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QUASI–COXETER QUASITRIANGULAR QUASIBIALGEBRAS AND THE CASIMIR CONNECTION 6 1 VALERIOTOLEDANOLAREDO 0 2 n Abstract. Let g be a complex, semisimple Lie algebra. We prove the exis- a tence of a quasi–Coxeter quasitriangular quasibialgebra structure on the en- J velopingalgebraofg,whichbindsthequasi–Coxeteralgebrastructureunder- 5 lying the Casimir connection of g and the quasitriangular quasibialgebra one 1 underlyingitsKZequations. Thisimpliesinparticularthatthemonodromyof therationalCasimirconnection ofgisdescribedbythequantum Weylgroup ] operators ofthequantum groupU~g. A Q . h t Contents a m 1. Introduction 1 [ 2. Filtered algebras 9 3. Quasi–Coxeter algebras 10 1 4. Differential twists and quasi–Coxeter structures 20 v 5. The fusion operator 28 6 7 6. The differential twist 33 0 7. The centraliser property 38 4 8. Quasi–Coxeter quasitriangular quasibialgebra structure on Ug 42 0 9. The quantum group U g 43 . ~ 1 10. The monodromy theorem 45 0 Appendix A. The basic ODE 46 6 Appendix B. The constant C± revisited 48 1 : Appendix C. The dynamical KZ and Casimir equations 52 v References 55 i X r a 1. Introduction 1.1. Letgbeacomplex,semisimpleLiealgebra. DeConcini[7],andindependently the author [24], conjectured that the monodromy of the Casimir connection of g is describedby the quantum Weyl groupgroupoperatorsofthe quantumgroupU g, ~ in a way analogous to the Drinfeld–Kohno theorem [13]. This conjecture was proved in [24] for the Lie algebra sl . For an arbitrary g, n it was reduced in [25] to a structural statement about the enveloping algebra Ug, namely the existence of a quasi–Coxeter quasitriangular quasibialgebra structure on Ug[[~]] binding the quasi–Coxeter structure underlying the Casimir connection Date:January2016. WorksupportedinpartbyNSFgrantsDMS–0707212, DMS-0854792andDMS–1206305. 1 2 V.TOLEDANOLAREDO of g, to the quasitriangular quasibialgebra structures underlying the Knizhnik– Zamolodchikov connections of its standard Levi subalgebras. The goal of the present paper is to establish the existence of such a structure, and therefore prove the monodromy conjecture for any semisimple Lie algebra. 1.2. Let h g be a Cartan subalgebra, Φ h∗ the corresponding root system, ⊂ ⊂ andh =h Ker(α)thesetofregularelementsinh. Fixannon–degenerate, reg \ α∈Φ invariant bilinear form (, ) on g. The Casimir connection of g is a connection on S · · the holomorphically trivial bundle V on h with fibre a given finite–dimensional reg representationV of g. It is given by h dα =d C C α ∇ − 2 α · αX∈Φ+ where h C is a deformationparameter, α rangesovera chosensystem of positive ∈ rootsΦ ,1andC is the Casimiroperatorinduced bythe restrictionof(, )to the + α three–dimensionalsubalgebraslα gcorrespondingtothe rootα. The c·on·nection 2 ⊆ is flat for any h [22, 24, 7, 20], and can be made equivariant with respect to C ∇ the Weyl group W of g [22, 24]. Its monodromy defines a one–parameter family of actions µh of the braid group BW = π(hreg/W) on V depending on h, which deformsthe actionof(the Tits extensionof)W. We denote byµ:B G(V[[~]]) W → the formal Taylor series of µh with respect to the parameter ~=πιh. 1.3. Let U g be the Drinfeld–Jimbo quantum group corresponding to g, thought ~ of as a topological Hopf algebra over C[[~]]. Let be a quantum deformation of V V, that is a topologically free C[[~]]–module such that /~ is isomorphic to V V V as a g–module. Since is integrable, the braid group B acts on though the W V V quantum Weyl group operators of U g [21]. The main result of the present paper ~ is the following. Theorem. The monodromy µ:B GL(V[[~]]) of the Casimir connection on V W → is equivalent to the quantum Weyl group action of the braid group B on . W V 1.4. Recallthataquasitriangularquasibialgebrais analgebraAoveracommuta- tive ring k endowed with two morphisms, the coproduct ∆ : A A⊗2 and counit → ε : A k, and two distinguished invertible elements, the R–matrix R A⊗2 and → ∈ associatorΦ A⊗3 [12]. Therelationssatisfiedby∆,ε,RandΦaredesignedsoas ∈ toendowthecategoryofA–moduleswiththestructureofabraidedtensorcategory. Inparticular,foranyV Mod(A),thereisafamilyofactionsρ :B GL(V⊗n) b n ∈ → of the n–strand braid group on the n–fold tensor product of V, which are labelled bythe choiceofacomplete bracketingb ofthe non–associativemonomialx x . 1 n ··· The actions corresponding to different choices of b are canonically isomorphic, via intertwiners built out of the associator Φ. For example, for n =3, the action of Φ on V⊗3 intertwines ρ and ρ . ((x1x2)x3) (x1(x2x3)) 1.5. Inasimilarspirit,aquasi–CoxeteralgebraAisdesignedsothatamoduleover it carries a family of canonically equivalent representations of the braid group B W ofagivenirreducibleCoxetergroupW [25]. Centraltothisnotionarethemaximal nested sets on the Coxeter graph D of W, which generalise complete bracketings to an arbitrary Coxeter type. These were introduced by De Concini–Procesi[8, 9], 1theconnection isindependent ofthechoiceofΦ+ QUASI–COXETER ALGEBRAS AND THE CASIMIR CONNECTION 3 andconsist ofmaximalcollections of connected subgraphsof D which are pairwise compatible, that is such that either one is contained in the other, or they are orthogonal,inthesensethattheyhavedisjointvertexsetsandarenotlinkedbyan edge ofD. When W is of type A , with the verticesof D labelled 1,...,n 1 as n−1 − in[4,Planche1],mappingaconnectedsubdiagramB withvertices i,...,j 1 to { − } the pair of parentheses x x (x x )x x , and noting that B,B′ D 1 i−1 i j j+1 n ··· ··· ··· ⊆ arecompatiblepreciselywhenthecorrespondingpairsofparenthesesareconsistent, yields a bijection between maximal nested sets on D and complete bracketings of the monomial x x . 1 n ··· 1.6. Aquasi–Coxeteralgebraisendowedwiththreesetsofdata. First,acollection of subalgebras A labelled by the connected subgraphs B D, which are such B ⊆ that A A if B B , and [A ,A ] = 0 if B and B are orthogonal. B1 ⊆ B2 1 ⊆ 2 B1 B2 1 2 Next, invertible elements Φ A called associators. These are labelled by pairs GF ∈ of maximal nested sets on D, and satisfy in particular the transitivity relations Φ Φ =Φ . Finally, invertible elements S labelled by the vertices of D, and HG GF HF i called local monodromies. This data satisfies various compatibility relations, in particular a version of the braid relations defining B . They give rise to a family W of actions λ : B GL(V) on any V Mod(A), which are labelled by the F W → ∈ maximal nested sets on D, with Φ A intertwining λ and λ . GF F G ∈ 1.7. Aquasi–CoxeterquasitriangularquasibialgebraoftypeW isaquasi–Coxeter algebra A additionally endowed with a coproduct ∆ and counit ε, such that each diagrammatic subalgebra A has a quasitriangularquasibialgebra structure of the B form (∆,ε,R ,Φ ), for a given R–matrix R A⊗2 and associator Φ A⊗3. B B B ∈ B B ∈ B Thisgivesriseinparticulartoafamily ofcommutingrepresentationsofthe groups B ,B on the tensor power V⊗n of any V Mod(A), specifically n W ∈ ρ :B GL(V) and λ :B GL(V) B,b n F,b W → → ThefirstfamilyisdeterminedbythechoiceofasubdiagramB Dandabracketing ⊆ b of the monomial x x , and arises by restricting V to the quasitriangular 1 n ··· quasibialgebra A , with the representations π and π equivalent provided B B,b B′,b′ B =B′. The second arises from the action of A on V⊗n determined by the choice ofb,anddependsonthechoiceofamaximalnestedset onD,withλ equivalent b,F F to λ for any b,b′ and , . b′,G F G 1.8. A quasi–Coxeter quasitriangular quasibialgebra A possesses an additional piece of data, which binds the associators Φ coming from the quasitriangular B quasibialgebrastructureoneachdiagrammaticA , to the associatorsΦ coming B GF from the quasi–Coxeter structure on A. This welding of the quasi–Coxeter and quasitriangularquasibialgebrastructuresis whatgivesthe examples ofinterestthe rigidity required to determine the monodromy of the Casimir connection. The additional data consists of relative twists, which are elements F A⊗2 (B,α) ∈ B labelled by a connected subdiagram B and a vertex α B. These twists are ∈ required to satisfy two identities. The first is the twist equation (Φ ) =Φ (1.1) B F(B;α) B\α 4 V.TOLEDANOLAREDO togetherwiththerequirementthatF commutewith∆(A ).2 Thisamounts (B;α) B\α to asking that F defines a tensor structure on the restriction functor from (B;α) the monoidal category of A –modules with associativity constraints given by the B associator Φ , to that of A –modules with associativity constraints given by B B\α Φ . B\α By restricting in stages from A to A =k, the relative twists allow to define a D ∅ tensorstructure onthe forgetfulfunctor fromthe monoidalcategoryof A–modules with associativity constraints given by Φ to k–modules, which depends on the D choice of a maximal nested set on D, as follows. For any element B of , the F F collection of maximal elements of properly contained in B covers all but one of F the verticesαB of B (and consists in fact ofthe connectedcomponents ofB α 3). F \ Define the twist F A⊗2 by F ∈ −→ F = F (1.2) F (B;αB) B∈F F Y where the productis takenwith F to the rightof F if B′ B.4 Then, it (B,α) (B′,α′) ⊂ follows from (1.1) that (Φ ) =1⊗3. D FF Thesecondidentitysatisfiedbytherelativetwistsrequiresthatthetensorstruc- tures F corresponding to the choices of different maximal nested sets be isomor- F phic, with the isomorphism given by the associators Φ of the quasi–Coxeter GF structure, that is F =Φ⊗2 F ∆(Φ )−1 (1.3) G GF · F · GF 1.9. The quantum group U g associated to a complex semisimple Lie algebra g ~ is a quasi–Coxeter quasitriangular quasibialgebra in a very simple way. For any subdiagram B of the Dynkin diagram D of g, the subalgebra U g is the quan- ~ B tum group corresponding to the generators of U g labelled by the vertices of B, ~ endowed with its universal R–matrix R and trivial associator Φ = 1⊗3. The B B associatorsΦ arealltrivial,andthelocalmonodromiesS aregivenbyLusztig’s GF i quantum Weyl group group operators. It was proved in [25, Thm 8.3] that this structure can be transferred to an isomorphic one on Ug[[~]] (which, however, does not have trivial associators). Moreover, it was also proved in [25, Thm. 9.1] that quasi–Coxeterquasitriangularquasibialgebrastructures on Ug[[~]] are rigid, that is determined by their R–matrices R and local monodromies S .5 Thus, Theorem B i 1.3canbe provedbyshowingtheexistenceofaquasi–Coxeterquasitriangularqua- sibialgebrastructureonUg[[~]]whichbinds the quasi–Coxeterstructureunderlying the monodromy of the Casimir connection of g, to the quasitriangular quasibialgebra structure underlying that of the Knizhnik–Zamolodchikovconnections of its standard Levi subalgebras. 2ThetwistΦF ofanassociatorΦbyatwistF isequalto1⊗F·id⊗∆(F)·Φ·∆⊗id(F)−1·F−1⊗1. If B\α isnot connected, the associator ΦB\α is taken to be the (commuting) product QiΦBi, whereBi runsovertheconnected components ofB\α. 3In type An−1, if B is the diagram with vertices i,...,j −1, the elements of F properly containedinBgiveabracketingofthemonomialxi···xj,whichnecessarilycontainstwomaximal pairsofparentheses oftheform(xi···xk−1)(xk···xj),andαBF isthevertexk. 4thisdoesnotspecifytheorderofthefactorsuniquely,butanytwoorderssatisfyingtheabove requirementsareeasilyseentogiverisetothesameproduct. 5Thisisnotthecaseofquasi–Coxeter algebrastructures onUg[[~]]. QUASI–COXETER ALGEBRAS AND THE CASIMIR CONNECTION 5 1.10. It is a well–known, and beautiful observation of Drinfeld’s that the monodromy of the KZ equations of g gives rise to a quasitriangular quasibialgebra structureonUg[[~]][13]. ThecorrespondingR–matrixisthemonodromyR =e~Ω KZ of the KZ equations on n = 2 points, and the associator Φ the ratio Ψ−1 KZ (x1x2)x3 · Ψ of the solutions of the KZ equations on n = 3 points corresponding to x1(x2x3) the asymptotic zones z z >>z z and z z >>z z respectively. The 1 3 1 2 1 3 2 3 − − − − associativity constraints relating the copies of the n–fold tensor power V⊗n of a representation V corresponding to two bracketings b,b′ can be expressed in terms of the associator Φ , as in any monoidal category, or more directly obtained as KZ the ratio Ψ−1 Ψ of the solutions of the KZ equations on n points corresponding b′ · b to the asymptotic zones determined by b and b′. 1.11. Similarly, the monodromy of the Casimir connection of g gives rise to a quasi–CoxeterstructureonUg[[~]]. This reliesonthe De Concini–Procesiconstruc- tion of a compactification of h , where the root hyperplanes are replaced by a reg normalcrossingsdivisor[8, 9]. The irreduciblecomponentsofthe divisorwhichin- tersectthe closureofthe Weylchamber arelabelledby the connectedsubdiagrams of D. The maximal nested sets on the Dynkin diagram D of g label the points at infinity,thatisthe non–emptyintersectionofamaximalcollectionofthesecompo- nents. Near each of those, one can construct a canonical fundamental solution Ψ F having good asymptotics. The associators Φ then arise as the ratios Ψ−1 Ψ . GF G · F 1.12. The previous paragraphssuggestthat the relativetwists of a quasi–Coxeter quasitriangularquasibialgebrastructure onUg[[~]] which binds the structures coming from the KZ and Casimir equations might also arise by comparing appropriate solutions of a flat connection. This is indeed the case, as we explain below. Since the associatorΦ of the quasi–Coxeterstructure underlying the Casimir GF connection is equal to Ψ−1 Ψ , where the latter are the De Concini–Procesi ∇C G · F fundamental solutions of , the compatibility relation (1.3) may be rewritten as C ∇ Ψ⊗2 F ∆(Ψ )−1 =Ψ⊗2 F ∆(Ψ )−1 (1.4) G · G · G F · F · F EithersidedefinesaholomorphicfunctionF :h Ug⊗2[[~]]whichisindependent reg → ofthechoiceofamaximalnestedsetonDby(1.4),satisfiesthedifferentialequation h dα dF = (C (1)+C (2))F F∆(C ) (1.5) α α α 2 α − αX>0 (cid:16) (cid:17) and the twist relation (Φ ) = 1⊗3. We shall call such an F a differential twist. KZ F Given F, the twists F may be recoveredas F F =(Ψ⊗2)−1 F ∆(Ψ ) F F · · F 1.13. We show in Section 4 that the requirement that the twists F possess the F factorised form (1.2), where the factors F satisfy the twist relation (1.1), is (B;α) equivalent to the following centraliser property of the differential twist F. This assumes that a differential twist F is given for any diagrammatic subalgebra gB g g,andexpressesacompatibilitybetweenF andF ,foranyvertexαofB. B ⊆ gB gB\α Specifically,considertheasymptoticsrlim F (µ)ofF (µ),asthecoordinate α→∞ gB gB α of µ h goes to .6 These asymptotics are a function of the image µ of µ in B ∈ ∞ 6the existence rlimα→∞FgB relies on the fact that the Casimir connection has regular singularities. 6 V.TOLEDANOLAREDO h ,thoughtofasaquotientofh . Theysolvethe Casimirequation(1.5)forthe B\α B Liealgebrag ,whicharethelimitoftheCasimirequationsofg asα . The B\α B →∞ centraliser property is the requirement that the element F Ug[[~]]⊗2 defined (B;α) ∈ by the equation r lim F (µ)=F F (µ) (1.6) α→∞ gB (B;α)· gB\α be invariant under g . This implies in particular that F does not depend on B (B;α) µ. 1.14. The construction of a differential twist satisfying the centraliser property can, in turn, be obtained from that of an appropriate fusion operator.7 The latter is a solution of a dynamical version of the KZ equations in n=2 points, namely dJ Ω = h +adµ(1) J (1.7) dz z (cid:18) (cid:19) where z = z z , and µ h . The dynamical KZ equations arise naturally 1 2 reg − ∈ in Conformal Field Theory (see, e.g., [15]). They were studied in more detail by Felder–Markov–Tarasov–Varchenko in [20], where it was shown that they are bispectralto,thatiscommute with, adynamicalversionofthe Casimirconnection with respect to the variable µ. The presence of the dynamical term adµ(1) creates an irregular singularity at z = of Poincarérank 1. Assuming µ to be real, so that the Stokes rays of (1.7) ∞ all lie on the real axis, we construct in Section 5 two canonical solutions J (z,µ) ± with values in Ug⊗2[[~]], which have the form J (z,µ)=H (z,µ) zhΩh ± ± · where H (z,µ) is a holomorphic function on the upper (resp. lower) half–plane ± whichpossessesanasymptotic expansionof the form1⊗2+O(z−1) as z with →∞ 0 << argz << π, and Ω = t ta, with t , ta dual bases of h with respect a a | | ⊗ { } { } to (, ), is the projection of Ω on the kernel of ad(µ(1)). The construction of J ± · · and the study of its analytic properties require some care, since the equation (1.7) takesvaluesin the infinite–dimensionalalgebraUg⊗2[[~]], andis the maintechnical contribution of the paper. Given the fusion operatorJ (z,µ), a differentialtwist canbe obtainedas either ± of the ratios F (µ)=J (z,µ)−1 J (z,µ) ± ± 0 · whereJ (z,µ)istheuniquesolutionof (1.7)whichisasymptoticto(1⊗2+O(z))z~Ω 0 · near z =0.8 1.15. TheproofthatF (µ)killstheassociatorisfairlystandard(seee.g.,[16,14]), ± provided an analogue of the fusion operator can be constructed for the dynamical KZ equation in n = 3 points. Even though the corresponding KZ equations have irregular singularities at z = , a solution can still be constructed by simulta- i ∞ neously scaling all variables z ,z ,z , and sending the scale to infinity. Here, we 1 2 3 crucially exploit a beautiful fact, which is that the dynamical KZ equations in n variables abelianise at infinity. Roughly speaking, this means that the connection 7TheterminologyisborrowedfromtheworkofEtingof(seee.g.,[16],andreferencestherein). Therelationofourconstruction to[17]isdiscussedin1.18. 8Theexistence ofJ0 isstraightforwardsincez=0isaregularsingularityof (1.7). QUASI–COXETER ALGEBRAS AND THE CASIMIR CONNECTION 7 satisfiedby the asymptotics ofsolutions onthe divisorwhere all z z are infinite i j − is the abelian KZ equations d h dlog(z z )Ωh − i− j ij i<j X Contrary to its non-abelian analogue, this equation possesses a canonical solution, namely i<j(zi −zj)hΩhij, which leads to the construction of a multicomponent fusion operator of the form Q J± =H±(z1,...,zn) (zi zj)hΩhij · − i<j Y 1.16. ThefactthatthetwistsF (µ)satisfytheCasimirequations1.5followsfrom ± thefactthatJ (z,µ)satisfiesadynamicalversionoftheCasimirequations,namely ± h dα d J = ∆(C )J J(C (1)+C (2)) +zad(dµ(1))J h α α α 2 α − αX>0 (cid:16) (cid:17) whichexpressesthefactthat,whenz z = ,thedynamicalCasimirconnection 1 2 − ∞ on the tensor product V V of two representations becomes the tensor product 1 2 ⊗ of the (non–dynamical) Casimir connections on each factor. The above equation is a consequence of the bispectrality of the dynamical KZ and Casimir equations, together with the fact that J varies smoothly in µ h , which follows from our reg ∈ analysis. 1.17. InSection7,weshowthatthedifferentialtwistsF (µ)satisfythecentraliser ± property. Thisfollowsbyrelatingthe(irregular)asymptoticsofthefusionoperator ofgwhenoneofthesimple rootscoordinatesα tends to ,tothe fusionoperator i ∞ of the corank 1 subalgebra g . D\αi WerevisitthesecalculationsinAppendixB,andgiveadirectconstructionofthe relativetwistsarisingfromthefusionoperatorwhichissimilarinspirittoDrinfeld’s construction of the KZ associator. This gives, to the best of our knowledge, the firstcanonicaltranscendentalconstructionofatwistkilling theKZassociatorΦ . KZ Specifically, we show that the twist F defined by the factorisation relation (D;αi) (1.6)canberealisedastheconstantrelatingthecanonicalfundamentalsolutionsat 0 and of an ODE with regular singularities at 0, 1 and an irregularsingularity ∞ − at . The ODE is defined with respect to the blowup coordinate v = zα , and is i ∞ given by dG = hΩ + h∆(KD−KD\αi)−2Ω +adλ∨(1) G dv v 2 v+1 i (cid:18) (cid:19) where, for any subdiagram B D, is the truncated (Cartan–less) Casimir B ⊆ K operator of g , and λ∨ is the coroot corresponding to α . B i i 1.18. Our construction of the differential twist is very close, in spirit at least, to Etingof and Varchenko’s study of the fusion operator J(w,λ) of the affine Lie algebra g [17]. The latter satisfies the trigonometric KZ equations with respect to w, together with the dynamical difference equations wbith respect to λ, where the lattebr are a system of difference equations which degenerates to the Casimir connection. The regularised limit Jλ) of J(z,λ) as z 1 kills the KZ associator → in the shifted sense, that is satisfies J (λ)J (λ bh(3))=bJ (λ)J (λ)Φ 12,3 1,2 − 1,23 2,3 KZ b b b b 8 V.TOLEDANOLAREDO (see, e.g., [14, 16]). A construction of a differential twist might therefore arise by taking an appropriate scaling limit of J(λ) as λ goes to infinity (a process which would kill the shift in the above equation). Controlling the asymptotics of J at λ = seems difficult, however, sincebJ(λ) only satisfies a system of difference ∞ equations with respect to λ. b Rather than purse this path, we givebin this paper a direct construction of a solution of the rational dynamical KZ equations, which can be thought of as (and probablyis)a degenerationofthe fusionoperatorofˆg. Our J (z,µ)shouldinfact ± be a fusionoperatorfor the currentalgebrag[t]. One further difference with [17] is that, unlike J(w,λ), J (z,µ) is not constructed via representation theory, specif- ± ically the fusion construction for loop modules of g, but purely using differential equations. Itbseems an interesting question to construct our fusion operator from representationtheory. Such a construction should bbe obtained by replacing Verma modulesbytheirregularWakimotomodulesforgconsideredin[19,18],sincethese give rise to the Casimir connection [18]. b 1.19. The monodromytheorem provedin this paper is extended to the case of an arbitrary symmetrisable Kac–Moody algebra in [1, 2, 3]. The approach is close in spirittothat[25],butdiffersverysignificantlyinthedetailsoftwooutofthe three steps of the construction, namely the transfer of braided quasi–Coxeter structure fromthe category int ofintegrable,highestweightU g–modulestoanisomorphic O~ ~ structureonthe correspondingcategory int[[~]]forUg[[~]],andtheproofthatsuch O structures are rigid. The last step, namely the construction of a braided quasi– Coxeter structure on int[[~]] which accounts for the monodromy of the Casimir O equations of g and that of the KZ equations of its Levi subalgebras is carried out in [3] by using the construction of the fusion operator and differential twist given in this paper. 1.20. Outline of the paper. Section 2 contains some preliminary material required to study differential equations with values in infinite–dimensional filtered vector spaces. In Section 3, we review the definition of quasitriangular quasibialgebras and of quasi–Coxeter algebras together with the fact that the monodromy of the KZ and Casimir connections respectively define such structures on the enveloping algebra Ug of a complex, semisimple Lie algebra g. In Section 4, we introduce the notion ofdifferential twist for g, andshow that it givesrisetoaquasi–CoxeterquasitriangularquasibialgebrastructureonUg,which interpolates between the quasitriangular quasibialgebra structure underlying the monodromyofthe KZconnectionandthe quasi–Coxeterstructureunderlyingthat of the Casimir one. InSection5,weconstructafusionoperatorforgasajointsolutionofthecoupled KZ–Casimirequationson2points,withprescribedasymptoticswhenz z . 1 2 − →∞ In Section 6, we obtain a differential twist for g as the regularised limit of the fusion operator when z 0, and prove that it kills the KZ associator. → InSection7, werelatethe differentialtwists for g andfor acorank1Levisubal- gebra, and use this to prove that the differential twist of g satisfies the centraliser property described in 1.13. ThispropertyisusedinSection8,toshowthatthedifferentialtwistarisingfrom the fusion operator it gives rise to a quasi–Coxeter quasitriangular quasibialgebra QUASI–COXETER ALGEBRAS AND THE CASIMIR CONNECTION 9 structureonUginterpolatingbetweenthe quasitriangularquasibialgebrastructure underlying the KZ equations and the quasi–Coxeter algebra one underlying the Casimir connection. Section 9 collects some facts about the quantum group U g, and in particular ~ the fact that it possesses a quasi–Coxeter quasitriangular quasibialgebra structure which accounts for both its R–matrix and quantum Weyl group representations. Section10containsthe mainresultofthis paper,namely the equivalenceofU g ~ and Ug[[~]] as quasi–Coxeter quasitriangular quasibialgebras and the immediate corollary that the monodromy of the Casimir connection is described by quantum Weyl group group operators. AppendixAcontainsadetaileddiscussionofthesolutionsofalinear,scalarODE with an irregular singularity at , which plays a similar role in this paper than Drinfeld’s ODE df =(A + B )f∞does for the construction of the KZ associators. dz z z−1 Appendix B gives an alternative proof that the differential twist obtained in Section 6 possesses the centraliser property. As a corollary, we obtain a canonical, transcendental construction of a twist killing the KZ associator Φ . KZ The final Appendix C gives an alternative proof that the coupled KZ–Casimir equations are integrable. 1.21. Acknowledgments. This paper was a very long time in the making, as its main result was first announced in 2005. I am very grateful to my late colleague Andrei Zelevinsky, and to Edward Frenkel for amicably, but firmly, encouraging me to complete it. I am also grateful to Claude Sabbah for correspondence and discussions on irregular singularities in 2 dimensions. 2. Filtered algebras 2.1. Let A be an algebra over C endowed with an ascending filtration C=A A 0 1 ⊂ ⊂··· suchthatA A A . Leto= o beasequenceofnon–negativeintegers, m n m+n k k∈N · ⊂ { } ~ a formal variable, and consider the subspace A[[~]]o A[[~]] defined by ⊂ A[[~]]o = a ~k a A { k | k ∈ ok} k≥0 X Note that: (1) A[[~]]o is a (closed) C[[~]]–submodule of A[[~]] if o is increasing.9 (2) A[[~]]o isasubalgebraofA[[~]]ifoissubadditive,thatissuchthato +o k l ≤ o for any k,l N. This implies in particular that o is increasing, and k+l ∈ that o =0. 0 If the subspaces A A are finite–dimensional, so are the quotients k ⊂ A[[~]]o/(~p+1A[[~]] A[[~]]o)=A ~A ~pA ∩ ∼ o0 ⊕ o1 ⊕···⊕ op Assuming this, we shall say that a map F : X A[[~]]o, where X is a topologi- → cal space (resp. a smooth or complex manifold), is continuous (resp. smooth or holomorphic) if each of its truncations F :X A[[~]]o/(~p+1A[[~]] A[[~]]o) are. p → ∩ 9A[[~]]o isthen the ~–adiccompletion ofthe Rees algebraof Acorresponding tothe filtration Ao0 ⊂Ao1 ⊂··· 10 V.TOLEDANOLAREDO 2.2. We shall mainly be interested in the following situation: A=Ug⊗n endowed with the standard order filtration given by deg(x(i))=1 for x g, where ∈ x(i) =1⊗(i−1) x 1⊗(n−i) ⊗ ⊗ The sequence o will be chosen subadditive, and such that o 2 in order for 1 ≥ ~Ω ,~∆(n)( ) A[[~]]o. Note that g Ug[[~]]o = 0 since o = 0, but that the ij α 0 K ∈ ∩ { } adjointaction of g on Ug⊗n induces one on by derivations on Ug⊗n[[~]]o. Note also that Ug[[~]]o is not a Hopf algebra, since ∆:Ug[[~]]o Ug⊗2[[~]]o )(Ug[[~]]o)⊗2. → 2.3. Let be a C[[~]]–module and consider the natural map ı : lim /~n . A A → ←−A A Recall that is separated if ı is injective, and complete if it is surjective. By A definition, is topologically free if it is separated, complete and torsion–free. A Consider now the map ı:End ( ) limEnd ( /~n ) C[[~]] A → ←− C[[~]] A A Lemma. Assume that is separated. Then, A (1) ı is injective. (2) If is complete, ı is surjective. A Proof. (1) If T End is such that ıT = 0, then T( ) ~n = 0. ∈ C[[~]] A ⊂ n A (2) Let T lim End ( /~n ). For any a , the sequence T a lies in { n} ∈ n C[[~]] A A ∈ A T{ n } lim /~n andisthereforetheimageofauniqueelementa′ . Theassignment n A A ∈A a Ta = a′ is easily seen to define an element of End ( ) which projects to → C[[~]] A each of the T . (cid:3) n Corollary. If istopologicallyfree,themapı:End ( ) limEnd ( /~n ) A C[[~]] A → ←− C[[~]] A A is an isomorphism. 3. Quasi–Coxeter algebras Wereviewinthissectionthedefinitionofquasitriangularquasibialgebrasfollow- ing [12], and of quasi–Coxeter algebras following [25], to which we refer for more details. WealsoexplainhowthemonodromyoftheKnizhnik–Zamolodchikov(KZ) and Casimir connections of a complex, semisimple Lie algebra g respectively give risetoaquasitriangularquasibialgebraandquasi–Coxeteralgebrastructureonthe enveloping algebra Ug[[~]] of g. 3.1. Quasitriangular quasibialgebras [12]. 3.1.1. Recallthataquasibialgebra(A,∆,ε,Φ)isanalgebraAendowedwithalge- bra homomorphisms ∆:A A⊗2 and ε:A k called the coproduct and counit, → → andaninvertibleelementΦ A⊗3 calledtheassociatorwhichsatisfy,foranya A ∈ ∈ id ∆(∆(a))=Φ ∆ id(∆(a)) Φ−1 ⊗ · ⊗ · id⊗2 ∆(Φ) ∆ id⊗2(Φ)=1 Φ id ∆ id(Φ) Φ 1 ⊗ · ⊗ ⊗ · ⊗ ⊗ · ⊗ ε id ∆=id ⊗ ◦ id ε ∆=id ⊗ ◦ id ε id(Φ)=1 ⊗ ⊗ A twist of a quasibialgebra A is an invertible element F A⊗2 satisfying ∈ ε id(F)=1=id ε(F) ⊗ ⊗