Drinfeld coproduct, quantum fusion tensor category and applications
a r X i v :m a t h /0504269v 3 [m a t h .Q A ] 27 M a r 2006DRINFELD COPRODUCT,QUANTUM FUSION TENSOR CATEGORY AND APPLICATIONS DAVID HERNANDEZ Abstract.The class of quantum a?nizations (or quantum loop algebras,see [Dr2,CP3,GKV,VV2,Mi1,N1,Jin,H3])includes quantum a?ne algebras and quantum toroidal algebras.In general they have no Hopf algebra structure,but have a “coproduct”(the Drinfeld coproduct)which does not produce tensor products of modules in the usual way because it is de?ned in a completion.In this paper we propose a new process to produce quantum fusion modules from it :for all quantum a?nizations,we construct by deformation and renormalization a new (non semi-simple)tensor category Mod.For quantum a?ne algebras this process is new and di?erent from the usual tensor product.For general quantum a?nizations,for example for toroidal algebras,so far,no process to produce fusion modules was known.We derive several applications from it :we construct the fusion of (?nitely many)arbitrary l -highest weight modules,and prove that it is always cyclic.We establish exact sequences involving fusion of Kirillov-Reshetikhin modules related to new T -systems extending results of [N4,N3,H5].Eventually for a large class of quantum a?nizations we prove that the subcategory of ?nite length modules of Mod is stable under the new monoidal bifunctor.2000Mathematics Subject Classi?cation:Primary 17B37,Secondary 20G42,81R50.Contents 1.Introduction 12.Background 53.Construction of the quantum fusion tensor category Mod 94.A -forms and specializations 165.Finite length property and quantum fusion tensor category
216.Applications
27References 36
1.Introduction
In this paper q ∈C ?is not a root of unity.
Drinfeld [Dr1]and Jimbo [Jim]associated,independently,to any symmetrizable Kac-Moody alge-bra g and q ∈C ?a Hopf algebra U q (g )called quantum Kac-Moody algebra.The quantum algebras of ?nite type U q (g )(g of ?nite type)and their representations have been intensively studied (see for example [CP3,L2,R2]and references therein).The quantum a?ne algebras U q (?g )(?g a?ne
1
2DA VID HERNANDEZ
algebra)are also of particular interest :they have two realizations,the usual Drinfeld-Jimbo real-ization and a new realization (see [Dr2,B])as a quantum a?nization of a quantum algebra of ?nite type U q (g ).Quantum a?ne algebras and their representations have also been intensively studied (see among others [AK,CP1,CP2,CP3,EM,FR,FM,H5,Kas2,N1,N4,VV3]and references therein).
The quantum a?nization process (that Drinfeld [Dr2]described for constructing the second realization of a quantum a?ne algebra)can be extended to all symmetrizable quantum Kac-Moody algebras U q (g )(see [Jin,N1]).One obtains a new class of algebras called quantum a?nizations :the quantum a?nization of U q (g )is denoted by U q (?g )and contains U q (g )as a subalgebra.The quantum a?ne algebras are the simplest examples and have the singular property of being also quantum Kac-Moody algebras.The quantum toroidal algebras (a?nizations of quantum a?ne algebras)are also of particular importance,see for example [GKV,Mi1,Mi2,N1,N2,Sa,Sc,STU,TU,VV2].In analogy to the Frobenius-Schur-Weyl duality between quantum groups of ?nite type and Hecke algebras [Jim2],and between quantum a?ne algebras and a?ne Hecke algebras
[CP4],quantum toroidal algebras have a close relation [VV1]to double a?ne Hecke algebras and their degenerations (Cherednik’s algebras [Che2])which have been recently intensively studied (for example see [BE,Che1,GGOR,GS,V,VV4]and references therein).In general a quantum a?nization (for example a quantum toroidal algebra)is not isomorphic to a quantum Kac-Moody algebra and has no Hopf algebra structure.For convenience of the reader,we ”describe”in the following diagram the relations between these di?erent algebras (in particular with the two main classes of algebras considered in this paper,quantum Kac-Moody algebras on the left and quantum a?nizations on the right).? Tensor category ?
Drinfeld Quantum Affinization Process
[Dr2,B]
Algebras Quantum Kac?Moody Quantum affinizations
[Dr2][GKV, N1, Jin]
[Jim2][CP3][VV1]
Hecke algebras Affine Hecke
algebras
Cherednik algebras
Quantum affine algebras Quantum affine algebras Quantum algebras of finite type Quantum toroidal algebras
? int. l?highest weight rep.? int. highest weight rep. [R1, L2]
[CP1, Mi1, N1, H4]? Not Hopf algebras in general
? Hopf algebras in general ? Semi?simple category O of int. rep.
? Tensor category ? Category O of int. rep. not semi?simple ? Combinatorial fusion product [H4]We denote by h (resp.U q (h ))the Cartan subalgebra of g (resp.U q (g )).U q (?h )?U q (?g )is the
quantum a?ne analog of the Cartan subalgebra.
DRINFELD COPRODUCT,QUANTUM FUSION TENSOR CATEGORY AND APPLICATIONS3 In analogy to the generalization of the representation theory of quantum algebras of?nite type to general quantum Kac-Moody algebras(see[L2]),a natural question is to extend the representation theory of quantum a?ne algebras to general quantum a?nizations.By extending the results of [CP3](quantum a?ne algebras),[Mi1](quantum toroidal algebras of type A)and[N1](general simply-laced quantum a?nizations),we established in[H3]a triangular decomposition of U q(?g) which allowed us to begin the study of the representation theory of general quantum a?nizations: one can de?ne the notion of l-highest weight representations of U q(?g)(analog to the usual notion of highest weight module corresponding to U q(?h)).One says that a U q(?g)-module is integrable(resp. in the category O)if it is integrable(resp.in the category O)as a U q(g)-module.Let Mod(U q(?g)) be the category of integrable U q(?g)-module which are in the category O.It appears that U q(?g) has numerous l-highest weight representations in Mod(U q(?g))and that one can classify the simple ones.The category Mod(U q(?g))is not semi-simple.
No Hopf algebra structure is known for U q(?g)except in the case of quantum a?ne algebras. However in[H3]we proved via Frenkel-Reshetikhin q-characters the existence of a combinatorial fusion product in the Grothendieck group of Mod(U q(?g)),that it to say a product in the semi-simpli?ed category.As Mod(U q(?g))is not semi-simple,this fusion product can not a priori be directly translated in terms of modules.
Each quantum a?nization has a”coproduct”(the Drinfeld coproduct)which is de?ned in a com-pletion of U q(?g)?U q(?g).Although it can not be directly used to construct tensor products of integrable representations(it involves in?nite sums),we propose in the present paper a new pro-cess to produce quantum fusion modules from it:for all quantum a?nizations we construct by deformation of the Drinfeld coproduct and renormalization a new(non semi-simple)tensor cate-gory Mod.In particular it gives a representation theoretical interpretation of the combinatorial fusion product and we get several applications from it.For quantum a?ne algebras this process is new and di?erent from the usual tensor product(we prove that it has di?erent properties),and for general quantum a?nizations(as general quantum toroidal algebras)no process to produce fusion of modules was known so far.
For this construction,the?rst technical point that we solve is a rationality problem for the depen-dence in the deformation parameter u of the Drinfeld coproduct.Let U′q(?g)=U q(?g)?C(u).We consider a”good”category Mod(U′q(?g))of U′q(?g)-modules V with an integrable U q(h)-submodule W?V satisfying V?(W?C C(u)),U q(?h)(W?C[u±])?(W?C[u±]),and some additional technical properties.It appears that the action of U q(?g)has a nice regularity property which is compatible with the Drinfeld coproduct and leads to this rationality,that is to say that Mod(U′q(?g)) is stable under a tensor product.The second point is the associativity:the Drinfeld coproduct is coassociative,but the deformation parameter u breaks the symmetry and the u-deformed Drin-feld coproduct is not coassociative in the usual sense.However there is a”twisted”coassociative property which allows us to get the associativity of a new monoidal structure on the category Mod=Mod0⊕(Mod(U′q(?g))⊕Mod(U′q(?g))⊕···)(here Mod0is an abelian semi-simple category with a unique simple object corresponding to a neutral object of the category).
Besides we prove that for a large class of quantum a?nizations(including quantum a?ne algebras and most quantum toroidal algebras)the subcategory of modules with a?nite composition series is stable under the monoidal bifunctor.This proof uses the specialization process described below and an investigation of the compatibility property between generalizations of Frenkel-Reshetikhin q-characters and the monoidal bifunctor.We get in particular that all l-highest weight modules of the category Mod(U q(?g))have a?nite composition series.
4DA VID HERNANDEZ
In order to go back to the usual category Mod(U q(?g)),we prove the existence of certain forms for all cyclic modules(A-forms):such forms can be specialized at u=1and give cyclic U q(?g)-modules; the proof uses the new rationality property.As a consequence,we can construct an l-highest weight fusion U q(?g)-module V1?f V2from two l-highest weight U q(?g)-modules V1,V2.In particular a simple module of Mod(U q(?g))is a quotient of a fusion module of fundamental representations(analog to a result of Chari-Pressley for the usual coproduct of quantum a?ne algebras).We establish a cyclicity property of the fusion of any l-highest weight modules:it is always an l-highest weight module.For quantum a?ne algebras this property is very di?erent from the properties of the usual tensor product.It allows to control the”size”of the modules obtained by?f;in particular we can produce”big”integrable l-highest weight modules whose existence was a priori not known.We get moreover another bifunctor?d:Modf(U q(?g))×Modf(U q(?g))→Modf(U q(?g))where Modf(U q(?g)) is the category of?nite dimensional representations(in general?d does not coincide with?f). Let us describe an application.One important aspect of the representation theory of quantum a?ne algebras is the existence of the Kirillov-Reshetikhin modules whose characters are given by the certain explicit”fermionic”formulas.The proof of the corresponding statement(the Kirillov-Reshetikhin conjecture)can be obtained from certain T-systems originated from the theory of integrable systems.These T-systems can be interpreted in the form exact sequences involving tensor products of Kirillov-Reshetikhin modules;they were?rst established in[N3]for simply-laced quantum a?ne algebras(with the main result of[N4])and then in[H5]for all quantum a?ne algebras(with a di?erent proof).In the present paper we establish for a large class of general quantum a?nizations exact sequences involving fusion of Kirillov-Reshetikhin modules: the usual tensor product can be replaced in the general situation by the fusion constructed in this paper.It is related to new generalized T-systems that we de?ne and establish.The existence of such new T-systems is a new particular regularity property of the representation theory of general quantum a?nizations.
We would like to mention other possible applications and new developments:for example the quantum fusion tensor category should lead to the description of the blocks of the cate-gory Mod(U q(?g)).Besides the generalized T-systems established in this paper should lead to new fermionic formulas in analogy to the case of quantum a?ne algebras and should have a nice interpretation in terms of integrable systems.It would be interesting to relate the quantum fusion to the Feigin-Loktev fusion procedure de?ned for classical a?ne algebras[FL].We believe that the deformation of the Drinfeld coproduct used is a particular case of a more general framework in-volving some deformed Hopf structure on Z-graded algebras(”quantum Hopf vertex algebras”that we hope to describe in another paper).Eventually the tensor category constructed in this paper should be related to certain new Hopf algebras in the Tannaka-Krein reconstruction philosophy.
Let us describe the organization of this paper:
In section2we give the background for general quantum a?nizations and their representations and we recall the main results of[H3].In section3we construct the”quantum fusion”tensor category Mod(theorem3.12)with the help of the intermediate category Mod(U′q(?g))(de?nition 3.6)and establish the rationality property in u.The case of fusion of Kirillov-Reshetikhin modules of the quantum a?ne algebra U q(?sl2)is studied explicitly in subsection3.4.In section4we de?ne and study the notion of A-form(de?nition4.1).In particular we prove in theorem4.6 that one can de?ne A-forms of cyclic modules of the category Mod(U′q(?g)).It gives rise to the two following constructions:a fusion l-highest weight module from two l-highest weight modules (de?nition4.10),and in the case of a quantum a?ne algebra a new bifunctor for the category of?nite dimensional representations(theorem4.13).In section5we prove for a large class of
DRINFELD COPRODUCT,QUANTUM FUSION TENSOR CATEGORY AND APPLICATIONS5 quantum a?nizations that the subcategory of?nite length modules is stable under the monoidal bifunctor(theorem5.27)by proving the compatibility with the q-characters(theorem5.12)and with the combinatorial fusion product introduced in[H3];as a by-product we get a representation theoretical interpretation of this combinatorial fusion product(corollary5.15).In section6we describe several applications:?rst we prove that a simple module of Mod(U q(?g))is a quotient of the fusion of fundamental representations(proposition6.1);we establish a cyclicity property which allows to control the”size”of this fusion module(theorem6.2and corollary6.3).Then we establish an exact sequence involving fusion of Kirillov-Reshetikhin modules(theorem6.7).It is related to new generalized T-systems(theorem6.10).In subsection6.3we address complements on additional questions and further possible applications.
The main results of this paper were?rst announced in the”s′e minaire quantique”in Strasbourg and at the conference”Representations of Kac-Moody Algebras and Combinatorics”in Ban?respectively in January and March2005.
Acknowledgments:The author would like to thank V.Chari,E.Frenkel,A.Moura,N. Reshetikhin,M.Rosso,O.Schi?mann and M.Varagnolo for useful discussions.This paper was completed as the author visited the university La Sapienza in Rome as a Liegrits visitor;he would like to thank C.De Concini and C.Procesi for their hospitality.
2.Background
In this section we give backgrounds on general quantum a?nizations and their representations (we also remind results of[H3]).In the following for a formal variable u,C[u±],C(u),C[[u]],C((u)) are the standard notations.
2.1.Cartan matrix.A generalized Cartan matrix is a matrix C=(C i,j)1≤i,j≤n satisfying C i,j∈Z,C i,i=2,(i=j?C i,j≤0)and(C i,j=0?C j,i=0).We denote I={1,...,n}and l=rank(C).For i,j∈I,we putδi,j=0if i=j,andδi,j=1if i=j.
In the following we suppose that C is symmetrizable,that means that there is a matrix D= diag(r1,...,r n)(r i∈N?)such that B=DC is symmetric.In particular if C is symmetric then it is symmetrizable with D=I n.In the following B,C and D are?xed.
q∈C?is not a root of unity and is?xed.We put q i=q r i and for l∈Z,we set[l]q=q l?q?l
6DA VID HERNANDEZ
Denote Q= i∈I Zαi?P the root lattice and Q+= i∈I Nαi?Q.De?ne h:Q+→N such that h(l1α1+...+l nαn)=l1+...+l n.
Forλ,μ∈h?,writeλ≥μifλ?μ∈Q+.Forλ∈h?,denote S(λ)={μ∈h?|μ≤λ}.
2.2.Quantum Kac-Moody algebra.In the following g is the symmetrizable Kac-Moody alge-bra associated with the data(h,Π,Π∨)?xed in section2.1(see[Kac]).
De?nition2.1.The quantum Kac-Moody algebra U q(g)is the C-algebra with generators k h(h∈
h),x±
i
(i∈I)and relations:
k h k h′=k h+h′,k0=1,k h x±j k?h=q±αj(h)x±j,
x+i x?j?x?j x+i=δi,j k r
iα∨i
?k?r
iα∨i
w?q±B i,j z
x±j(w)φ+i(z), (4)φ?i(z)x±j(w)=
q±B i,j w?z
DRINFELD COPRODUCT,QUANTUM FUSION TENSOR CATEGORY AND APPLICATIONS7
(5)x+i(z)x?j(w)?x?j(w)x+i(z)=δi,j
z
)φ+i(w)?δ(
z
w?q±B i,j z
is viewed as
a formal power series in z/w expanded around∞(resp.0).
When C is of?nite type,U q(?g)is called a quantum a?ne algebra,and when C is of a?ne type, U q(?g)is called a quantum toroidal algebra.There is a huge amount of very interesting papers on quantum a?ne algebras(see the introduction for references).They have the very particular property to be also quantum Kac-Moody algebras[Dr2,B].The quantum toroidal algebras are also of particular importance and are closely related via a Frobenius-Schur-Weyl duality[VV1]to double a?ne Hecke algebras(Cherednik’s algebras,see the introduction for references).In general a quantum a?nization(for example a quantum toroidal algebra)is not isomorphic to a quantum Kac-Moody algebra.
Note that formulas(3),(4)are equivalent to(see for example[H3]):
(8)h i,m x±j,r?x±j,r h i,m=±
1
8DA VID HERNANDEZ
De?nition2.4.A U q(h)-module V is said to be integrable if V is U q(h)-diagonalizable and for all
ω∈h?,Vωis?nite dimensional,and for i∈I there is R≥0such that(r≥R?Vω±rα
i ={0}).A
U q(?g)-module(resp.a U q(g)-module)V is said to be integrable if V is integrable as a U q(h)-module.
An integrable representation is not necessarily?nite dimensional.
We have the natural analog of the classical category O of Kac-Moody algebras[Kac]:
De?nition2.5.A U q(h)-module V is said to be in the category O(U q(h))if
i)V is U q(h)-diagonalizable,
ii)for allω∈h?,dim(Vω)<∞,
iii)there is a?nite number of elementλ1,...,λs∈h?such that the weights of V are in j=1,···,s S(λj).
De?nition2.6.For m respectively equal to?h,g,?g,we denote by Mod(U q(m))the category of U q(m)-modules which are integrable and in the category O(U q(h))as a U q(h)-module.
The category Mod(U q(g))(in a slightly more general form)is considered in[L2].De?nitions for U q(?g)-modules are given in[N1,section1.2].
Forλ∈P,denote by L(λ)the simple highest weight U q(g)-module of highest weightλ.We have (see for example[R1,L2]):
Theorem2.7.L(λ)∈Mod(U q(g))if and only ifλ∈P+.
The category Mod(U q(g))is semi-simple.But Mod(U q(?g))is not semi-simple.
By generalizing the approach of?nite dimensional representations of quantum a?ne algebras developed in[CP3],and de?nitions in[N1,Mi1],we de?ned in[H3]for general quantum a?nization U q(?g):
De?nition2.8.An l-weight is a couple(λ,Ψ)whereλ∈P andΨ=(Ψ±
i,±m
)i∈I,m≥0,such that
Ψ±i,±m∈C?andΨ±i,0=q±λ(α∨i)
i
for i∈I and m≥0.
A U q(?g)-module V is said to be of l-highest weight(λ,Ψ)if there is v∈Vλ(l-highest weight vector) such that V=U q(?g).v and x+i,r.v=0,φ±i,±m.v=Ψ±i,±m v for i∈I,r∈Z and m≥0.
An l-weight(λ,Ψ)is said to be dominant if there is an n-tuplet of(Drinfeld)polynomials(P i(u))i∈I∈(C[u])n satisfying for i∈I,P i(0)=1and the relation in C[[z]](resp.in C[[z?1]]):
(9) m≥0Ψ±i,±m z±m=q deg(P i)i P i(zq?1i)
DRINFELD COPRODUCT,QUANTUM FUSION TENSOR CATEGORY AND APPLICATIONS9 algebras of type A,in[N1,Proposition1.2.15]for general simply-laced quantum a?nizations and
in[H3,Theorem4.9]for general quantum a?nizations.
Examples.For i∈I,a∈C?,r≥1,consider the l-weight(Λi,Ψi,a,r)whereΨi,a,r is given by
)and P j(u)=1for equation(9)with the n-tuplet P i(u)=(1?ua)(1?uaq2i)···(1?uaq2(r?1)
i
j=i.The U q(?g)-module W(i)r,a=L(Λi,Ψi,a,r)∈Mod(U q(?g))is called a Kirillov-Reshetikhin module.
In the particular case r=1,the U q(?g)-module L i,a=W(i)1,a is called a fundamental representation. Forλ∈P satisfying(?j∈I,λ(α∨j)=0),consider the l-weight(λ,Ψ0)whereΨ0is given by equation(9)with the n-tuplet P j(u)=1for all j∈I.The U q(?g)-module Lλ=L(λ,Ψ0)∈Mod(U q(?g))is also called a fundamental representation in this paper.
3.Construction of the quantum fusion tensor category Mod
In general a quantum a?nization U q(?g)has no Hopf algebra structure(except in the case of quantum a?ne algebras where we have the coproduct of the Kac-Moody realization).However Drinfeld(unpublished note,see also[DI,DF])de?ned for U q(?sl n)a map which behaves like a new coproduct adapted to the a?nization realization,but it is de?ned in a completion and can not directly be used to de?ne tensor product of representations.In this paper we construct a corresponding tensor category(section3)by deforming the Drinfeld coproduct,see how to go back to the category Mod(U q(?g))by specialization(section4),and give applications of these results(section6).We also prove in section5that it induces a tensor category structure on the subcategory of modules of?nite length.For quantum a?ne algebras this process is new and di?erent from the usual tensor product(we prove that it has di?erent properties),and for general quantum a?nizations(as general quantum toroidal algebras)there was no process to construct fusion of modules so far.For this construction,the?rst technical point that we solve is a rationality problem for the dependence in the deformation parameter u of the Drinfeld coproduct (lemma3.10).The second point is the associativity:the u-deformed Drinfeld coproduct has a”twisted”coassociative property which allows us to get the associativity of the new monoidal structure(lemma3.4).
In this section we de?ne the category Mod(de?nition3.6)and the tensor structure(theorem3.12). The case of fusion of Kirillov-Reshetikhin modules of the quantum a?ne algebra U q(?sl2)is studied explicitly in subsection3.4.
3.1.Deformation of the Drinfeld coproduct.In this section we study a u-deformation of the Drinfeld coproduct,in particular a”twisted”coassociativity property(lemma3.4).
Let U′q(?g)=U q(?g)?C(u)and U′q(?g)??U′q(?g)=(U q(?g)?C U q(?g))((u))be the u-topological completion of U′q(?g)?C(u)U′q(?g).Let?U q(?g)be the algebra de?ned as U q(?g)without a?ne quantum Serre relations(7).We also de?ne?U′q(?g)=?U q(?g)?C(u)and?U′q(?g)???U′q(?g)=(?U q(?g)?C?U q(?g))((u)).
In[H3,Proposition6.3]we introduced a u-deformation of the Drinfeld coproduct: Proposition3.1.There is a unique morphism of C(u)-algebra?u:?U′q(?g)→?U′q(?g)???U′q(?g)such that for i∈I,r∈Z,m≥0,h∈h:
(10)?u(x+i,r)=x+i,r?1+ s≥0u r+l(φ?i,?s?x+i,r+s),
10DA VID HERNANDEZ
(11)?u(x?i,r)=u r(1?x?i,r)+ s≥0u s(x?i,r?s?φ+i,s),
(12)?u(φ±i,±m)= 0≤s≤m u±s(φ±i,±(m?s)?φ±i,±s),
(13)?u(k h)=k h?k h.
If(i=j?C i,j C j,i≤3),then?1is compatible with a?ne quantum Serre relations(7)(see [DI,E,G]).By de?nition of?u,this can be easily generalized to:
Corollary3.2.If(i=j?C i,j C j,i≤3),then the map?u induces a morphism?u:U′q(?g)→U′q(?g)??U′q(?g).
This case includes quantum a?ne algebras and most quantum toroidal algebras(except A(1)1,A(2)
2l ,
l≥0).
Remark3.3.Note that the simple l-highest weight modules of?U q(?g)are the same as for U q(?g) (via the projection?U q(?g)→U q(?g)).From corollary3.2,if(i=j?C i,j C j,i≤3),the results in this paper can be indi?erently stated and proved for U q(?g)or?U(?g).So in some sections,we use the following notation:
1)If(i=j?C i,j C j,i≤3),U q(?g)(resp.U q(g),U′q(?g))means the algebra with a?ne quantum Serre relations(7).
2)Otherwise,U q(?g)(resp.U q(g),U′q(?g))means the algebra without a?ne quantum Serre rela-tions(7).
This point will be explicitly reminded in the rest of this paper by references to this remark.
Observe that for i∈I,the subalgebra?U′i=?U i?C C(u)?U′q(?g)is a”Hopf subalgebra”of U′q(?g) for?u,that is to say?u(?U′i)??U′i???U′i.
The usual coassociativy property of a coproduct?is(Id??)??=(??Id)??.This relation is not satis?ed by?u.However,we have a”twisted”coassociativity property satis?ed by?u and obtained by replacing the parameter u by some power of u(at the”limit”u=1we recover the usual coassociativity property).This relation will be crucial in the construction of the quantum fusion tensor category(note that we do not use the quasi-Hopf algebras point of view):
Lemma3.4.Let U q(?g)as in remark3.3.Let r,r′≥1.As algebra morphisms U q(?g)→(U q(?g)?U q(?g)?U q(?g))((u)),the two following maps are equal:
(Id??u r′)??u r=(?u r?Id)??u r+r′.
Proof:It su?ces to check the equality on the generators.Because of u,the images of both applications are of the form
u r(deg(g(2))+deg(g(3)))+r′deg(g(3))g(1)?g(2)?g(3)
where g(1),g(2),g(3)are homogeneous.We just give the results of the maps on generators and leave it to the reader to check it.For x+i,p,x?i,p,φ±i,±m,k h,respectively the two maps give:
x+i,p?1?1+ s≥0u r(p+s)(φ?i,?s?x+i,p+s?1)+ s,s′≥0u r(p+s)+r′(p+s+s′)(φ?i,?s?φ?i,?s′?x+i,p+s+s′), u(r+r′)p(1?1?x?i,p)+ s≥0u rp+r′s(1?x?i,p?s?φ+i,s)+ s,s′≥0u r(s+s′)+s′r′(x?i,p?s?s′?φ+i,s?φ+i,s′),
DRINFELD COPRODUCT,QUANTUM FUSION TENSOR CATEGORY AND APPLICATIONS11
{s1,s2,s3≥0|s1+s2+s3=m}u±((s2+s3)r+s′3r′)(φ±i,±s1?φ±i,±s2?φ±i,±s3),
k h?k h?k h.
3.2.The category Mod(U′q(?g)).In this section we introduce and study the category Mod(U′q(?g)) which is”a building graded part”of the quantum fusion tensor category.
First let us give a u-deformation of the notion of l-weight:
De?nition3.5.An(l,u)-weight is a couple(λ,Ψ(u))whereλ∈h?,Ψ(u)=(Ψ±i,±m(u))i∈I,m≥0,
Ψ±i,±m(u)∈C[u±1]satisfyingΨ±i,0(u)=q±λ(α∨i)
i
.
De?nition3.6.Let Mod(U′q(?g))be the category of U′q(?g)-modules V such that there is a C-vector subspace W?V satisfying:
i)V?W?C C(u),
ii)W is stable under the action of U q(h)and is an object of Mod(U q(h)),
iii)the image of W?C[u±]in V by the morphism of i)is stable under the action of U q(?h),
iv)for allω∈h?,all i∈I,all r∈Z,there are a?nite number of C-linear operators f±
k,k′,λ:
Wω→V(k≥0,k′≥0,λ∈C?)such that for all m≥0and all v∈Wω:
x±i,r±m(v)= k≥0,k′≥0,λ∈C?λm u km m k′f±k,k′,λ(v),
v)we have a decomposition
V= (λ,Ψ(u))(l,u)-weight V(λ,Ψ(u))
where V(λ,Ψ(u))={v∈Vλ|?p≥0,?i∈I,m≥0,(φ±i,±m?Ψ±i,±m(u))p.v=0}.
We have a?rst result:
Lemma3.7.Let V∈Mod(U′q(?g)).For allω∈h?,all i∈I,there are a?nite number of C-linear
operators g±
k,k′,λ
:Wω→V(k≥0,k′≥0,λ∈C?)such that for all m≥1and all v∈Wω:
φ±i,r±m(v)= k≥0,k′≥0,λ∈C?λm u km m k′g±k,k′,λ(v).
Proof:Formula(5)gives for m≥1:
φ?i,?m=(q?1i?q i)(x+i,0x?i,?m?x?i,?m x+i,0)andφ+i,m=(q i?q?1i)(x+i,m x?i,0?x?i,0x+i,m)
and so property iv)of de?nition3.6gives the result.
Now we give a typical example of an object of Mod(U′q(?g)).For V a module in Mod(U q(?g)), consider i(V)the U′q(?g)-module obtained by extension:i(V)=V?C(u).
Proposition3.8.i de?nes an injective faithful functor i:Mod(U q(?g))→Mod(U′q(?g)).
In particular Mod(U q(?g))can be viewed as a subcategory of Mod(U′q(?g)).
Proof:Let us prove that the functor is well-de?ned.Consider V∈Mod(U q(?g))and let us prove that i(V)∈Mod(U′q(?g))where we choose W=V for the C-vector space of de?nition3.6.The properties i),ii),iii)are clear.The property v)is clear because the base?eld for V is C.Let us prove the property iv):let i∈I,ω∈h?and let us consider the operators x+i,r,r∈Z(the
12DA VID HERNANDEZ
proof for the x?i,r is analog).As W is integrable,there is a?nite dimensional?U i-submodule W′such that Wλ?W′?W.Considerρ:?U i→End(W′)the action.Consider the linear map Φ:End(W′)→End(W′)de?ned byΦ(p)=1
(ρ(h i,?1)p?pρ(h i,?1)).So i is well-de?ned.It is clearly injective and faithful.
[2r i]q
3.3.Construction of the tensor structure.The aim of this section is to de?ne a tensor cate-gory Mod by using?u and Mod(U′q(?g)).It is the main tool used in this paper;the stability inside the?eld C(u)is one of the crucial points that make the category Mod useful for the purposes of the present paper.
3.3.1.The category Mod.Let Mod0be the full abelian semi-simple subcategory of Mod(U′q(?g))-modules with a unique simple object i(L0)(we recall that L0=L(0,Ψ)whereΨis de?ned with Drinfeld polynomials equal to1).
De?nition3.9.We denote by Mod the direct sum of categories
Mod=Mod0⊕(Mod(U′q(?g))⊕Mod(U′q(?g))⊕···).
For r≥1,the r-th summand in the second sum is denoted by Mod r:Mod= r≥0Mod r.
Note that with the identi?cation Mod(U′q(?g))?Mod1,we can also consider that i is an injective faithful functor Mod(U q(?g))→Mod.So one can view Mod(U q(?g))as a subcategory of Mod.
3.3.2.Rationality property.We use the notations of remark3.3.Fix r≥ 1.Let V1,V2∈Mod(U′q(?g)).One de?nes an action of U′q(?g)on(V1?C(u)V2)((u))by the following formula (g∈U′q(?g),v1∈V1,v2∈V2):
(14)g.(v1?v2)=?u r(g)(v1?v2).
(A priori this formula only makes sense for r≥1,but we will see bellow that in some particular situations we can also use it for r=0).
We have the following”rationality property”of the action given by formula(14),which is the crucial point for the construction of the tensor structure.
Lemma3.10.The subspace(V1?C(u)V2)?(V1?C(u)V2)((u))is stable under the action of U′q(?g) de?ned by formula(14).The induced U′q(?g)-module structure on(V1?C(u)V2)is an object of Mod(U′q(?g)).
Proof:From de?nition3.6we have subspaces W1?V1,W2?V2.We choose W=W1?C W2?(V1?C(u)V2)and we prove that the properties of de?nition3.6are satis?ed.properties i),ii)are clear.Properties iii),v)follow directly from above formulas(12),(13).Let us prove property iv). Letλ,μ∈h?,p∈Z and suppose that for m≥0,v1∈(W1)λ,x±i,p±m(v1)=u mk1,±λm1,±m k′1,±f±1(v1),
DRINFELD COPRODUCT,QUANTUM FUSION TENSOR CATEGORY AND APPLICATIONS13
for m≥0,v2∈(W2)μ,x±i,p±m(v2)=u mk2,±λm2,±m k′2,±f±2(v2).It follows from lemma3.7,that we can suppose that for t≥1,φ?i,?t(v1)=u k3tλt3t k′3f3(v).Let v1∈(W1)λ,v2∈(W2)μand v=v1?v2. Formulas(10)and(14)give x+i,p+m(v)=A+B where
v1?x+i,p+m(v2)
A=x+i,p+m(v1)?v2+u r(p+m)q?λ(α∨i)
i
v1?u mk2,+λm2,+m k′2,+f+2(v2), =u mk1,+λm1,+m k′1,+f+1(v1)?v2+u r(p+m)q?λ(α∨i)
i
and
B= t≥1u r(p+m+t)(φ?i,?t(v1)?x+i,p+m+t(v2))
= t≥1u r(p+m+t)u k3tλt3t k′3f3(v1)?u k2,+(m+t)λm+t2,+(m+t)k′2,+f+2(v2)
= s=0,···,k′2,+ k′2,+s λm2,+u r(p+m)u mk2,+m k′2,+?s R s(u)f3(v1)?f+2(v2),
where
R s(u)= t≥1u rt u k3tλt3u tk2,+λt2,+t s+k′3.
As R s(u)∈C(u),x+i,p+m(v)makes sense in V1?C(u)V2.Moreover R s(u)does not depend of m, and so it follows from the above formulas for A and B that property iv)of de?nition3.6is satis?ed (the study is analog for x?i,m).So the U′q(?g)-module V1?C(u)V2is an object of Mod(U′q(?g)).
3.3.3.De?nition of the tensor structure.We use the notations of remark3.3.First let us study how formula(14)behaves with the representation i(L0).
Lemma3.11.Let r≥1and V∈Mod r.Then formula(14)de?nes a structure of U′q(?g)-module on V?C(u)i(L0)(resp.on i(L0)?C(u)V)which is isomorphic to V.
Proof:For i∈I,s∈Z we have x±i,s.i(L0)={0}and for i∈I,s>0we haveφ±i,±s.i(L0)={0}. For i∈I,the action ofφ±i,0on i(L0)is the identity.So for V?C(u)i(L0),the action given by formula(14)is g?1which makes sense for g∈U′q(?g)(direct computation on generators).For i(L0)?C(u)V the action given by formula(14)is1?g which makes sense for g∈U′q(?g).
Fix r,r′≥0.Let V1∈Mod r and V2∈Mod r′.If r,r′≥1,one de?nes an action of U′q(?g) on V1?C(u)V2by formula(14).From lemma3.10we get an object in Mod(U′q(?g))(this can be extended to the cases r=0or r′=0by lemma3.11).We consider this tensor product as an object in the(r+r′)-th summand of Mod.So we have de?ned a bifunctor?f:Mod×Mod→Mod. Theorem3.12.The bifunctor?f de?nes a tensor structure on Mod.
The category Mod together with the tensor product?f is called quantum fusion tensor category (see[Ma]for the de?nition and complements on tensor categories).The aim of this section is to prove this theorem.We warn that objects of the category Mod are not necessarily of?nite length.Sometimes it is required that the objects of a tensor category have a?nite composition series(for example see[CE]),so Mod is not a tensor category in this sense.However,we will prove in section5that for a large class of quantum a?nizations(including quantum a?ne algebras and most quantum toroidal algebras)the subcategory of?nite length modules is stable under the monoidal bifunctor?f,and so we get a tensor category in this sense.
Proof of theorem3.12.We already have proved the well-de?nedness in lemma3.10.We have to show the associativity and existence of a neutral element.This will be formulated in the following two lemmas.
14DA VID HERNANDEZ
Lemma3.13.?f is associative.
Proof:Let r1,r2,r3≥1,V1∈Mod r1,V2∈Mod r2,V3∈Mod r3.Let us prove that the identity de?nes an isomorphism between the modules V1?f(V2?f V3)and(V1?f V2)?f V3as objects of Mod r1+r2+r3.The action of g∈U′q(?g)is given in the?rst case by:(Id??u r2)??u r1and in the second case by:(?u r1?Id)??u r
1+r2
.But it follows from lemma3.4that these maps are equal. If one r i is equal to0,V1?f(V2?f V3)?(V1?f V2)?f V3is a direct consequence of lemma3.11. The pentagon axiom is clearly satis?ed as for the usual tensor category of vector spaces. Lemma3.14.i(L0)object of Mod0is a neutral object of(Mod,?f).
Proof:It is a direct consequence of lemma3.11(the triangle axiom is clearly satis?ed as for the usual tensor category of vector spaces).
3.3.
4.Application to(l,u)-highest weight simple modules of the category Mod(U′q(?g)).We use the notations of remark3.3.As an application of lemma3.10we prove a”deformed version”of the ”if”part of theorem2.9.
De?nition3.15.An(l,u)-weight(λ,Ψ(u))is said to be dominant if for i∈I there exists a
polynomial P i,u(z)=(1?za i,1u b i,1)...(1?za i,N
i u b i,N i)(N i≥0,a i,j∈C?,b i,j≥0)such that in
C[u±1][[z]](resp.in C[u±1][[z?1]]):
r≥0Ψ±i,±r(u)z±r=q N i i P i,u(zq?1i)
DRINFELD COPRODUCT,QUANTUM FUSION TENSOR CATEGORY AND APPLICATIONS15 where0≤j≤r and we denote v?1=v r+1=0.In particular we have as a power series in z(resp.
in z?1):
φ±(z).v j=q r?2j
(1?zaq?r)(1?zaq r+2)
(1?zaq r?2j+2)(1?zaq r?2j)(1?zubq r′?2k+2)(1?zubq r?2k)
(v j?v′k), whereαj,k=[r?j+1]q,γj,k=[k+1]q,andβj,k is equal to
[r′?k+1]q(q2j?r+(q?1?q)ua?1bq r′?2k+1?r+2j(q[r?j]q[j+1]q
1?ua?1bq r′?2k?r+2j
)),
andμj,k is equal to
[j+1]q(q r′?2k+(q?q?1)ua?1bq r′?2k+1?r+2j(q?1[r′?k]q[k+1]q
1?ua?1bq r′?2k?r+2j+2
)).
Note thatαj,k,βj,k,γj,k,μj,k are independent of m.
(Observe that on this example the action is rational,as proved in lemma3.10).
Remark3.17.For the usual tensor product of quantum a?ne algebras,certain tensor products of l-highest weight modules are l-highest weight(see[Cha,Theorem4]and[Kas2,Theorem9.1]) but not all.For example for U q(?sl2),L a?L aq2is l-highest weight but L aq2?L a is not l-highest weight(where the L b are the fundamental representations de?ned in section2.4).
In contrast to this well-known situation,in the following examples the fusion modules are always of l-highest weight(we will see in theorem6.2that this observation is a particular case of a more general picture for the quantum fusion tensor category).
Proposition3.18.The fusion module V=i(W r(a))?f i(W r′(b))is a simple(l,u)-highest weight module.
Proof:First let us prove that V is of(l,u)-highest weight.Let W be the sub U′q(?g)-module of V generated by v0?v′0.It su?ces to prove by induction on K≥0that((j+k=K)?(v j?v′k∈W)). For K=0it is clear.Let K≥1and j,k≥0satisfying j+k=K?1.By de?nition,γj,k,μj,k are not equal to zero.As ubq r′?2k=aq r?2j,equation(16)implies that v j+1?v′k and v j?v′k+1are in r∈Z C(u)x?r.(v j?v′k)?W.So {(a,b)|a+b=K}C(v a?v′b)?W.
Let us prove that V is simple.Suppose that V′is a proper submodule of V.Suppose that the
eigenvalues given in equation(17)for v j
1?v′k
1
and v j
2
?v′k
2
are equal.Then we have
(1?zaq r?2j1+2)(1?zaq r?2j1)=(1?zaq r?2j2+2)(1?zaq r?2j2) and
(1?zubq r′?2k1+2)(1?zubq r?2k1)=(1?zubq r′?2k2+2)(1?zubq r?2k2).
16DA VID HERNANDEZ
So j1=j2and k1=k2as the conditions aq r?2j1=aq r?2j2+2and aq r?2j1+2=aq r?2j2(resp. aq r?2k1=aq r?2k2+2and aq r?2k1+2=aq r?2k2)can not be simultaneously satis?ed.So the op-eratorsφ±±m have a diagonalizable action on V with common l-weight spaces of dimension1.As V′is stable under the action of theφ±±m,it is of the form V′= (j,k)∈J C(u)(v j?v′k)where J?{0,...,r}×{0,...,r′}.Consider v j?v′k∈V′such that j+k is minimal for this property.If
V′=V,we have j+k>0and V′∩( {(s,t)|s+t 4.A-forms and specializations The aim of this section is to explain how to go back from Mod to the usual category Mod(U q(?g)). Consider the ring A={f(u) DRINFELD COPRODUCT,QUANTUM FUSION TENSOR CATEGORY AND APPLICATIONS17 Proof:1)and2):for allλ∈h?,a basis of?V∩Vλas an A-module is also a basis of the C(u)-vector space Vλ. 3)It su?ces to consider the case v∈Vλ.Let us write it vλ= k=1,···,K f i(u)w i where {w i}i=1,···,K is an A-basis of Vλ∩?V.Then n(v)is the unique n(v)∈Z satisfying:?i∈{1,...,K}, (1?u)n(v)f i(u)∈A and there exists i∈{1,...,K}such that(1?u)n(v)?1f i(u)/∈A. 4)Denote?W=W∩?V,and let us check the properties of de?nition4.1:i)is clear.For ii), let v∈W.There is P∈C[u±]?{0}such that v′=P v∈?V.So v′∈?W and?W generates the C(u)-vector space W.For the property iii),we have?W∩Wλ=Vλ∩W∩?V??V∩Vλis a?nitely generated A-module. We have directly from de?nition4.1: Lemma4.3.Let V∈Mod(U′q(?g))and?V?V(resp.?V′?V)be an A-form of U′q(?g).?V(resp.of U′q(?g).?V′).Then?V+?V′is an A-form of U′q(?g).(?V+?V′). 4.1.2.Specialization.One can consider the specialization of an A-form at u=1: De?nition4.4.Let V∈Mod(U′q(?g))and?V an A-form of V.Then we denote by(?V)u=1the U q(?g)-module?V/((u?1)?V). Lemma4.5.We have(?V)u=1∈Mod(U q(?g)). Proof:It follows from lemma4.2that dim C(((?V)u=1)λ)=rk A(?V∩Vλ)=dim C(u)(Vλ). So(?V)u=1∈Mod(U q(?g)). In general the specialization at u=1of two A-forms of an U′q(?g)-module are not necessarily isomorphic,as illustrated in the following examples. 4.1.3.Examples.Let L1=L1,1=C v0⊕C v1and L2=L1,q2=C w0⊕C w1be two U q(?sl2)-fundamental representations.The fusion modules V=i(L1)?f i(L2)and V′=i(L2)?f i(L1) can be described explicitly(see section3.4;they have also been studied in[H3,section6.6]with a di?erent formalism).Consider the C(u)-basis of V(resp.of V′):f0=v0?w0,f1=v1?w0,f2= v0?w1,f3=v1?w1(resp.f′0=w0?v0,f′1=w1?v0,f′2=w0?v1,f′3=w1?v1).The action of U′q(?sl2)on V is given by: f1f3 0q2r?1u r1?q4u f2+q1+2r u r1?u x?r 1?q2u f1q?11?q4u0 q2(1?q?2z)(1?uz)(1?q2z)(1?uz)(1?q?2z)(1?q4uz)q?2(1?q2z)(1?q4uz) f′0f′2 x+r q2r f′0 1?uq?2f′0 1?uq?2 f′1 u r f′2+q2r+11?q?4u u r f′3 1?q?2u f′3 φ±(z) (1?q2z)(1?uz)f′0 (1?q2z)(1?uz) f′1 (1?q2z)(1?uz) f′2 (1?q2z)(1?uz) f′3 . It follows from proposition3.18that V and V′are simple(l,u)-highest weight modules. 18DA VID HERNANDEZ In particular ?V =U u q (?g ).f 0=A .f 0⊕A (1?u ).f 1⊕A .f 2⊕A f 3, ?V ′=U u q (?g ).f ′0=A .f ′0⊕A f ′1 ⊕A .f ′2⊕A f ′3,are respectively A -forms of V and V ′.We use the same notations for the vectors in ?V and in (?V )u =1(resp.in ?V ′and in (?V ′)u =1).The U q (?sl 2)-modules (?V )u =1and (?V ′)u =1are l -highest weight modules,but not simple :C .(1?u )f 1(resp.C .f ′2)is a U q (?sl 2)-submodule of (?V )u =1(resp.of (?V ′)u =1)of dimension 1.Note that we have an isomorphism of U q (?sl 2)-modules σ:(?V )u =1→(?V ′)u =1de?ned by σ(f 0)=f ′0,σ((1?u )f 1)=(q ?1?q )f ′2,σ(f 2)=(q +q ?1)f ′1,σ(f 3)=f ′3. Consider the following respective A -forms of V and V ′:?V f = i =0,1,2,3U u q (?sl 2).f i =A f 0⊕A f 1⊕A f 2⊕A f 3 ?V , i =0,1,2,3 U u q (?sl 2).f ′i =A f ′0⊕A f ′1⊕A f ′2⊕A f ′3=?V ′.We use the same notation for the vectors in ?V f and in (?V f )u =1.(?V f )u =1is not an l -highest weight module,is cyclic generated by f 1and has a submodule of dimension 3,namely C f 0⊕C f 2⊕C f 3.So (?V f )u =1and (?V )u =1are not isomorphic.4.2.A -form of cyclic modules.In this section we study A -forms of cyclic modules which will be used later (in particular we will study the interestin g properties of their specializations in other sections).The main result of this section is : Theorem 4.6.Let V ∈Mod (U ′q (?g ))and v ∈V ?{0}.Then ?V (v )=U u q (?g ).v is an A -form of U ′q (?g ).v .Moreover (?V (v ))u =1∈Mod (U q (?g ))is a non zero cyclic U q (?g )-module generated by v . This theorem is proved in this section.We can suppose that V is a non zero cyclic U ′q (?g )-module generated by v . Lemma 4.7.Let V ∈Mod (U ′q (?g ))and F ?V be a ?nitely generated A -submodule of V .Then A .U q (?h ).F is a ?nitely generated A -module.Proof: Let W ?V as in de?nition 3.6and write F = j =1,...,m A .f j where f j ∈V .For each j ∈{1,...,m },we can write a ?nite sum f j = k Ψj,k f j,k where f j,k ∈W λj,k (λj,k ∈h ?)and Ψj,k ∈C (u ).But A .U q (?h )f j,k ?A .W λj,k ,and so U q (?h ).F ? j,k Ψj,k A .W λj,k is a ?nitely generated A -module. Let us study the case of (l,u )-highest weight modules,which is a ?rst step in the proof of theorem 4.6: Lemma 4.8.Let V be an (l,u )-highest weight module in the category Mod (U q (?g ))and let v be an highest weight vector.Then ?V =U u q (?g ).v is an A -form of V . Proof: Properties i),ii)of de?nition 4.1are clear.Let us prove property iii):let λbe the weight of v .For μa weight of V ,we have μ∈λ?Q +.Let us prove the result by induction on h ′(μ)=h (λ?μ).For h ′(μ)=0it is clear,and in general let us prove that Λl +1= {μ|h ′(μ)=l +1}?V ∩V μis a ?nitely generated A -module.But we have Λl +1= {i ∈I,m ∈Z ,μ|h ′(μ)=l }x ?i,m .(V μ∩?V ).It follows from formula (8)that for i ∈I,m =0,we have x ?i,m =?m DRINFELD COPRODUCT,QUANTUM FUSION TENSOR CATEGORY AND APPLICATIONS19 U q(?h).(Vμ∩?V)=Vμ∩?V.In particularΛl+1? {i∈I,μ|h′(μ)=l}U q(?h).x?i,0(Vμ∩?V).We can conclude with lemma4.7. Note that the rationality in u was a crucial point of this proof. (Note that in[H3]we considered C[u±]U q(?g).v for a simple(l,u)-highest weight module L and called it a C[u±]-form of L.As U u q(?g).v=A.C[u±]U q(?g).v,it is a particular case of the point of view of this paper.) Proof of theorem4.6: Let us prove that we get an A-form:properties i),ii)of de?nition4.1are clear.Let us check property iii).First consider d(v)=Max{h(wt(v′)?wt(v))|v′∈V and wt(v′)?wt(v)∈Q+}. Let us prove by induction on d(v)≥0,that for allλ∈h?,U u q(?g).v∩(U′q(?g).v)λis a?nitely generated A-module.For d(v)=0,the result is proved as in lemma4.8.In general:the (?g)U u q(?h).v triangular decomposition of U′q(?g)(theorem2.3)gives U u q(?g).v=A+B where A=U u,? q and B= i∈I,m∈Z U u q(?g).x+i,m.v.Moreover forλ∈h?,(U u q(?g).v)∩Vλ=A∩Vλ+B∩Vλ.We see as in the proof of lemma4.8that A∩Vλis a?nitely generated A-module.It follows from formula(8)that for i∈I,m=0,we have x+i,m=m 20DA VID HERNANDEZ De?nition4.10.The module(?W(v))u=1is denoted by V1?f V2?f···?f V r and is called the fusion module of V1,V2,···,V r. Examples:in section4.1.3,for the U q(?sl2)-l-highest weight modules L1,L2we computed explicitly L1?f L2=(?V)u=1and L2?f L1=(?V′)u=1and we de?ned an isomorphismσ:L1?f L2?L2?f L1. Moreover we noticed that L1?f L2is not semi-simple. Other examples and applications will be studied in the section6. 4.4.Fusion of?nite dimensional representations.We use the notations of remark3.3.In this subsection we study another application of the quantum fusion tensor category:it allows to de?ne a bifunctor on the category of?nite dimensional representations of U q(?g).This bifunctor will not be used in the rest of this paper,we hope to study it in more details in another paper. We recall that i is a functor from Mod(U q(?g))to Mod(U′q(?g))(see proposition3.8). Corollary4.11.Let V1,V2be two?nite dimensional representations of U q(?g).Then U u q(?g).(V1?C V2)?i(V1)?i(V2) is an A-form of i(V1)?i(V2). Proof:Let{vα}1≤α≤p,{wβ}1≤β≤p′be C-basis respectively of V1and V2.We have (18)U u q(?g).(V1?C V2)= 1≤α≤p,1≤β≤p′(U u q(?g).(vα?wβ))?(V1?C V2)?C C(u). From theorem4.6,each U u q(?g).(vα?wβ)is an A-form of U′q(?g).(vα?wβ).As the sum in equation (18)is?nite,we can conclude with lemma4.3. Because of corollary4.11,it makes sense to de?ne: De?nition4.12.We denote by V1?d V2the U q(?g)-module(U u q(?g).(V1?C V2))u=1. Examples:in section4.1.3,for the U q(?sl2)l-highest weight modules L1and L2we computed explicitly L1?d L2=(?V f)u=1and L2?d L1=(?V′f)u=1.Note that L1?d L2is not isomorphic to L2?d L1,and that L1?d L2is not isomorphic to L1?f L2.Note that in general if V1and V2are semi-simple,V1?d V2is not necessarily semi-simple. Let Modf(U q(?g))be the subcategory of?nite dimensional representations in Mod(U q(?g)).If U q(?g)is a quantum a?ne algebra,the simple integrable l-highest weight modules are objects of Mod(U q(?g)) (see[CP3]). Theorem4.13.?d de?nes a bifunctor?d:Modf(U q(?g))×Modf(U q(?g))→Modf(U q(?g)). Proof:As i(V1)?f i(V2)is a?nite dimensional C(u)-vector space,V1?d V2is a?nite dimensional C-vector space,and so necessarily is an object of Modf(U q(?g)).Consider V1,V2,V′1,V′2objects of Modf(U q(?g))and f1:V1→V′1,f2:V2→V′2two morphisms of U q(?g)-module.From theorem 3.12we have a morphism of U′q(?g)-module f1?f f2:i(V1)?f i(V2)→i(V′1)?f i(V′2).As(f1?f f2)(U u q.(V1?C V2))?U u q.(V′1?C V′2)and(f1?f f2)((1?u)U u q.(V1?C V2))?(1?u)U u q.(V′1?C V′2), we get a morphism f1?d f2:V1?d V2→V′1?d V′2. For quantum a?ne algebras,it should be interesting to relate?d to the usual tensor category structure.