iD,−)

With respect to this topology, Ump+ is closed inside Unp+ for m < n. N+

is therefore equipped with the topology of the direct limit, i.e. U ⊂ N+ isopen iff U ∩ Unp+ is open for each n.

In order to define the structure of g-module on N∗+ we need the followinglemma.

Lemma. Let g ∈ g. The map πN+(g) : N+ → N+ is a continuous operator.


Proof. Let g ∈ g. We need to show that for any neighborhood of theorigin U ⊂ N+, there exists a neighborhood of zero U ′ ⊂ N+ such thatπN+(g)U ′ ⊂ U . Since the topology on SigD,− is discrete, U is a subset of

the form U ⊗ 1 and the construction of U ′ follows from [[EK96], Lemma7.3].

As vector spaces, then, we have

N∗+ ' (Up+)∗ ' lim(Unp+)∗

Define a filtration of N∗+, (N∗+)n, by

0→ (N∗+)n → (Up+)∗ → (Unp+)∗ → 0

So we have

N∗+ ⊃ (N∗+)0 ⊃ (N∗+)1 ⊃ · · · ⊃ (N∗+)n ⊃ (N∗+)n+1 ⊃ · · ·

and

N∗+ ' limN∗+/(N∗+)n

Lemma. πN∗+(g)g∈g is an equicontinuous family of operators.

Proof. Let fn ∈ (N∗+)n, i.e. f ∈ HomC(N+,C) such that f |Unp+ = 0. Sinceg acts on N+ simply by multiplication, clearly, if g ∈ p+, g.fn restricted toUn−1p+ is the zero map. So

g.(N∗+)n ⊂ (N∗+)n−1

If g ∈ m− and xi ∈ Up+ for i = 1, . . . , n, then in Ug we have

gx1 · · ·xn = x1 · · ·xng −n∑i=0

x1 · · ·xi−1[xi, g]xi+1 · · ·xn

where [xi, g] ∈ g. Iterating, we get (g.f)(x1 · · ·xn) = 0, therefore

g.(N∗+)n ⊂ (N∗+)n ⊂ (N∗+)n−1

Then, for any neighborhood of zero U = (N∗+)n, it is enough to take U ′ =(N∗+)n+1 to get

πN∗+(g)(N∗+)n+1 ⊂ (N∗+)n ∀g ∈ g

Lemma. The map πN∗+ : g→ End(N∗+) is a continuous map.


Proof. Since g− is discrete, it is enough to check that, for any f ∈ N∗+ andn ∈ N , the subset

Y (f, n) = b ∈ g+| b.f ∈ (N∗+)nis open in g+, i.e.

bi.f ∈ (N∗+)n for a.a. i ∈ ISince f ∈ N∗+ ' limN∗+/(N

∗+)n, we have f = fn where fn is the class of f

modulo (N∗+)n. Therefore bi.f ∈ (N∗+)n iff

(bi.f)n = bi.fn+1 = 0

Now, for any x1 · · ·xn ∈ Unp+, we have

bi.fn+1(x1 · · ·xn) = −fn+1(bix1 · · ·xn) = 0

for a.a. i ∈ I and the lemma is proved (it is enough to exclude the indicescorresponding to the generators involved in the expression of fn+1).

As a vector spaces, we can identifies

p∗+ = g∗+ ⊕ g∗D,− ' g− ⊕ gD,+ = p−

where (∗ is always the continuous dual). We can give as a basis for p+ andp−

p+ ⊃ bii∈I , arr∈I(D) p− ⊃ aii∈I , brr∈I(D)and obvious relations

(bi, aj) = δij (bi, br) = 0

(ar, aj) = 0 (ar, bs) = δrs

with i, j ∈ I, r, s ∈ I(D). We can then identify fn+1 with an element inUn+1p−. Call Tn+1(f) the set of indices of all ai involved in the expressionof fn+1. Excluding these finite set of indices we get the result.

4.5. Properties of relative Verma modules. Let (g, g+, g−) be an arbi-trary Manin triple, (gD, gD,+, gD,−) a sub-Manin triple and RepΦ(Ug[[~]])be the Drinfeld category associated to g. Let N+, L− be the relative Vermamodules defined as before. The module are identified, respectively as topo-logical vector spaces with

N+ ' Up+ L− ' Um−

Thus, we can define the maps i+ : N+ → N+⊗N+, i− : L− → L−⊗L−given by comultiplication in the universal enveloping algebras Up+, Um−.These maps are Ug-intertwiners, since they are Up+-intertwiner and Um−-intertwiner, respectively, and they map the vector 1± to the m−-invariantvector 1+ ⊗ 1+ and the p+-invariant vector 1− ⊗ 1−, respectively.

Lemma. We have

Φ(1⊗3− ) = 1⊗3

− Φ(1⊗3+ ) = (1⊗3

+ )ΦD


Proof. To prove the first identity it is enough to notice that, since g+1− = 0and Ω =

∑(ai ⊗ bi + bi ⊗ ai),

Ωij(1⊗3− ) = 0

Then Φ(1⊗3− ) = 1⊗3

− . To prove the second identity, we should notice thatm−1+ = 0 and that we can rewrite

Ω =∑j∈ID

(aj⊗bj+bj⊗aj)+∑

i∈I\ID

(ai⊗bi+bi⊗ai) = ΩD+∑

i∈I\ID

(ai⊗bi+bi⊗ai)

where ajj∈ID is a basis of gD,− and bjj∈ID is the dual basis of gD,+.Then

Ωij(1⊗3+ ) = ΩD,ij(1

⊗3+ )

and, since for any element g ∈ gD, the right and the left gD-action coincideon 1+, i.e. g.1+ = 1+.g, we have

Ωij(1⊗3+ ) = (1⊗3

+ )ΩD,ij

and consequently Φ(1⊗3+ ) = (1⊗3

+ )ΦD.

Following [Dri90, Prop. 1.2], we consider the invertible element T ∈Ug⊗ Ug[[~]] satisfying relations:

(S ⊗ S ⊗ S)(Φ321) · (T ⊗ 1) · (∆⊗ 1)(T ) = (1⊗ T )(1⊗∆)(T ) · ΦT∆(S(a)) = (S ⊗ S)(∆(a))T

Let N∗+ be as before and f, g ∈ N∗+. Consider the linear functional N+ →C defined by

v 7→ (f ⊗ g)(T · i+(v))

Since the action of T on N+⊗N+ is continuous, this functional is continuous,so it belongs to N∗+ and allow us to define the map

i∨+ ∈ HomC(N∗+ ⊗N∗+, N∗+)[[~]] , i∨+(f ⊗ g)(v) = (f ⊗ g)(T · i+(v))

This map is continuous and it extends to a map from N∗+⊗N∗+ to N∗+. Wehave

i∨+(a(f ⊗ g))(v) = (f ⊗ g)((S ⊗ S)(∆(a))T.i+(v)) =

= (f ⊗ g)(T∆(S(a)).i+(v)) = i∨+(f ⊗ g)(S(a).v) = (a.i∨+(f ⊗ g))(v)

and then i∨+ ∈ Homg(N∗+⊗N∗+, N∗+)[[~]].

Lemma. We have the following relations:

(a) (i− ⊗ 1)i− = (1⊗ i−)i− in Homg(L−, L⊗3− );

(b) for any f, g, h ∈M∗+, we have

i∨+(1⊗ i∨+)Φλ(f ⊗ g ⊗ h) = i∨+(i∨+ ⊗ 1)ΦρD(f ⊗ g ⊗ h)


Proof. The comultiplication in Um− is coassociative, i.e. (i− ⊗ 1)i−(x) =(1⊗ i−)i−(x). It is enough to show that for any vector x ∈ L−

Φ(i− ⊗ 1)i−(x) = (1⊗ i−)i−(x)

This is a consequence of Lemma 4.5 and (a) is proved.To prove (b), consider v ∈ N+, then

i∨+(1⊗ i∨+)(Φ(f ⊗ g ⊗ h))(v1+) =

= (h⊗ g ⊗ f)((S⊗3(Φ321) · (T ⊗ 1) · (∆⊗ 1)(T )) · (i+ ⊗ 1)i+(v1+)) =

= (h⊗ g ⊗ f)((1⊗ T )(1⊗∆)(T ) · Φ(i+ ⊗ 1)i+(v1+)) =

= (h⊗ g ⊗ f)((1⊗ T )(1⊗∆)(T )(1⊗ i+)i+(v1+)ΦD) =

= ((h⊗ g ⊗ f)ΦD)((1⊗ T )(1⊗∆)(T )(1⊗ i+)i+(v1+)) =

= i∨+(i∨+ ⊗ 1)((f ⊗ g ⊗ h)Φ321D )(v) =

= i∨+(i∨+ ⊗ 1)(Φ−1D )ρ(f ⊗ g ⊗ h)(v)

and (b) is proved.

4.6. The fiber functor over gD. To any representation V [[~]] ∈ Meg[[~]],

we can associate the C[[~]]-module

Γ(V ) = Homg(L−, N∗+⊗V )[[~]]

denoting by Homg the set of continuous homomorphisms, as usual. Equipthis space of the weak topology.

Lemma. The complete vector space Homg(L−, N∗+⊗V ) is isomorphic to V

as topological vector space. The isomorphism is given by

αV : f 7→ (1+ ⊗ 1)f(1−)

for any f ∈ Homcg(L−, N

∗+⊗V ).

Proof. Since L− is discrete, Homcg(L−, N

∗+⊗V ) ' Homg(L−, N

∗+⊗V ). By

Frobenius reciprocity

Homg(L−, N∗+⊗V ) ' (N∗+⊗V )p+

as a topological vector space via f 7→ f(1−). Consider the space HomcC(N+, V )

of continuous linear functionals from N+ to V equipped with the weak topol-ogy and the map

φ : (N∗+⊗V )→ HomC(N+, V ) φ(f ⊗ v)(x) = f(x)v

for f ∈ N∗+, x ∈ N+, v ∈ V . Clearly, φ is injective and continuous and

φ((N∗+⊗V )p+) ⊂ Homcp+

(N+, V )


Indeed, for any p ∈ p+, f ⊗ v ∈ (N∗+⊗V )p+ , we have 0 = p.(f ⊗ v) =p.f ⊗ v + f ⊗ p.v and consequently

φ(f ⊗ v)(p.1+) = f(p.1+)v = (−p.f)(1+)v = φ(−p.f ⊗ v) =

= φ(f ⊗ p.v) = f(1−)p.v = p.f(1−)v = p.φ(f ⊗ v)(1+)

We want to show that φ is surjective. Recall that V ' limU⊂V V/U , where Uis an open linear subspace in V . For any map g ∈ Homc

p+(N+, V ), g−1(U)∩

(Up+)n is open in (Up+)n, i.e. it is a complement of a finite dimensionalsubspace (we are using the assumption that dim gD < ∞, otherwise, beinggD,− discrete, it is in general possible to consider open subset that arecomplement of an infinite dimensional subspace). Then, composing withthe projection πU : V → V/U , the map gU := πU g sends (Up+)n into afinite dimensional linear subspace in V/U and, taking the limit in

(N∗+⊗V )p+ = limn

limU⊂V

((Up+)∗n⊗V/U)

Homcp+

(N+, V ) = limn

limU⊂V

Homcp+

((Up+)n, V/U)

it is possible to determine g′ ∈ (N∗+⊗V )p+ such that g = φ(g′).

By Frobenius reciprocity, the space Homcp+

(N+, V ) is isomorphic to V as

a topological vector space via f 7→ f(1+). The lemma is proved.

Unless otherwise specified, we will denote with Homg the set of continuousmorphisms. We can easily define a gD-action on F (V ) using the right gD-action on N∗+. In particular, for f ∈ Γ(V ) and g ∈ UgD, g.f ∈ Γ(V ) isdefined by

(g.f)(x) = f(x)(S(g)⊗ 1)

for x ∈M+.

Lemma. The isomorphism αV : Γ(V ) ' V [[~]] is an isomorphism of gD-modules.

Proof. For any g ∈ gD, we have g.1− = 0 and

0 = f(g.1−) = ∆(g)f(1−) = (g ⊗ 1 + 1⊗ g)f(1−)

for any f ∈ F (V ). Then, if f(1−) =∑fr ⊗ vr ∈ N∗+⊗V ,

αV (g.f) = (1+ ⊗ 1)(g.f)(1−)= −(1+ ⊗ 1)f(1−)(g ⊗ 1)= −(1+ ⊗ 1)

∑(fr.g)⊗ vr

=∑fr(1+.g)vr

=∑fr(g.1+)vr

= −(1+ ⊗ 1)∑

(g.fr)⊗ vr= −(1+ ⊗ 1)(g ⊗ 1)f(1−)= (1+ ⊗ 1)(1⊗ g)f(1−)= g.(1+ ⊗ 1)f(1−)= g.αV (f)


As a consequence, Γ(V ) is isomorphic to V as equicontinuous gD-module.

For any continuous ϕ ∈ Homg(V, V′), define a map Γ(ϕ) : Γ(V )→ Γ(V ′)

byΓ(ϕ) : f 7→ (1⊗ ϕ)f

This map is clearly continuous and for all g ∈ gD

Γ(ϕ)(g.f)(1−) = −(1⊗ ϕ)f(1−)(g ⊗ 1) = −(Γ(ϕ)f)(1−)(g ⊗ Γ(ϕ)f)(1−)

then Γ(ϕ) ∈ HomgD(Γ(V ),Γ(V ′)).

Finally, it is clear that the diagram

Γ(V )

αV

Γ(ϕ)// Γ(V ′)

αV ′

V [[~]]ϕ// V ′[[~]]

is commutative for all ϕ ∈ Homg(V, V′). Then, we have a well-defined

functorΓ : Repeq Ug[[~]]→ Repeq UgD[[~]]

that is naturally isomorphic to the pullback functor induced by the inclusioniD : gD → g via the natural transformation

αV : Γ(V ) ' i∗DV [[~]]

4.7. Tensor structure on Γ. From now on, when no confusion is possible,we will denote the tensor product in the categories RepΦ(Ug[[~]]),RepΦD(UgD[[~]])by ⊗. Denote by

B1234 : (V1 ⊗ V2)⊗ (V3 ⊗ V4)→ V1 ⊗ ((V2 ⊗ V3)⊗ V4)

the composition

(V1⊗V2)⊗(V3⊗V4)Φ1,2,34−−−−→ V1⊗(V2⊗(V3⊗V4))

1⊗Φ−12,3,4−−−−−→ V1⊗((V2⊗V3)⊗V4)

and by

B′1234 : (V1 ⊗ V2)⊗ (V3 ⊗ V4)→ (V1 ⊗ (V2 ⊗ V3))⊗ V4

the composition

(V1⊗V2)⊗ (V3⊗V4)Φ−1

12,3,4−−−−→ ((V1⊗V2)⊗V3)⊗V4Φ1,2,3−−−→ (V1⊗ (V2⊗V3))⊗V4

For any v ∈ Γ(V ), w ∈ Γ(W ), define JVW (v ⊗ w) to be the composition ofmorphisms

L−i−−→ L− ⊗ L−

v⊗w−−−→ (N∗+ ⊗ V )⊗ (N∗+ ⊗W )A−→ N∗+ ⊗ ((V ⊗N∗+)⊗W )→

β23−−→ N∗+⊗ ((N∗+⊗ V )⊗W )A′−→ (N∗+⊗N∗+)⊗ (V ⊗W )

i∨+⊗1−−−→ N∗+⊗ (V ⊗W )


where the pair (A,A′) can be chosen to be (BN∗+,V,N∗+,W , B−1N∗+,N

∗+,V,W

) (as

in the above diagram) or (B′N∗+,V,N∗+,W, B′−1

N∗+,N∗+,V,W

).

The map JVW (v⊗w) is clearly a continuous g-morphism from L− to N∗+⊗(V ⊗W ), so we have a well-defined map

JVW : Γ(V )⊗ Γ(W )→ Γ(V ⊗W )

Proposition. The maps JVW are isomorphisms and define a tensor struc-ture on the functor Γ.

Proof. It is obvious that the map JVW is an isomorphism, since it is anisomorphism modulo ~. Indeed,

JVW (v ⊗ w) ≡ (i∗+ ⊗ 1)(i⊗ s⊗ 1)(v ⊗ w)i− mod ~

To prove that JVW define a tensor structure on Γ we need to show that, forany Vi ∈Me

g the following diagram is commutative:

(Γ(V1)⊗ Γ(V2))⊗ Γ(V3)

ΦD,123

JV1V2⊗1// Γ(V1 ⊗ V2)⊗ Γ(V3)

JV1⊗V2,V3 // Γ((V1 ⊗ V2)⊗ V3)

Γ(Φ123)

Γ(V1)⊗ (Γ(V2)⊗ Γ(V3))1⊗JV2V3 // Γ(V1)⊗ Γ(V2 ⊗ V3)

JV1,V2⊗V3 // Γ(V1 ⊗ (V2 ⊗ V3))

For brevity, we will denote Jij the map JVi,Vj and Ji⊗j,k the map JVi⊗Vj ,Vk .For every vi ∈ Γ(Vi), i = 1, 2, 3, the map Γ(Φ)J1⊗2,3(J12⊗ 1) corresponds tothe composition:

Γ(Φ)J1⊗2,3(J12 ⊗ 1)(v1 ⊗ v2 ⊗ v3) =

= (1⊗ Φ)(i∗+ ⊗ 1⊗3)A4(1⊗ β1⊗2,N∗+⊗ 1)A3((i∗+ ⊗ 1)⊗ 1⊗3)(A2 ⊗ 1⊗ 1)

(1⊗ βN∗+,1 ⊗ 1⊗3)(A1 ⊗ 1⊗ 1)(v1 ⊗ v2 ⊗ v3)(i− ⊗ 1)i−

where

A1 = BN∗+,1,N∗+,2 A3 = BN∗+,1⊗2,N∗+,3

A2 = B−1N∗+,N

∗+,1,2

A4 = B−1N∗+,N

∗+,1⊗2,3


illustrated by the diagram

L−i−

// L− ⊗ L−i−⊗1

// (L− ⊗ L−)⊗ L−

v1⊗v2⊗v3 // ((N∗+ ⊗ V1)⊗ (N∗+ ⊗ V2))⊗ (N∗+ ⊗ V3)A1⊗1⊗1

// (N∗+ ⊗ ((V1 ⊗N∗+)⊗ V2))⊗ (N∗+ ⊗ V3)

1⊗βN∗+

,1⊗1⊗3

// (N∗+ ⊗ ((N∗+ ⊗ V1)⊗ V2))⊗ (N∗+ ⊗ V3)A2⊗1⊗1

// ((N∗+ ⊗N∗+)⊗ (V1 ⊗ V2))⊗ (N∗+ ⊗ V3)

(i∗+⊗1)⊗1⊗3

// (N∗+ ⊗ (V1 ⊗ V2))⊗ (N∗+ ⊗ V3)A3 // N∗+ ⊗ (((V1 ⊗ V2)⊗N∗+)⊗ V3)

1⊗β1⊗2,N∗+⊗1

// N∗+ ⊗ ((N∗+ ⊗ (V1 ⊗ V2))⊗ V3)A4 // (N∗+ ⊗N∗+)⊗ ((V1 ⊗ V2)⊗ V3)

i∗+⊗1⊗3

// N∗+ ⊗ ((V1 ⊗ V2)⊗ V3)1⊗Φ

// N∗+ ⊗ (V1 ⊗ (V2 ⊗ V3))

By functoriality of associativity and commutativity isomorphisms, we have

A3(i∗+ ⊗ 1⊗4) = (i∗+ ⊗ 1⊗4)A5

where A5 = BN∗+⊗N∗+,1⊗2,N∗+,3,

(1⊗ β1⊗2,N∗+⊗ 1)(i∗+ ⊗ 1⊗4) = (i∗+ ⊗ 1⊗4)(1⊗2 ⊗ β1⊗2,N∗+

⊗ 1⊗2)

and

A4(i∗+ ⊗ 1⊗4) = (i∗+ ⊗ 1⊗4)A6

where A6 = B−1N∗+⊗N∗+,N∗+,1⊗2,3. Then

(1⊗ Φ123)(i∗+(i∗+ ⊗ 1)⊗ 1⊗3) = (i∗+(i∗+ ⊗ 1)⊗ 1⊗3)(1⊗3 ⊗ Φ123)

Finally, we have

Γ(Φ)J1⊗2,3(J12 ⊗ 1)(v1 ⊗ v2 ⊗ v3) =

= ((i∗+(i∗+ ⊗ 1))⊗ 1⊗3)(1⊗3 ⊗ Φ123)A6(1⊗2 ⊗ β1⊗2,N∗+⊗ 1⊗2)A5(A2 ⊗ 1⊗2)

(1⊗ βN∗+,1 ⊗ 1⊗3)(A1 ⊗ 1⊗ 1)(v1 ⊗ v2 ⊗ v3)(i− ⊗ 1)i−

On the other hand, the map J1,2⊗3(1 ⊗ J23)ΦD corresponds to the com-position:

J1,2⊗3(1⊗ J23)ΦD(v1⊗v2 ⊗ v3) =

= (i∗+ ⊗ 1⊗3)A′4(1⊗ βN∗+,1 ⊗ 1⊗2)A′3(1⊗2 ⊗ i∗+ ⊗ 1⊗2)(1⊗ 1⊗A′2)

(1⊗3 ⊗ β2,N∗+⊗ 1)(1⊗ 1⊗A′1)ΦD(v1 ⊗ v2 ⊗ v3)(1⊗ i−)i−

where

A′1 = BN∗+,2,N∗+,3 A′3 = BN∗+,1,N∗+,2⊗3

A′2 = B−1N∗+,N

∗+,2,3

A′4 = B−1N∗+,N

∗+,1,2⊗3


illustrated by the diagram

L−i−

// L− ⊗ L−1⊗i−

// L− ⊗ (L− ⊗ L−)

ΦD(v1⊗v2⊗v3)// (N∗+ ⊗ V1)⊗ ((N∗+ ⊗ V2)⊗ (N∗+ ⊗ V3))

1⊗1⊗A′1// (N∗+ ⊗ V1)⊗ (N∗+ ⊗ ((V2 ⊗N∗+)⊗ V3))

1⊗3⊗β2,N∗+⊗1

// (N∗+ ⊗ V1)⊗ (N∗+ ⊗ ((N∗+ ⊗ V2)⊗ V3))1⊗1⊗A′2// (N∗+ ⊗ V1)⊗ ((N∗+ ⊗N∗+)⊗ (V2 ⊗ V3))

1⊗2⊗i∗+⊗1⊗2

// (N∗+ ⊗ V1)⊗ (N∗+ ⊗ (V2 ⊗ V3))A′3 // N∗+ ⊗ ((V1 ⊗N∗+)⊗ (V2 ⊗ V3))

1⊗β1,N∗+⊗1

// N∗+ ⊗ ((N∗+ ⊗ V1)⊗ (V2 ⊗ V3))A′4 // (N∗+ ⊗N∗+)⊗ (V1 ⊗ (V2 ⊗ V3))

i∗+⊗1⊗3

// N∗+ ⊗ (V1 ⊗ (V2 ⊗ V3))

By functoriality of associativity and commutativity isomorphisms, we have

A′3(1⊗2 ⊗ i∗+ ⊗ 1⊗2) = (1⊗2 ⊗ i∗+ ⊗ 1⊗2)A′5

where A′5 = BN∗+,1,N∗+⊗N∗+,2⊗3,

(1⊗ β1,N∗+⊗ 1⊗2)(1⊗2 ⊗ i∗+ ⊗ 1⊗2) = (1⊗ i∗+ ⊗ 1⊗3)(1⊗ β1,N∗+⊗N∗+ ⊗ 1⊗2)

and

A′4(1⊗ i∗+ ⊗ 1⊗3) = (1⊗ i∗+ ⊗ 1⊗3)A′6

where A′6 = B−1N∗+,N

∗+⊗N∗+,1,2⊗3. Then we have

J1,2⊗3(1⊗ J23)ΦD(v1⊗v2 ⊗ v3) =

= (i∗+ ⊗ 1⊗3)((1⊗ i∗+)⊗ 1⊗3)A′6(1⊗ β1,N∗+⊗N∗+ ⊗ 1⊗2)A′5(1⊗2 ⊗A′2)

(1⊗3 ⊗ β2,N∗+⊗ 1)(1⊗ 1⊗A′1)ΦD(v1 ⊗ v2 ⊗ v3)(1⊗ i−)i−

Call A : ((N∗+⊗V1)⊗ (N∗+⊗V2))⊗ (N∗+⊗V3)→ ((N∗+⊗N∗+)⊗N∗+)⊗ ((V1⊗V2)⊗ V3) the map

A = A6(1⊗2 ⊗ β1⊗2,N∗+⊗ 1⊗2)A5(A2 ⊗ 1⊗2)(1⊗ βN∗+,1 ⊗ 1⊗3)(A1 ⊗ 1⊗ 1)

and B : (N∗+⊗V1)⊗ ((N∗+⊗V2)⊗ (N∗+⊗V3))→ (N∗+⊗ (N∗+⊗N∗+))⊗ (V1⊗(V2 ⊗ V3)) the map

B = A′6(1⊗ β1,N∗+⊗N∗+ ⊗ 1⊗2)A′5(1⊗2 ⊗A′2)(1⊗3 ⊗ β2,N∗+⊗ 1)(1⊗ 1⊗A′1)


We have to show that the following diagram is commutative:

L−

(i−⊗1)i−

xx

(1⊗i−)i−

&&

(L− ⊗ L−)⊗ L−

v1⊗v2⊗v3

L− ⊗ (L− ⊗ L−)

ΦD(v1⊗v2⊗v3)

((N∗+ ⊗ V1)⊗ (N∗+ ⊗ V2))⊗ (N∗+ ⊗ V3)

A

Φ // (N∗+ ⊗ V1)⊗ ((N∗+ ⊗ V2)⊗ (N∗+ ⊗ V3))

B

((N∗+ ⊗N∗+)⊗N∗+)⊗ ((V1 ⊗ V2)⊗ V3)

(i∗+(i∗+⊗1))⊗1⊗3

Φ⊗Φ// (N∗+ ⊗ (N∗+ ⊗N∗+))⊗ (V1 ⊗ (V2 ⊗ V3))

(i∗+(1⊗i∗+))⊗1⊗3

N∗+ ⊗ ((V1 ⊗ V2)⊗ V3)1⊗Φ

// N∗+ ⊗ (V1 ⊗ (V2 ⊗ V3))

Using the pentagon and the hexagon axiom, we can show that

(Φ⊗ Φ)A = BΦ

We have to show that

Γ(Φ)J1⊗2,3(J12 ⊗ 1)(v1 ⊗ v2 ⊗ v3) = J1,2⊗3(1⊗ J23)ΦD(v1 ⊗ v2 ⊗ v3)

in Homg(L−, N∗+ ⊗ (V1 ⊗ (V2 ⊗ V3))). Now,

J1,2⊗3(1⊗ J23)ΦD(v1 ⊗ v2 ⊗ v3) =

= (i∗+(1⊗ i∗+)⊗ 1⊗3)BΦD(v1 ⊗ v2 ⊗ v3)(1⊗ i−)i−(1−)

= (i∗+(1⊗ i∗+)⊗ 1⊗3)B((v1 ⊗ v2 ⊗ v3)(1⊗ i−)i−(1−)

).(Φ−1

D )′

= (i∗+(1⊗ i∗+)⊗ 1⊗3)(B(v1 ⊗ v2 ⊗ v3)Φ−1(1⊗ i−)i−(1−)

).(Φ−1

D ⊗ 1)

= (i∗+(1⊗ i∗+)⊗ 1⊗3)(BΦ(v1 ⊗ v2 ⊗ v3)(1⊗ i−)i−(1−)

).(Φ−1

D ⊗ 1)

= (i∗+(1⊗ i∗+)⊗ 1⊗3)((Φ⊗ Φ)A(v1 ⊗ v2 ⊗ v3)(i− ⊗ 1)i−(1−)

).(Φ−1

D ⊗ 1)

= (1⊗ Φ)((i∗+(1⊗ i∗+)Φ)⊗ 1⊗3)(A(v1 ⊗ v2 ⊗ v3)(i− ⊗ 1)i−(1−)

).(Φ−1

D ⊗ 1)

Set

A(v1 ⊗ v2 ⊗ v3)(i− ⊗ 1)i−(1−) =∑

(f ′ ⊗ f ′′ ⊗ f ′′′)⊗ (v′ ⊗ v′′ ⊗ v′′′)


with (f ′ ⊗ f ′′ ⊗ f ′′′) ∈ (N∗+)⊗3 and (v′ ⊗ v′′ ⊗ v′′′) ∈ (V1 ⊗ V2 ⊗ V3). Then

J1,2⊗3(1⊗ J23)ΦD(v1 ⊗ v2 ⊗ v3) =

= (1⊗ Φ)((i∗+(1⊗ i∗+)Φ)⊗ 1⊗3)(∑

(f ′ ⊗ f ′′ ⊗ f ′′′)⊗ (v′ ⊗ v′′ ⊗ v′′′)).(Φ−1

D ⊗ 1)

= (1⊗ Φ)∑

((i∗+(1⊗ i∗+))Φ(f ′ ⊗ f ′′ ⊗ f ′′′)Φ−1D )⊗ (v′ ⊗ v′′ ⊗ v′′′)

= (1⊗ Φ)∑

((i∗+(i∗+ ⊗ 1))(f ′ ⊗ f ′′ ⊗ f ′′′))⊗ (v′ ⊗ v′′ ⊗ v′′′)

= (1⊗ Φ)((i∗+(i∗+ ⊗ 1))⊗ 1⊗3)A(v1 ⊗ v2 ⊗ v3)(i− ⊗ 1)i−(1−)

= Γ(Φ)J1⊗2,3(J12 ⊗ 1)(v1 ⊗ v2 ⊗ v3)

This complete the proof of Theorem 4.1.

5. Quantization of Verma modules

This section and the next contain results about the quantization of clas-sical Verma modules, necessary to construct the morphism of D–categoriesbetween the representation theory of Ug[[~]] and that of U~g. In partic-ular, from now on, we will assume the existence of a finite N–grading ong, that induces on g a Lie bialgebra structure and allows to consider thequantization of g through the Etingof–Kazhdan functor, UEK

~ g.

5.1. Quantum Verma Modules. Because of the functoriality of the quan-tization defined by Etingof and Kazhdan in [EK98], in the category ofDrinfeld-Yetter modules over UEK

~ g+ we can similarly define quantum Vermamodules.

The standard inclusions of Lie bialgebras g± ⊂ g ' Dg− lift to UEK~ g± ⊂

UEK~ g ' DUEK

~ g−, and we can define the induced modules of the trivial

representation over UEK~ g±

M~± = Ind

UEK~ g

UEK~ g±

C[[~]]

Similarly, we have Hopf algebra maps UEK~ p± ⊂ UEK

~ g and UEK~ p± →

UEK~ gD, and we can define induced modules

L~− = IndUEK~ g

UEK~ p+

C[[~]] N~+ = Ind

UEK~ g

UEK~ p−

UEK~ gD

We want to show that the equivalence FEK : YDUg− [[~]] → YDUEK~ g−

matches these modules. We start proving the statement for M−,M∗+.

5.2. Quantization of M±. We denote by (M~+)∗ the UEK

~ g–module

HomC(IndUEK~ g

UEK~ g−

C[[~]],C[[~]])

Theorem. In the category of left UEK~ g–modules,


(a) F (M−) 'M~−

(b) F (M∗+) ' (M~+)∗

Proof. The Hopf algebra UEK~ g− is constructed on the space F (M−) with

unit element u ∈ F (M−) defined by u(1−) = ε+ ⊗ 1−, where ε+ ∈ M∗+ isdefined as ε+(x1+) = ε(x) for any x ∈ Ug+. Consequently, the action ofUEK~ g− on u ∈ F (M−) is free, as multiplication with the unit element. The

coaction of UEK~ g− on F (M−) is defined using the R-matrix associated to

the braided tensor functor F , i.e.

π∗M− : F (M−)→ F (M−)⊗ F (M−), π∗(x) = R(u⊗ x)

where x ∈ F (M−) and RVW ∈ EndUEK~ g(F (V )⊗F (W )) is given by RVW =

σJ−1WV F (βVW )JVW , JV,W V,W∈YDUg−

being the tensor structure on F . It

is easy to show that J(u⊗u)|1− = ε+⊗ 1−⊗ 1−, and, since Ω(1−⊗ 1−) = 0,we have

R(u⊗ u) = u⊗ u

For a generic V ∈ YDUg− [[~]], the action of UEK~ g∗− is defined as

F (M−)∗ ⊗ F (V )→ F (M−)∗ ⊗ F (M−)⊗ F (V )→ F (V )

This means, in particular that, for every φ ∈ I ⊂ UEK~ g∗−, where I is the

maximal ideal corresponding to u⊥, we have φ.u = 0. This proves (a).

The module M∗+ satisfies the following universal property: for any V inthe Drinfeld category of equicontinuous Ug-modules, we have

HomUg(V,M∗+) ' HomUg−(V,C)

Indeed, to any map of Ug-modules f : V → M∗+, we can associate f :

V → C, f(v) = 〈f(v), 1+〉. It is clear that f factors through V/g−.V . The

equicontinuity property is necessary to show the continuity of f with respectto the topology on V .Since F defines an equivalence of categories, we have

HomUEK~ g(F (V ), F (M∗+)) ' HomUg(V,M

∗+)[[~]] ' HomUg−(V,C)[[~]]

Using the natural isomorphism αV : F (V )→ V [[~]], defined by

αV (f) = 〈f(1−), 1+ ⊗ id〉

we obtain a map HomUg−(V,C)[[~]] → HomC(F (V ),C[[~]]). Consider now

the linear isomorphism α : UEK~ g− → Ug−[[~]] and for any x ∈ Ug− consider

the g-intertwiner ψx : M− → M∗+ ⊗M− defined by ψx(1−) = ε+ ⊗ x1−. It


is clear that, if f(1−) = f(1) ⊗ f(2) in Swedler’s notation,

αV (ψx.f) = 〈(i∨+ ⊗ id)Φ−1(id⊗f)(ε+ ⊗ x.1−), 1+ ⊗ id〉= 〈Φ−1(ε+ ⊗ id⊗ id)(id⊗∆(x))(id⊗f(1) ⊗ f(2)), (T ⊗ id)(1+ ⊗ 1+ id)〉= 〈∆(x)(f(1) ⊗ f(2)), 1+ ⊗ id〉= 〈f(1), 1+〉x.f(2)

= x.αV (f)

using the fact that (ε ⊗ 1 ⊗ 1)(Φ) = 1⊗2 and (ε ⊗ 1)(T ) = 1. So, clearly, ifφ ∈ HomUg−(V,C), then φ αV ∈ HomUEK

~ g−(F (V ),C[[~]]). Then F (M∗+)

satisfies the universal property of HomC(IndUEK~ g

UEK~ g−

C[[~]],C[[~]]) and (b) is

proved.

5.3. Quantization of relative Verma modules. The proof of (b) showsthat the linear functional F (M∗+)→ C[[~]] is, in fact, the trivial deformationof the functional M∗+ → C. These results extend to the relative case andhold for the right gD–action on L−, N

∗+.

Theorem. In the category YDUEK~ g−

(a) F (L−) ' L~−

(b) F (N∗+) ' (N~+)∗

Moreover, as right UEK~ gD–module

(c) FD(L−) ' L~−

(d) FD(N∗+) ' (N~+)∗

The proof of (a) and (b) amounts to constructing the morphisms

C[[~]]→ FEK(L−) F (N∗+)→ UEK~ g∗D

equivariant under the action of UEK~ p+ and UEK

~ p− respectively.

A direct construction along the lines of the proof of Theorem 5.2 is how-ever not straightforward. We prove this theorem in the next section byusing a description of the modules L−, N

∗+ and their images through FEK

via Prop categories. These descriptions show that the classical intertwiners

C→ L− N∗+ → Ug∗D

satisfy the required properties and yield canonical identifications

FEK(L−) ' L~− FEK(N∗+) ' (N~+)∗


6. Universal relative Verma modules

In this section, we complete the proof of Theorem 5.3, using suitableProp categories (product-permutation) compatible with the EK universalquantization functor [EK98, EG].

6.1. Prop description of the EK quantization functor. We will brieflyreview the construction of Etingof–Kazhdan in the setting of Prop cate-gories [EK98].A Prop is a symmetric tensor category generated by one object. Moreprecisely, a cyclic category over S is the datum of

• a symmetric monoidal k–linear category (C,⊗) whose objects arenon–negative integers, such that [n] = [1]⊗n and the unit object is[0]• a bigraded set S =

⋃m,n∈Z≥0

Snm of morphism of C, with

Snm ⊂ HomC([m], [n])

such that any morphism of C can be obtained from the morphisms in S andpermutation maps in HomC([m], [m]) by compositions, tensor products orlinear combinations over k. We denote by FS the free cyclic category overS. Then there exists a unique symmetric tensor functor FS → C, and thefollowing holds (cf. [EK98])

Proposition. Let C be any cyclic category generated by a set S of mor-phisms. Then C has the form FS/I, where I is a tensor ideal in FS.

Let N be a symmetric monoidal k–linear category, and X an object in N .A linear algebraic structure of type C on X is a symmetric tensor functorGX : C → N such that GX([1]) = X. A linear algebraic structure of typeC on X is a collection of morphisms between tensor powers of X satisfyingcertain consistency relations.

We mainly consider the case of non–degenerate cyclic categories, i.e.,symmetric tensor categories with injective maps k[Sn] → HomC([n], [n]).We first consider the Karoubian envelope of C obtained by formal additionto C of the kernel of the idempotents in k[Sn] acting on [n]. Furthermore, weconsider the closure under inductive limits. In this category, denoted S(C),every object is isomorphic to a direct sum of indecomposables, correspondingto irreducible representations of Sn (cf. [EK98, EG]). In particular, in S(C),we can consider the symmetric algebra

S[1] =⊕

Sn[1]

If N is closed under inductive limits, then any linear algebraic structure oftype C extends to an additive symmetric tensor functor

GX : S(C)→ NWe introduce the following fundamental Props .


• Lie bialgebras. In this case the set S consists of two elements ofbidegrees (2, 1), (1, 2), the universal commutator and cocommuta-tor. The category C = LBA is FS/I, where I is generated by theclassical five relations.

• Hopf algebras. In this case, the set S consists of six elements ofbidegrees (2, 1), (1, 2), (0, 1), (1, 0), (1, 1, ), (1, 1), the universal prod-uct, coproduct, unit, count, antipode, inverse antipode. The cate-gory C = HA is FS/I, where I is generated by the classical fourrelations.

The quantization functor described in Section 3 can be described in thisgenerality, as stated by the following (cf. [EK98, Thm.1.2])

Theorem. There exists a universal quantization functor Q : HA→ S(LBA).

Let g− be the canonical Lie dialgebra [1] in LBA with commutator µ andcocommutator δ. Let Ug− := Sg− ∈ S(LBA) be the universal envelopingalgebra of g−. The construction of the Etingof–Kazhdan quantization func-tor amounts to the introduction of a Hopf algebra structure on Ug−, whichcoincides with the standard one modulo 〈δ〉, and yields the Lie bialgebrastructure on g− when considerd modulo 〈δ2〉. This Hopf algebra defines theobject Q[1], where [1] is the generating object in HA. The formulae usedto defined the Hopf structure coincide with those defined in [EK96, Part II]and described in Section 3. In particular, they rely on the construction ofthe Verma modules

M− := Sg−,M∗+ = Sg−

realized in the category of Drinfeld–Yetter modules over g− as object ofLBA .

6.2. Props for split pairs of Lie bialgebras. Let (g−, gD,−) be a splitpair of Lie bialgebras, i.e., there are Lie bialgebra maps

gD,−i−→ g−

p−→ gD,−

such that p i = id. These maps induce an inclusion DgD,− ⊂ Dg− andconsequently an inclusion of Manin triple (gD, gD,−, gD,+) ⊂ (g, g−, g+), asdescribed in Section 4.2.

Definition. We denote by PLBA the Karoubian envelope of the multicol-ored Prop, whose class of objects is generated by the Lie bialgebra objects[g−], [gD,−], related by the maps i : [gD,−] → [g−] , p : [g−] → [gD,−], suchthat p i = id[gD,−].

The Karoubian envelope implies that [m−] := ker(p) ∈ PLBA.

Proposition. The multicolored Prop PLBA is endowed with a pair of func-tors U,L

U,L : LBA→ PLBA U [1] := [g−], L[1] := [gD,−]


and natural transformations i, p, induced by the maps i, p in PLBA,

LBA

U

((

L

66PLBAp

i

KS

such that p i = id. Moreover, it satisfies the following universal property:for any tensor category C, closed under kernels of projections, with the sameproperty as PLBA , there exists a unique tensor functor PLBA → C suchthat the following diagram commutes

LBA

U

((

L

66

##

;;PLBAp

i

KS

// C

6.3. Props for split pairs of Hopf algebras. We can analogously definesuitable Prop categories corresponding to split pairs of Hopf algebras. Inparticular, we consider the Prop PHA characterized by functors U~, L~ andnatural transformations p~, i~ satisfying

HA

U~''

L~

77

""

<<PHAp~

i~

KS∃! // C

where HA denotes the Prop category of Hopf algebras. These also satisfy

HAQEK

//

S(LBA)

PHAQPLBA

// S(PLBA)

where QPLBA is the extension of the Etingof–Kazhdan quantization functorto PLBA, obtaine by the universal property described above with C =S(PLBA).

6.4. Props for parabolic Lie subalgebras. In order to describe the mod-ule N∗+ it is necessary to deal with the Lie bialgebra object p− or, in otherwords to introduce the double of gD,− and the Prop D⊕(LBA) [EG]. Wethen introduce the multicolored Prop as a cofiber product of PLBA andD⊕(LBA) over LBA .

Proposition. The multicolored Prop PLBAD is endowed with canonicalfunctors

D⊕(LBA)→ PLBAD ← PLBA


and satisfies the following universal property:

LBAdouble //

D⊕(LBA)

PLBA

00

// PLBAD∃!

(( C

where double is the Prop map introduced in [EG].

In PLBAD we can consider the Lie bialgebra object [p−].

6.5. Props for parabolic Hopf subalgebras. Similarly, we introduce themulticolored Prop PHAD , endowed with canonical functors (cf. [EG])

D⊗(HA)→ PHAD ← PHA

and satisfying an analogous universal property:

HAdouble //

D⊗(HA)

PHA

00

// PHAD∃!

(( CMoreover, we then have a canonical functor

QPLBAD : PHAD → S(PLBAD)

obtained applying such universal property with C = S(PLBAD) and satis-fying

HA

LHAxx

double //

QEK

D⊗(HA)

Q2

vv

PHA

QPLBA

// PHAD

QPLBAD

S(LBA)

xx

S(double)// S(D⊕(LBA))

vv

S(PLBA) // S(PLBAD)


The commutativity of the square on the back is given by the compatibilityof the quantization functor with the doubling operations, proved in [EG].

6.6. Prop description of L−, N∗+. The modules L−, N

∗+ can be realized in

PLBAD . The module L− is constructed over the object Sm− ∈ PLBA. Thestructure of Drinfeld-Yetter module over g− is determined in the followingway:

(i) the free action of the Lie algebra object m− is defined by the mul-tiplication maps

Sm− ⊗ Sm− → Sm−

given by Campbell-Hausdorff series.

(ii) we define the action of gD,− to be trivial on 1→ Sm−.

(iii) The actions of m−, gD,−, the relation

π ([, ]⊗ 1) = π (1⊗ π)− π (1⊗ π) σ12

and the map [, ] : gD,− ⊗m− → m− define the action of g−.

(iv) We then impose the trivial coaction on 1→ Sm− and the compat-ibility condition

π∗|m− = π∗ π = (1⊗ π)σ12(1⊗ π∗)− (1⊗ π)(δ ⊗ 1) + (µ⊗ 1)(1⊗ π∗)determines it for Sm−.

(v) The action defined at (iv) is compatible with [, ] : gD,−⊗m− → m−

Similarly, the module N∗+ can be realized on Sp−, formally added toPLBAD .

6.7. Proof of Theorem 5.3. The relative Verma module

N+ = indgm− C ' indg

p− UgD

satisfies

HomUg(N+, V ) ' HomUp−(UgD, V )

for every Ug-module V . We have a canonical map of p−-modules ρD :UgD → N+ corresponding to the identity in the case V = N+. We get amap of p−-modules ρ∗D : N∗+ → Ug∗D inducing an isomorphism

HomUg(V,N∗+) ' HomUp−(V,Ug∗D)

The morphism ρ∗D can indeed be thought as

Up− ⊗N∗+

// N∗+

ρ∗D

UgD ⊗ Ug∗D// Ug∗D


Assuming the existence of a suitable finite N–grading, a split pair of Liebialgebras (g−, gD,−), gives rise to a functor

PLBAD → Vect

Consider now the trivial split pair given by (gD,−, gD,−). We have a naturaltransformation

PLBAD

(g−,gD,−)

((

(gD,−,gD,−)

77Vectp

where p naturally extends to the projection p− → gD.

The module U(gD)∗ is indeed the module N∗+ with respect to the trivialpair (gD,−, gD,−). Consequently, the existence of the p−–intertwiner ρ∗D canbe interpreted as a simple consequence of the existence of natural transfor-mation p.

The quantization functor QPLBAD extends the natural transformation pto

PHAD""

<<// S(PLBAD)

(g−,gD,−)

((

(gD,−,gD,−)

66VectS(p)

and shows that

F (N∗+) ' (N~+)∗

Similarly, we can consider the natural transformation S(i) and the dia-gram

PHAD""

<<// S(PLBAD)

(g−,gD,−)

((

(gD,−,gD,−)

66VectS(i)

KS

implying

F (L−) ' L~−We can make analogous consideration for the right gD–action on L−, N

∗+.

This leads to isomorphisms of right UEK~ gD–modules

FEKD (N∗+) ' (N~+)∗ FEKD (L−) ' L~−


7. Chains of Manin triples

In this section, we apply the previous results to chains of Manin triples.

7.1. Chains of length 2. In section 5, given an inclusion of Manin triplesiD : gD ⊆ g, we introduced the relative quantum Verma modules

L~− = IndUEK~ g

UEK~ p+

C[[h]] N~+ = Ind

UEK~ g

UEK~ p−

UEK~ gD

These modules allow to define the functor

Γ~ : Rep(UEK~ g)→ Rep(UEK

~ gD)

by

Γ~(V) = HomUEK~ g(L

~−, (N

~+)∗ ⊗ V)

This functor is naturally isomorphic to UEK~ (iD)∗. Moreover, this is trivially

a tensor isomorphism.

We will now prove the following

Theorem. Let g, gD be Manin triples with a finite Z–grading and iD : gD ⊆g an inclusion of Manin triples compatible with the grading.Then, there existsan algebra isomorphism

Ψ : UEK~ g→ Ug[[~]]

restricting to ΨEKD on UEK

~ gD, where the completion is given with respect tothe Drinfeld-Yetter modules.

Proof. In the previous section, we showed that the quantization of the(UEK~ g, UEK

~ gD)–modules N∗+, L− gives

FEK(N∗+)UEK~ g−−−→ (N~

+)∗UEK~ gD←−−−− FEKD (N∗+)

FEK(L−)UEK~ g−−−→ L~−

UEK~ gD←−−−− FEKD (L−)

Recall that the standard natural transformations αV : FEK(V ) ' V [[~]],(αD)V : FEKD (V ) ' V [[~]] give isomorphisms of right UgD[[~]]–modules

FEK(N∗+) ' N∗+[[~]] FEK(L−) ' L−[[~]]

and isomorphisms of Ug[[~]]–modules

FEKD (N∗+) ' N∗+[[~]] FEKD (L−) ' L−[[~]]

In particular, we get isomorphisms of right UEK~ gD–modules

FEKD FEK(N∗+) ' FEKD (N∗+) ' (N~+)∗ FEKD FEK(L−) ' FEKD (L−) ' L~−

and isomorphisms of UEK~ g–modules

FEKD FEK(N∗+) ' FEK(N∗+) ' (N~+)∗ FEKD FEK(L−) ' FEK(L−) ' L~−


We have a natural isomorphism through J :

HomUEK~ g(F (L−), F (N∗+)⊗ F (V )) ' Homg(L−, N

∗+ ⊗ V )[[~]]

This is indeed an isomorphism of UgD[[~]]–modules, since, for x ∈ UgD,φ ∈ HomUEK

~ g(F (L−), F (N∗+)⊗ F (V )), we have

x.φ := (F (x)⊗ id) φ J (F (x)⊗ id)) = F (x⊗ id) J

Quantizing both side and using the isomorphism FEKD F (N∗+) ' (N~+)∗,

we obtain a natural transformation

γD : Γ~ FEK ' FEKD Γ

making the following diagram commutative

RepΦ(Ug[[~]]) FEK //

Γ

Rep(U~g)

Γ~

γD

rzRepΦD(UgD[[~]])

FEKD // Rep(U~gD)

Applying the construction above to the algebra of endomorphisms of thefiber functor, we get the result.

7.2. Chains of arbitrary length. For any chain

C : g0 ⊆ g1 ⊆ · · · ⊆ gn ⊆ g

of inclusions of Manin triples, by composition of the natural transformations

γi,i+1 ∈ Nat⊗(Γ~i,i+1 FEKi+1 , FEKi Γi,i+1)

we obtain a natural transformation inducing a filtered isomorphism

UEK~ g→ Ug[[~]]

We proved the following

Theorem. For any chain of Manin triples

C : g0 ⊆ g1 ⊆ · · · ⊆ gn ⊆ g

there exists an isomorphism of algebras

ΨC : UEK~ g→ Ug[[~]]

such that ΨC(UEK~ gi) = Ugi[[~]] for any gi ∈ C.

Corollary. Given two chains C,C′, the natural transformation

ΦCC′ := γ−1C′ γC

satisfies

Ad(ΦCC′)ΨC′ = ΨC


8. An equivalence of quasi–Coxeter categories

In this section, we give the proof of the main theorem

Theorem. Let g be a symmetrizable Kac–Moody algebra. Then the comple-

tion U~g is isomorphic to a quasi-Coxeter quasitriangular quasibialgebra oftype Dg on the quasitriangular Dg-quasibialgebra

(Ug[[~]], UgD[[~]],∆0, ΦKZD , RKZ

D )where the completion is taken with respect to the integrable modules in cat-egory O.

8.1. D–structures on Kac–Moody algebras. Let A = (aij)i,j∈I be anirreducible, generalized, symmetrizable Cartan matrix, g = g(A) the corre-sponding Kac–Moody algebra defined in Section 3. Let Dg = D(A) be theDynkin diagram of g, that is, the connected graph having I as vertex setand an edge between i and j if aij 6= 0. For any i ∈ I, let sli2 ⊂ g be thethree–dimensional subalgebra spanned by ei, fi, hi.

Let J be a nonempty subset of I. Consider the generalized Cartan sub-matrix of A defined by

AJ = (aij)i,j∈J

We recall the following proposition from [Kac, Ex.1.2]

Proposition. Let

ΠJ := αj | j ∈ J Π∨J := hj | j ∈ JLet h′J be the subspace of h generated by Π∨J and

kJ =⋂j∈J

Kerαj = h ∈ h | 〈αj , h〉 = 0 ∀j ∈ J

Let h′′J be a supplementary subspace of h′J + kJ in h and let

hJ = h′J ⊕ h′′J

Then,

(i) (hJ,ΠJ,Π∨J) is a realization of the generalized Cartan matrix AJ.

(ii) The subalgebra gJ ⊂ g, generated by ej , fjj∈J and hJ, is the Kac–Moody algebra associated to the realization (hJ,ΠJ,Π

∨J) of AJ.

SetQJ =

∑j∈J

Zαj ⊂ Q g = g(A) =⊕α∈Q

gα

Then,

(iii)

gJ = hJ ⊕⊕

α∈QJ\0

gα


Any connected subdiagram D ⊆ Dg defines a subset JD ⊂ I. The assign-ment J→ gJ defines a structure of D–algebra on g.

Remark. The derived algebra g′J is generated by ej , fj , hjj∈J, where hj =[ej , fj ]. Therefore, it does not depend of the choice of the subspace h′′J andthe D–structure described above coincides with the one provided in [TL4].

Remark. If A is a generalized Cartan matrix of finite type, h′′J = 0 for anysubset J ⊂ I. Therefore, the Dg–algebra structure on g = g(A) is uniquely

defined by the subalgebras Usli2i∈I and the Cartan subalgebra is definedfor any subdiagram D ⊂ Dg by

hD = hi | i ∈ V(D)

Similarly, if A is a generalized Cartan matrix of affine type, we obtain dia-grammatic Cartan subalgebras hD, where

hD =

hi | i ∈ V(D) if D ⊂ Dg

h if D = Dg

8.2. qCqtqba structure on U~g. The quantum enveloping algebra U~gis naturally endowed with a quasi–Coxeter quasitriangular quasibialgebrastructure of type Dg defined by

(i) Dg-algebra: for any D ∈ SD(Dg), let gD ⊂ g be the correspondingKac–Moody subalgebra. The Dg-algebra structure is given by thesubalgebras U~gD.

(ii) Quasitriangular quasibialgebra: the universalR-matrices R~,D,with trivial associators ΦD = 1⊗3 and structural twists FF = 1⊗2.

(iii) Quasi-Coxeter: the local monodromies are the quantum Weylgroup elements. The gauge transformations ΦGF are trivial.

We transfer this qCqtqba structure on Ug[[~]]. More precisely, we de-fine an equivalence of quasi–Coxeter categories between the representationtheories of U~g and Ug[[~]].

8.3. Gauge transformations for g(A). For any D ⊂ Dg, the inclusiongD ⊂ g, defined in the previous section, lifts to an inclusion of Manin triples

gD ⊕ hD ⊂ g⊕ h

We denote by gD = (gD ⊕ hD, bD,+, bD,−) the Manin triple attached to gD,for any D ⊆ Dg.

In this setting, the natural transformations γBB′ , B ⊆ B′ ⊆ Dg con-structed in Section 7, define, by vertical composition, a natural transforma-tion

γCBB′ ∈ Nat⊗(Γ~BB′ FEKB′ , FEKB ΓBB′)

for any chain of maximal length

C : B = C0 ⊂ C1 ⊂ · · · ⊂ Cr = B′


Any chain of maximal length defines uniquely a maximal nested set FC ∈Mns(B,B′), but this is not a one to one correspondence. For example, forD = A3, the maximal nested set

F = α1, α3, α1, α2, α3

corresponds to two different chains of maximal length

C1 : α1 ⊂ α1 t α3 ⊂ A3 C2 : α3 ⊂ α1 t α3 ⊂ A3

In order to prove that the natural transformations γ define a morphism ofbraided Dg–monoidal categories, we need to prove that the transformationγCBB′ depend only on the maximal nested set corresponding to C.

In particular, we have to prove that, for any B1 ⊥ B2 in I(D), the con-struction of the fiber functor

CB1tB2

FB1,B1tB2

##

FB2,B1tB2

CB1

FB1 ##

CB2

FB2

C∅is independent of the choice of the chain. In our case,

CB1tB2 = Rep(U gB1 [[~]]⊗ U gB2 [[~]])

and the braided tensor structure is given by product of the braided tensorstructures on

CB1 = RepΦB1(U gB1 [[~]]) CB2 = RepΦB2

(U gB2 [[~]])

Similarly, the tensor structure on the forgetful functor

CB1tB2 → CBi i = 1, 2

is obtained killing the tensor structure on CBi , i = 1, 2, i.e., applyingthe tensor structure on CBi → C∅. In particular, the tensor structure onFB1 FB1,B1tB2 and FB2 FB2,B1tB2 coincide, since [gB1 , gB2 ] = 0.

Analogously we have an equivality of natural transformation

γB1 γB1,B1tB2 = γB2 γB2,B1tB2

Therefore, for any maximal nested set F ∈ Mns(B,B′), it is well defineda natural transformation

γFBB′ ∈ Nat⊗(Γ~BB′ FEKB′ , FEKB ΓBB′)

so that the data (FEKB , γFBB′) define an isomorphism of D–categoriesfrom RepΦB (U gB[[~]]) to Rep(U~gB). More precisely,


Theorem. There exists an isomorphism of braided Dg–monoidal categoriesfrom

((RepΦB (U gB[[~]]),⊗B,ΦB, σRB), (ΓBB′ , JBB′

F ))

to

((Rep(U~gB),⊗B, id, σR~B), (Γ~BB′ , id))

given by (FEKB , γFBB′).

8.4. Reduction to category Oint. We showed in Section 3, the the Etingof–Kazhdan functor gives rise, by restriction, to an equivalence of categories

FEK : Og[[~]]→ OU~g

We will show now that this equivalence can be further restricted to integrablemodules in category O, i.e., modules in category O with a locally nilpotentaction of the elements ei, fii∈I (respectively Ei, Fi).

Proposition. There exists an equivalence of braided tensor categories

FEK : Ointg [[~]]→ Oint

U~g

which is isomorphic to the identity functor at the level of h–graded k[[~]]–modules.

Proof. Let V ∈ Ointg . Then, the elements ei, fi for i ∈ I act nilpotently on

V . Then, by [Kac], for all λ ∈ P(V ), there exist p, q ∈ Z≥0 such that

t ∈ Z | λ+ tαi ∈ P(V ) = [−p, q]

Since the Cartan subalgebra h is not deformed by the quantization, thefunctor FEK preserves the weight decomposition. In U~g, for any h ∈ h andi ∈ I, we have

[h,Ei] = αi(h)Ei

Therefore the action of the Ei’s on V is locally nilpotent. The action of theFi’s is always locally nilpotent, since

P(V ) ⊂r⋃s=1

D(λs)

The result follows.

Corollary. There exists an isomorphism of Dg–algebras

ΨF : U~g→ Ug[[~]]

such that ΨF (U~gDi) = UgDi [[~]] for any Di ∈ F , where the completion istaken with respect to the integrable modules in category O.


8.5. Quasi–Coxeter structure. The previous isomorphism of braidedDg–monoidal categories induces on

((OintgB

[[~]],⊗B,ΦB, σRB), (ΓBB′ , JBB′

F ))a structure of quasi–Coxeter category of tipe D, given, for any B ⊆ B′ inI(D), F ,G ∈ Mns(B,B′) and i ∈ D, by

HB(ΦGF ) = (γFBB′)−1 γGBB′

Si = Ψi(S~i )

where Ψi : End(Γ~i )→ End(Γi).

Theorem. U~g is isomorphic to a quasi-Coxeter quasitriangular quasibial-gebra of type Dg on the quasitriangular Dg-quasibialgebra

(Ug[[~]], UgD[[~]],∆0, ΦKZD , RKZ

D )

8.5.1. Local monodromies. Clearly, the local monodromies Si satisfy

∆F (Si) = (Ri)21F · (Si ⊗ Si)

since ΨF is given by an isomorphism of braided D–monoidal categories andtherefore

ΨF ((R~i )F ) = (Ri)FSimilarly, the braid relations are easily satisfied, since

Ad(ΦGF )ΨF = ΨG

8.5.2. Associators. The associators ΦFG defined above satisfy the requiredproperties.

(i) Orientation For any elementary pair (F ,G) in Mns(B,B′)

HB(ΦFG) = (γFBB′)−1 γGBB′ =

(HB(ΦGF )

)−1

(ii) Coherence For any F ,G,H ∈ Mns(B,B′)

HB(ΦFG) = (γFBB′)−1γHBB′ (γHBB′)

−1 γGBB′ = HB(ΦFH) HB(ΦHG)

This property implies the coherence.

(iii) Support The natural transformations γF , γG coincide above supp(F ,G)and below z supp(F ,G). The support condition of ΦFG follows.

(iv) Forgetfulness Given two equivalent elementary pairs (F ,G), (F ′,G′)in Mns(B,B′), ΦFG and ΦF ′G′ are defined by the same elemen-tary operation from supp(F ,G) = supp(F ′,G′) and z supp(F ,G) =z supp(F ′,G′) and satisy therefore the forgetfulness property.

Moreover, the associators clearly are gauge transformations of the tensorfiber functors, because they are given by composition of gauge transforma-tions. This concludes the proof of Theorem 8.5.


References

[ATL1] A. Appel, V. Toledano Laredo, A rigidity theorem forquasi–Coxeter quasitriangular quasibialgebra structure onaffine Kac–Moody algebras.(Preliminary version available athttp://www.math.neu.edu/aappel/Andrea Appel/Preliminary.html)

[ATL2] , Monodromy theorems in the affine setting. (in preparation)[DCP1] C. De Concini, C. Procesi, Wonderful models of subspace arrangements. Selecta

Math. (N. S.) 1 (1995), 459-494.[DCP2] Hyperplane arrangements and holonomy equations. Selecta Math. (N. S.) 1

(1995), 495-535.[Dri86] , Quantum groups, Proceedings of the I.C.M., Berkeley (1986), 798-820.[Dri89] , Quasi Hopf algebras and the KZ equations, in Problem of modern QFT

(1989), 1-13.[Dri90] , Quasi-Hopf Algebras, Leningrad Math. J. 1 (1990), no. 6.[Dri91] , On quasi triangular quasi-Hopf algebras and on a group that is closely

connected with Gal(Q/Q). Leningrad Math. J. 2 (1991): 829-60.[EG] B. Enriquez, N. Geer, Compatibility of quantization functors of Lie bialgebras

with duality and doubling operations, Selecta Math. 15 (2009), 1-59.[EK96] P. Etingof, D. Kazhdan, Quantization of Lie bialgebras, I, Selecta Math. 2 (1996),

no. 1, 1-41.[EK98] , Quantization of Lie bialgebras, II, Selecta Math. 4 (1998), 213-231.[EK08] , Quantization of Lie bialgebras. VI. Quantization of generalized Kac–Moody

algebras, Transform. Groups 13 (2008), no. 3-4, 527-539.[Kac] V. Kac, Infinite-dimensional Lie algebras, Cambridge University Press (1991),

3rd edition.[KR] A.N. Kirillov, N. Reshetikin, q-Weyl group and a multiplicative formula for uni-

versal R-matrices, Comm. Math. Phys. (1990), no. 3, 537-595.[K87] T. Kohno, Monodromy representations of braid groups and Yang-Baxter equa-

tions, Ann. Inst. Fourier (Grenoble) 37 (1987), no. 4, 139-160.[L] G. Lusztig, Introduction to quantum groups, Birkhauser Boston, 1993.[S] Y. Soibelman, Algebra of functions on a compact quantum group and its repre-

sentations, Leningrad Math. J. (1991), no. 2, 161-178.[TL1] V. Toledano Laredo, Flat connections and quantum groups, Acta Appl. Math.

73 (2002), no. 1-2, 155-173.[TL2] , A Kohno-Drinfeld theorem for quantum Weyl groups, Duke Math. J. 112

(2002), no. 3, 421-451.[TL3] , Cohomological construction of relative twists, Adv. Math. 210 (2007), 375-

403.[TL4] , Quasi Coxeter algebras, Dynkin diagram cohomology and quantum groups,

Math. Res. Pap. IMRP 2008, art. ID rpn009, 167 pp.[TL5] , Quasi-Coxeter quasi triangular quasibialgebras and the Casimir connec-

tion. (forthcoming)[TL6] , Affine extension of rational Casimir connection. (forthcoming)

Department of Mathematics, Northeastern University, 360 Huntington Av-enue, Boston, MA 02115

E-mail address: [email protected]

Department of Mathematics, Northeastern University, 360 Huntington Av-enue, Boston MA 02115

E-mail address: [email protected]

http://www.math.neu.edu/aappel/Andrea_Appel/Preliminary.html

QUASI{COXETER CATEGORIES AND A RELATIVE …mathserver.neu.edu/.../Preliminary_files/ATL1.pdf ·...

Documents

Transcript of QUASI{COXETER CATEGORIES AND A RELATIVE …mathserver.neu.edu/.../Preliminary_files/ATL1.pdf ·...