Quasi-Coxeter quasitriangular quasibialgebras and the Casimir connection

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by  Valerio Toledano-Laredo

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 quasibial-
gebra 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 mon-
odromy 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 com-
ing 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 Poincar´erank 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 re-
quired 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)
⊗ ⊗