Regular representations of the quantum groups at roots of unity
aa r X i v : . [ m a t h . QA ] D ec REGULAR REPRESENTATIONS OF THE QUANTUM GROUPSAT ROOTS OF UNITY
MINXIAN ZHU
Abstract.
We study the bimodule structure of the quantum function algebra atroots of 1 and prove that it admits an increasing filtration with factors isomorphicto the tensor products of the dual of Weyl modules V ∗ λ ⊗ V ∗− ω λ . As an applicationwe compute the 0-th Hochschild cohomology of the function algebra at roots of 1. Introduction
Let g be a simple complex Lie algebra, and let U be the quantized envelopingalgebra of g over Q ( v ), v an indeterminate. Let O be the linear span of matrixcoefficients of finite-dimensional U -modules (see [D]), then there is a perfect Hopfalgebra pairing between U and O . Moreover, as a U ⊗ U -module, O has a classicalPeter-Weyl type of decomposition. Following [L3], set A = Z [ v, v − ] and let U denote Lusztig’s A -form of U generated by the divided powers. Set A = Q [ v, v − ]and U A = U ⊗ A A . Let U ∗ A be the set of all A -linear maps U A → A , and set O = O ∩ U ∗ A . Then O is a Hopf algebra over A and the inclusion O ⊂ O induces anisomorphism of Q ( v )-algebras: O ⊗ A Q ( v ) ∼ = O . Now let q ∈ C be a primitive ℓ -throot of unity; set U q = U A ⊗ A Q ( q ) and O q = O ⊗ A Q ( q ), where Q ( q ) is made intoan A -algebra by specializing v to q . There is a Hopf algebra pairing between U q and O q , thus O q admits the structure of a U q × U q -module. The main goal of the presentpaper is to investigate this bimodule structure.Our motivation comes from a family of vertex operator algebras associated to themodified regular representations of the affine Lie algebra ˆ g (see [Z1] and referencestherein). Each one of these vertex operator algebras admits two commuting actionsof ˆ g in dual levels. When the dual central charges are generic, it decomposes intosummands corresponding to the dominant weights of g . However when the dualcentral charges are rational, the module structure is less well understood, and it shouldbe closely related to the regular representation of the corresponding quantum groupat a root of unity, by the equivalence of tensor categories established by Kazhdan andLusztig between representations of the affine Lie algebra and representations of thequantum group (see [KL1-4]).For simplicity, we assume g = sl n is of type A , and ℓ ≥ n is odd. The quantumcoordinate algebra O of sl n can be described by generators and relations. Let V be thequantization of the n -dimensinal natural representation of sl n , then O is generated bythe matrix coefficients X ij , ≤ i, j ≤ n of V subject to a list of relations (see [D], [T],[APW]). In Section 2, we will show that O , as an A -subalgebra of O , is generated by X ij ’s over A , which generalizes Proposition 1.3 of [CL]. A similar result was obtainedin the Appendix of [APW] by P. Polo, using some local ring as the basic ring. Another A -form of U , denoted by Γ( g ) and slightly different from U A , was introduced in [CL]. The dual of Γ( g ), appropriately defined, coincides with O . Specializing v to q ,a primitive ℓ -th root of 1, we get a perfect pairing between Γ q ( g ) = Γ( g ) ⊗ A Q ( q )and O q = O ⊗ A Q ( q ), hence it induces an embedding O q ֒ → Γ q ( g ) ∗ ([CL, Lemma6.1]). When U q = U A ⊗ A Q ( q ) and O q are concerned, we still have the embedding O q ֒ → U ∗ q , though the pairing between U q and O q is in general degenerate on the U q -side.Consider the non-semisimple category C f of finite dimensional U q -modules (of type ). The quantum function algebra O q can be realized as the linear span of matrixcoefficients of modules from C f . One of our main results is that O q , as a U q × U q -module, admits an increasing filtration with factors isomorphic to the tensor productsof the dual of Weyl modules V ∗ λ ⊗ V ∗− ω λ . It can be regarded as a generalization ofthe Peter-Weyl type of decomposition for O q , q not a root of unity. Similar resultsfor the regular representation of the affine Lie algebra in a rational level provide aproof of the conjecture, stated at the end of [Z1], about the bimodule structure of afamily of vertex operator algebras in rational levels (see [Z2]). As an application ofthis increasing filtration of O q , we show that the cocommutative elements of O q arelinear combinations of the “traces” of modules from C f , moreover as an algebra, it isisomorphic to the Grothendieck ring of C f extended to the field Q ( q ).The paper is organized as follows: Section 2 gives an overview of the quantizedenveloping algebra U , the quantum coordinate algebra O , Lusztig’s A -form U of U ,the corresponding A -form O of O , and their specializations U q , O q at a root of unity.In particular we describe O for type A (Proposition 2.1), and identify O q with thematrix coefficients of finite dimensional U q -modules (Proposition 2.4). In Section 3,we describe an increasing filtration of O q using tilting modules (Theorem 3.3), andcompute the 0-th Hochschild cohomology of O q as a coalgebra (Proposition 3.5). Wetreat the sl case more thoroughly in Section 4 and are able to obtain more explicitresults (Theorem 4.6).I am very grateful to my advisor Igor Frenkel for his guidance, and CatharinaStroppel for helpful discussions.2. The General Setting
Let ( a ij ) ≤ i,j ≤ n − be the Cartan matrix of a simply-laced simple Lie algebra g .The quantized enveloping algebra U is the Q ( v )-algebra defined by the generators E i , F i , K i , K − i (1 ≤ i ≤ n −
1) and the relations K i K − i = K − i K i = 1 , K i K j = K j K i ,K i E j = v a ij E j K i , K i F j = v − a ij F j K i ,E i F j − F j E i = δ ij K i − K − i v − v − ,E i E j = E j E i , F i F j = F j F i , if a ij = 0 ,E i E j − ( v + v − ) E i E j E i + E j E i = 0 , if a ij = − ,F i F j − ( v + v − ) F i F j F i + F j F i = 0 , if a ij = − . U is a Hopf algebra over Q ( v ) with comultiplication △ , counit ε and antipode S defined by △ ( E i ) = E i ⊗ K i ⊗ E i , △ ( F i ) = F i ⊗ K − i + 1 ⊗ F i , △ ( K i ) = K i ⊗ K i , EGULAR REPRESENTATIONS OF THE QUANTUM GROUPS AT ROOTS OF UNITY 3 ε ( E i ) = ε ( F i ) = 0 , ε ( K i ) = 1 ,S ( E i ) = − K − i E i , S ( F i ) = − F i K i , S ( K i ) = K − i . Following [L3, Sect 7], let F be the set of all two-sided ideals I in U such that I has finite codimension and there exists some r ∈ N such that for any i we have Q rh = − r ( K i − v h ) ∈ I . Let O be the set of all Q ( v )-linear maps f : U → Q ( v ) suchthat f | I = 0 for some I ∈ F . We call O the quantum coordinate (or function) algebra,which is equivalent to the linear span of matrix coefficients of finite dimensional U -modules with a weight decomposition. Moreover O is a Hopf algebra over Q ( v ),and there exists a perfect Hopf algebra pairing U × O → Q ( v ). Since all finitedimensional U -modules are completely reducible, O has a classical Peter-Weyl typeof decomposition as a U × U -module.For type A , we can describe O by generators and relations. Set g = sl n , andlet ( a ij ) ≤ i,j ≤ n − be the Cartan matrix with a ij = − | i − j | = 1; 2 if i = j ; 0otherwise. Let α , · · · , α n − be the simple roots of sl n associated to ( a ij ) ≤ i,j ≤ n − ,and let ω , · · · , ω n − be the corresponding fundamental weights. Let V denote thequantization of the n -dimensional natural representation of sl n with highest weight ω . Fix a highest weight vector x ∈ V , and set x i +1 = F i x i for all 1 ≤ i ≤ n − x i has weight ω i − ω i − (with the convention ω = ω n = 0), and { x , · · · , x n } form a Q ( v )-basis of V with E i x i +1 = x i . Let { δ , · · · , δ n } be the dual basis in V ∗ ,and define X ij ∈ U ∗ by X ij ( u ) = δ i ( u · x j ) for any u ∈ U . These functionals X ij belong to O and they satisfy the following relations: X il X jl − vX jl X il = 0 for all l, i < jX li X lj − vX lj X li = 0 for all l, i < jX li X mj − X mj X li = 0 if l < m and i > jX li X mj − X mj X li − ( v − v − ) X lj X mi = 0 if l < m and i < j X σ ∈ S n ( − v ) l ( σ ) X σ (1)1 · · · X σ ( n ) n = 1where l ( σ ) denotes the length of the permutation. In fact O is generated by X ij , ≤ i, j ≤ n subject to the above relations (see [D], [T]).Set A = Z [ v, v − ]. Given n ∈ Z , m ∈ N , we define [ n ] = v n − v − n v − v − ∈ A , [ m ]! =[ m ][ m − · · · [1] and (cid:20) nm (cid:21) = Q mj =1 v n − j +1 − v − n + j − v j − v − j ∈ A . Following [L1, L3], let U bethe A -subalgebra of U generated by the elements E ( N ) i = E Ni / [ N ]!, F ( N ) i = F Ni / [ N ]!, K i , K − i (1 ≤ i ≤ n − , N ≥ U is a free A -module and is itself a Hopfalgebra over A in a natural way. Let U + , U − , U be the A -subalgebras of U generatedby the elements E ( N ) i ; F ( N ) i ; K ± i , (cid:20) K i ; ct (cid:21) , where (cid:20) K i ; ct (cid:21) = t Y s =1 K i v c − s +1 − K − i v − c + s − v s − v − s , then multiplication induces an isomorphism of A -modules: U − ⊗ U ⊗ U + ∼ = U .We follow [L1] to construct an A -basis of U . Let s i , ≤ i ≤ n − W = S n of sl n . Let R , R + denote the root system andpositive roots. Set α ij = s j s j − · · · s i +1 α i = P jk = i α k ∈ R + for any 1 ≤ i < j ≤ n − MINXIAN ZHU and consider the following total order on R + : α n − < α n − ,n − < · · · < α ,n − <α n − < α n − ,n − < · · · < α ,n − < · · · < α < α < α . Let Ω : U → U opp bethe Q -algebra isomorphism and T i : U → U , 1 ≤ i ≤ n − Q ( v )-algebraisomorphism defined in [L1, Sect 1]. Set E ij = T j T j − · · · T i +1 E i , and define for any φ, φ ′ ∈ N R + , E φ = Y β ∈ R + E ( φ ( β )) β , F φ ′ = Ω( E φ ′ ) , where E α i = E i , E α ij = E ij , E ( N ) β = E Nβ / [ N ]! and the factors in E φ are written in thegiven order of R + . Then the elements E φ ; F φ ′ ; Q n − i =1 K δ i i (cid:20) K i ; 0 t i (cid:21) , t i ≥ , δ i = 0 or 1form an A -basis of U + ; U − ; U respectively. Hence the elements F φ ′ KE φ , with K in the above A -basis of U , form an A -basis of U . They also form a Q ( v )-basis of U , hence we have an isomorphism of Q ( v )-algebras: U ⊗ A Q ( v ) ∼ = U .Again following [L3, Sect 7], set A = Q [ v, v − ] and U A = U ⊗ A A . Let U ∗ A bethe set of all A -linear maps U A → A and let O = O ∩ U ∗ A . Then O is a Hopfalgebra over A , and the inclusion O ֒ → O induces an isomorphism of Hopf Q ( v )-algebras: O ⊗ A Q ( v ) ∼ = O . Let M be a U A -module, which is a free A -module offinite rank with a basis in which the operators K i , (cid:20) K i ; 0 t i (cid:21) act by diagonal matriceswith eigenvalues v m , (cid:20) mt (cid:21) . For any m ∈ M and ξ ∈ Hom A ( M , A ), the matrixcoefficient c m,ξ : u → ξ ( u · m ), an element of U ∗ A , belongs to O . Moreover O is exactlythe A -submodule of U ∗ A spanned by the matrix coefficients c m,ξ for various M , m, ξ as above.Another integral form of U , denoted by Γ( g ), was introduced in [CL]. By defini-tion Γ( g ) is the A -subalgebra of U generated by E ( N ) i , F ( N ) i , K ± i , (cid:18) K i ; ct (cid:19) , where (cid:18) K i ; ct (cid:19) = Q ts =1 K i v c − s +1 − v s − . This algebra is larger than U A because its Cartanpart Γ( t ) is larger than that of U A . The elements Q n − i =1 (cid:18) K i ; 0 t i (cid:19) K − [ t i / i ( t i ≥ A -basis of Γ( t ), where the Gauss symbol [ x ] denotes the largest integer thatis not greater than x . Let C be the full subcategory of Γ( g )-modules, which is free offinite rank as an A -module and has a basis in which the operators K i , (cid:18) K i ; 0 t (cid:19) actby diagonal matrices with eigenvalues v m , (cid:18) mt (cid:19) , where (cid:18) mt (cid:19) = Q ts =1 v m − s +1 − v s − .Then the dual of Γ( g ), defined to be the linear span of matrix coefficients of modulesfrom C , coincides with O ([CL, Remark 4.1]).Recall that for g = sl n , the quantum coordinate algebra O is generated by X ij , ≤ i, j ≤ n over Q ( v ) subject to some relations. We want to show that its subalgebra O is generated by X ij , ≤ i, j ≤ n over A . It generalizes Proposition 1.3 of [CL], andthe proof written below is pure calculation. Proposition 2.1.
The A -subalgebra O of O is generated by X ij , ≤ i, j ≤ n . EGULAR REPRESENTATIONS OF THE QUANTUM GROUPS AT ROOTS OF UNITY 5
Proof.
Let Ξ be the set of all matrices M = ( r ij ) ≤ i,j ≤ n such that r ij ∈ N and at leastone of r , · · · , r nn is zero. Fix a total order on { , · · · , n } , and set X M = Q ij X r ij ij for any M ∈ Ξ. Then { X M , M ∈ Ξ } form a Q ( v )-basis of O . Any element f ∈ O canbe represented uniquely as f = P M ∈ Ξ γ M X M with γ M ∈ Q ( v ). Since X ij ∈ O forall 1 ≤ i, j ≤ n , it suffices to prove that one of the nonzero coefficients γ M belongs to A . Define a set of nonnegative integers inductively as follows: s n = min { r n | γ M =( r ij ) = 0 } s n − , = min { r n − , | γ M =( r ij ) = 0 , r n = s n }· · · s = min { r | γ M =( r ij ) = 0 , r i = s i , < i ≤ n } s n = min { r n | γ M =( r ij ) = 0 , r i = s i , < i ≤ n }· · · s = min { r | γ M =( r ij ) = 0 , r j = s j , r i = s i , < j ≤ n, < i ≤ n }· · · s n,n − = min { r n,n − | γ M =( r ij ) = 0 , r ij = s ij , ≤ j < n − , j < i ≤ n } . Define φ ∈ N R + ; α i s i +1 ,i , α ij s j +1 ,i , then f ( F φ KE ) = P M ∈ Λ γ M X M ( F φ KE )for any K ∈ U , E ∈ U + , where Λ = { M = ( r ij ) ∈ Ξ | γ M = 0 , r ij = s ij , ≤ j < i ≤ n } . Similarly define another set of nonnegative integers: s n = min { r n | M = ( r ij ) ∈ Λ }· · · s = min { r | M = ( r ij ) ∈ Λ , r i = s i , < i ≤ n } s n = min { r n | M = ( r ij ) ∈ Λ , r i = s i , < i ≤ n }· · · s = min { r | M = ( r ij ) ∈ Λ , r j = s j , r i = s i , < j ≤ n, < i ≤ n }· · · s n − ,n = min { r n − ,n | M = ( r ij ) ∈ Λ , r ij = s ij , ≤ i < n − , i < j ≤ n } . Define ψ ∈ N R + ; α i s i,i +1 , α ij s i,j +1 , then f ( F φ KE ψ ) = P M ∈ Υ γ M X M ( F φ KE ψ )for any K ∈ U , where Υ = { M = ( r ij ) ∈ Λ | r ij = s ij , ≤ i < j ≤ n } . In other words,Υ is the collection of matrices M ∈ Ξ whose non-diagonal entries are s ij ’s and the co-efficient of X M in f ∈ O is nonzero, moreover f ( F φ KE ψ ) = P M ∈ Υ γ M v n M χ λ + µ M ( K )for some n M ∈ Z , where λ = P i>j s ij ( ω j − ω j − ) + P i MINXIAN ZHU that χ λ + µ M ( K ′ ) = 1 if M = M ; 0 otherwise. Hence f ( F φ K ′ E ψ ) = γ M v n M ∈ A ,hence γ M ∈ A . (cid:3) Let ℓ ≥ n (the Coxeter number of sl n ) be an odd integer, and let q be a prim-itive ℓ -th root of 1. Let p ℓ ( v ) denote the ℓ -th cyclotomic polynomial, then wehave an isomorphism of fields A / ( p ℓ ( v )) ∼ = Q ( q ). Set U q = U A ⊗ A Q ( q ) and O q = O ⊗ A Q ( q ). They are both Hopf algebras over Q ( q ) and inherit the comultipli-cations, counits and antipodes from U A and O respectively. We denote the imagesof E ( N ) i , F ( N ) i , K ± i , (cid:20) K i ; ct (cid:21) ∈ U A in U q by the same notations. Proposition 2.2. There is a pairing of Hopf algebras ( , ) : U q × O q → Q ( q ) , and itinduces an embedding O q ֒ → U ∗ q .Proof. This is basically Lemma 6.1 of [CL], where the specialization of the larger A -algebra Γ( g ), i.e. Γ q = Γ( g ) ⊗ A Q ( q ), is considered, and the pairing between Γ q and O q is non-degenerate. However if instead we consider the pairing between U q and O q , it is only non-degenerate on the O q -half, i.e. ( u, f ) = 0 for all u ∈ U q impliesthat f = 0 ∈ O q . To see why it is degenerate on the U q -half, consider the image of K ℓi − ∈ U A in U q , denoted by the same notation. It is not difficult to see that( K ℓi − , f ) = 0 for all f ∈ O q , but K ℓi = 1 in U q (instead K ℓi = 1 in U q ). Theinjectivity of the induced map O q → U ∗ q can also be proved using the arguments ofProposition 2.1. (cid:3) The dual space U ∗ q admits two commuting (left) actions of U q , which we denoteby ρ , ρ . By definiton, ρ ( u ) f ( u ′ ) = f ( u ′ u ) and ρ ( u ) f ( u ′ ) = f ( S ( u ) u ′ ) for any f ∈ U ∗ q , u, u ′ ∈ U q , where S denotes the antipode of U q . The Hopf algebra O q is a U q × U q -submodule of U ∗ q , and the U q -actions can be expressed as follows: ρ ( u ) g = X g (1) ( u, g (2) ) , ρ ( u ) g = X ( S ( u ) , g (1) ) g (2) , for any u ∈ U q , g ∈ O q , where △ ( g ) = P g (1) ⊗ g (2) . The question we want toinvestigate is how O q decomposes as a U q × U q -module, i.e. as a bicomodule of itself.Let V be a finite dimensional representation of U q , and let V ∗ be the dual represen-tation defined by uf ( v ) = f ( S ( u ) v ) for any u ∈ U q , f ∈ V ∗ , v ∈ V . It is obvious thatthe map φ V : V ⊗ V ∗ → U ∗ q ; v ⊗ f f ( · v ) is a U q × U q -morphism. We denote theimage of φ V by M ( V ), called the matrix coefficients of V . Usually φ V is not injectiveunless V is irreducible. Lemma 2.3. Let V, V ′ be finite-dimensional U q -modules and U ⊂ V be a submodule,then we have the following: (1) φ V ( U ⊗ V ∗ ) = M ( U ) and φ V ( V ⊗ ( V /U ) ∗ ) = M ( V /U ) . In particular M ( U ) , M ( V /U ) ⊂ M ( V ) . (2) M ( V ⊗ V ′ ) = M ( V ) · M ( V ′ ) and M ( V ⊕ V ′ ) = M ( V ) + M ( V ′ ) . (3) M ( V ∗ ) = S ( M ( V )) .Proof. (1) is easy to prove: in terms of matrix representations, M ( U ) and M ( V /U )are the matrix coefficients in the diagonal blocks. Also note that the multiplicationand the map S of U ∗ q are defined by taking the transposes of the comultiplication andthe antipode of U q . Hence (2) and (3) follow. (cid:3) EGULAR REPRESENTATIONS OF THE QUANTUM GROUPS AT ROOTS OF UNITY 7 We have a triangular decomposition U q = U − q U q U + q , where U − q = U − ⊗ A Q ( q )and similar definitions for U q and U + q . Denote by X, X + the weight lattice and thedominant weights of g . For λ ∈ X , the character χ λ : U → A induces a characterof U q to Q ( q ): K ± i q ±h λ,α ∨ i i , (cid:20) K i ; ct (cid:21) (cid:20) h λ, α ∨ i i + ct (cid:21) q , where the subscript q means evaluating an element of Z [ v, v − ] at v = q . Let C f be the category of finitedimensional U q -modules with a weight decomposition with respect to U q . We willshow that O q ⊂ U ∗ q is precisely the linear span of matrix coefficients of modules from C f . To prove it, let’s first recall some important modules in C f . For any dominantweight λ ∈ X + , we can associate four canonical modules: the Weyl module V λ , thedual of the Weyl module V ∗ λ , the irreducible module L λ and the tilting module T λ .The Weyl module V λ is generated by a vector of highest weight λ , and has theuniversal property that any module in C f generated by a vector of highest weight λ is a quotient of V λ . The character of V λ is given by Weyl’s character formula(see [APW, A2] for definitions of V λ and V ∗ λ in terms of some induction functor andits derived functors). We have the following property for these standard objects:Ext i C f ( V λ , V ∗ µ ) = Q ( q ) if i = 0 and λ = − ω µ ; 0 otherwise, where ω is the longestelement in the Weyl group.The irreducible module L λ is the head of V λ as well as the socle of V ∗− ω λ , and L ∗ λ ∼ = L − ω λ . Furthermore the modules L λ , λ ∈ X + give a complete list of non-isomorphic irreducible modules in C f .A module in C f is called tilting if it admits both a Weyl filtration and a dual Weylfiltration. Tilting modules are closed by taking the dual and the tensor product.For each λ ∈ X + , there exists a unique (up to isomorphism) indecomposable tiltingmodule T λ such that T λ admits a Weyl filtration starting with V λ ֒ → T λ , and anyother Weyl modules V µ entering the Weyl filtration of T λ satisfy that µ < λ , here ≤ is the usual partial order on X determined by a set of positive roots. The highestweight λ occurs with multiplicity 1 in T λ . By consideration of characters, we have T ∗ λ ∼ = T − ω λ . Moreover the modules T λ , λ ∈ X + form a complete list of inequivalentindecomposable tilting modules. It is easy to see that T λ = V λ if and only if V λ isirreducible. There are enough projectives in C f and all projective modules are tilting(see [APW], [A2]). Proposition 2.4. O q ⊂ U ∗ q is the linear span of matrix coefficients of modules from C f , i.e. O q = P V ∈ C f M ( V ) .Proof. By Proposition 2.1, the Q ( q )-algebra O q is generated by X ij , the matrix coef-ficients of V ω . By Lemma 2.3 (2), we have O q = P n M ( V ⊗ nω ) ⊂ P V ∈ C f M ( V ). Toprove the inverse inclusion, it suffices to show that O q contains the matrix coefficientsof all the tilting modules, since all projective modules are tilting.Note that the Weyl modules V ω i , i = 1 , · · · , n − V ω i lie in the W -orbit of ω i with multiplicity 1, hence we must have L ω i = V ω i = T ω i . It is well known that thefundamental representation of sl ( n, C ) with highest weight ω i can be realized as the i -th exterior power of the n -dimensional natural representation, in particular it is adirect summand of the i -th tensor power. Therefore by consideration of the characters, V ω i is either a direct summand of V ⊗ iω or a composition factor of a direct summand of MINXIAN ZHU V ⊗ iω (which is tilting). Either way, we have M ( T ω i ) ⊂ M ( V ω ) i by Lemma 2.3. Nowfor any λ ∈ X + with m i = h λ, α ∨ i i , the tilting module T λ must be a direct summand of T ⊗ m ω ⊗· · ·⊗ T ⊗ m n − ω n − , hence M ( T λ ) ⊂ M ( T ω ) m · · · M ( T ω n − ) m n − ⊂ P n M ( V ω ) n . (cid:3) an increasing filtration of O q Fix g = sl n , and q to be a primitive ℓ -th root of 1, where ℓ ≥ n is odd. Itfollows from Proposition 2.1 that the quantum coordinate algebra O q is generated by X ij , ≤ i, j ≤ n over Q ( q ) subject to a list of relations. Proposition 2.4 identifies O q with the linear span of matrix coefficients of finite dimensional U q -modules. In thissection, we will describe a canonical increasing filtration of O q as a U q × U q -module.Let R, R + , X, X + , W denote the root system, positve roots, weight lattice, dom-inant weights and Weyl group of g . The affine Weyl group W ℓ is generated by theaffine reflections s β,m , β ∈ R + , m ∈ Z given by s β,m · λ = s β · λ + mℓβ, λ ∈ X. Here s β is the reflection corresponding to the positive root β , and we are using thedot-action defined by s β · λ = s β ( λ + ρ ) − ρ , where ρ is the half sum of the positiveroots.Denote by C the first dominant alcove, i.e. C = { λ ∈ X + | h λ + ρ, β ∨ i < ℓ for all β ∈ R + } , and set ¯ C = { λ ∈ X | ≤ h λ + ρ, β ∨ i ≤ ℓ for all β ∈ R + } , then ¯ C is a fundamental domain for the action of W ℓ on X .The linkage principal (see [A1]) allows us to decompose any module from C f intosummands corresponding to the representatives in ¯ C , therefore it yields a decompo-sition of O q as well. Proposition 3.1. As a U q × U q -module, we have O q ∼ = L λ ∈ ( ℓ − ρ + ℓX + V λ ⊗ V ∗ λ ⊕ ( L µ ∈ ¯ C \{ ( ℓ − ρ + ℓX } Λ µ ) , where Λ µ = P ν ∈W ℓ · µ ∩ X + M ( T ν ) .Proof. Recall that O q is spanned by the matrix coefficients of (tilting) modules from C f . The linkage principal implies that O q = ⊕ µ ∈ ¯ C Λ µ , where Λ µ = P ν ∈W ℓ · µ ∩ X + M ( T ν ).The vertices of the simplex ¯ C are − ρ, ℓω i − ρ, i = 1 , · · · , n − 1, where ω i ’s are thefundamental weights of g . The W ℓ -orbits of these vertices consist of weights of theform ( ℓ − ρ + ℓX . By [APW, Corollary 7.6], if λ ∈ ( ℓ − ρ + ℓX + , the Weyl module V λ is irreducible, in which case V λ = T λ and M ( T λ ) ∼ = V λ ⊗ V ∗ λ . (cid:3) Lemma 3.2. Let V, V ′ ∈ C f . (1) Suppose V admits a Weyl filtration V ⊂ V ⊂ · · · ⊂ V m = V such that V i /V i − ∼ = V λ i for some λ i ∈ X + , then M ( V ) ⊂ P i M ( T λ i ) . (2) Suppose V ′ admits a dual Weyl filtration with factors isomorphic to V ∗ µ i forsome µ i ∈ X + , then M ( V ′ ) ⊂ P i M ( T ∗ µ i ) .Proof. Let f i be the composition of V i ։ V i /V i − ∼ = V λ i ֒ → T λ i . Apply Hom C f ( − , T λ i )to the short exact sequence 0 → V i → V → V /V i → 0, we get Hom C f ( V, T λ i ) → Hom C f ( V i , T λ i ) → Ext C f ( V /V i , T λ i ) which is exact. Since V /V i and T λ i admit a Weylfiltration and a dual Weyl filtration respectively, it follows that Ext C f ( V /V i , T λ i ) = 0, EGULAR REPRESENTATIONS OF THE QUANTUM GROUPS AT ROOTS OF UNITY 9 hence Hom C f ( V, T λ i ) → Hom C f ( V i , T λ i ) is surjective. Let g i : V → T λ i be a preim-age of f i , then V i ∩ Ker g i = V i − . Define g = P g i : V → ⊕ i T λ i , then Ker g = ∩ i Ker g i = 0, i.e. g is injective. Hence by Lemma 2.3 we have M ( V ) ⊂ P i M ( T λ i ).Analogously we can prove (2) by constructing a surjective map ⊕ i T ∗ µ i → V ′ , but wecan also argue as follows: by assumption, V ′∗ admits a Weyl filtration with factorsisomorphic to V µ i , hence by (1) we have M ( V ′∗ ) ⊂ P M ( T µ i ), hence it follows fromLemma 2.3 (3) that M ( V ′ ) = S ( M ( V ′∗ )) ⊂ P i S ( M ( T µ i )) = P i M ( T ∗ µ i ). (cid:3) Theorem 3.3. Let µ ∈ ¯ C \ { ( ℓ − ρ + ℓX } , and write W ℓ · µ ∩ X + = { ν i , i ≥ } sothat ν i ≤ ν j implies i ≤ j . Set P i = P j ≤ i M ( T ν j ) , then P ⊂ · · · ⊂ P i − ⊂ P i ⊂ · · · is an increasing filtration of U q × U q -submodules of Λ µ with subquotients P i /P i − ∼ = V ∗− ω ν i ⊗ V ∗ ν i as a U q × U q -module.Proof. Since the dual Weyl filtration of T ν i ends with T ν i ։ V ∗− ω ν i , there exists asubmodule W ⊂ T ν i such that T ν i /W ∼ = V ∗− ω ν i , and W admits a filtration withfactors isomorphic to V ∗ γ ’s with − ω γ < ν i and − ω γ ∈ W ℓ · ν i , i.e. − ω γ = ν j for some j < i . Hence we have φ T νi ( W ⊗ T ∗ ν i ) = M ( W ) ⊂ P i − by Lemma 2.3 andLemma 3.2; analogously we also have φ T νi ( T ν i ⊗ ( T ν i /V ν i ) ∗ ) = M ( T ν i /V ν i ) ⊂ P i − . Set N = W ⊗ T ∗ ν i + T ν i ⊗ ( T ν i /V ν i ) ∗ , then φ T νi induces a surjective map ψ : ( T ν i ⊗ T ∗ ν i ) /N ։ M ( T ν i ) / ( M ( T ν i ) ∩ P i − ) = P i /P i − . Note that ( T ν i ⊗ T ∗ ν i ) /N ∼ = V ∗− ω ν i ⊗ V ∗ ν i , the socleof which is L ν i ⊗ L ∗ ν i . Since ψ ( L ν i ⊗ L ∗ ν i ) = ( M ( L ν i ) + P i − ) /P i − ∼ = M ( L ν i ) = 0, ψ is also injective, hence it induces the isomorphism V ∗− ω ν i ⊗ V ∗ ν i ˜ → P i /P i − . (cid:3) As an application, we will compute HH ( O q , O q ), the 0-th Hochschild cohomologyof the coalgebra O q with coefficients in O q , which is equivalent to the algebra ofcocommutative elements in O q .Suppose f ∈ O q is cocommutative, then f ( uu ′ ) = f ( u ′ u ) for any u, u ′ ∈ U q , i.e. ρ ( u ) f = ρ ( S − u ) f . Lemma 3.4. For any λ ∈ X + , the subspace Y = { y ∈ V ∗− ω λ ⊗ V ∗ λ : ρ ′ ( u ) y = ρ ′ ( S − u ) y, ∀ u ∈ U q } is one-dimensional, where ρ ′ , ρ ′ denote the actions of U q on V ∗− ω λ and V ∗ λ respectively.Proof. Set B q = U q U − q , k = Q ( q ), and denote by k λ the one-dimensional B q -moduledefined by the character χ λ : U q → k and extended to a B q -module with triv-ial U − q -action. Recall from [APW, A1] that V ∗− ω λ is an integrable submodule ofHom B q ( U q , k λ ), where U q is considered a B q -module via left multiplication of B q on U q , and the U q -module structure on Hom B q ( U q , k λ ) is defined via the right multipli-cation of U q on itself. Choose a basis v , · · · , v s of V ∗− ω λ such that v ( uE ( r ) i ) = 0 forany i if r > v ( b ) = χ λ ( b ) for any b ∈ B q . Then v has weight λ (recall that λ occurs with multiplicity 1 in V ∗− ω λ ). Assume that v , · · · , v s are also homogeneousvectors (with weights less than λ ), then v i ( b ) = 0 for any b ∈ U − q and i = 2 , · · · , s .Similarly choose a homogeneous basis v ′ , · · · , v ′ s of V ∗ λ such that v ′ has weight − λ .For any y = P ij y ij v i ⊗ v ′ j ∈ Y , it is easy to check that in order for y to satisfythe equality ρ ′ ( u ) y = ρ ′ ( S − u ) y for all u = u ∈ U q , we must have y i = 0 for any i = 1 (since χ − λ S − = χ λ ). Define a linear map pr : Y → k ; y y , which we willshow is in fact injective. Suppose y = 0 for some y ∈ Y , then for any u +1 , u +2 ∈ U + q , we have y ( u +1 ⊗ u +2 ) = P ij y ij v i ( u +1 ) v ′ j ( u +2 ) = P ij y ij { ρ ′ ( u +1 ) v i } (1) v ′ j ( u +2 ) = P ij y ij v i (1) { ρ ′ ( S − u +1 ) v ′ j } ( u +2 ) = y v ′ ( u +2 S − u +1 ) = 0, hence y = 0. It means that pr is injective, hence dim k Y ≤ 1. On the other hand, let e i be a basis of L λ and let δ i be the dual basis of L ∗ λ , it is easy to check that P i e i ⊗ δ i ∈ L λ ⊗ L ∗ λ ⊂ V ∗− ω λ ⊗ V ∗ λ satisfies the condition of Y , hence dim k Y = 1. (cid:3) Proposition 3.5. HH ( O q , O q ) ∼ = Q ( q )[ X ] W .Proof. For a module V ∈ C f , we denote by [ V ] its image in the Grothendieck ring[ C f ]. It is clear that [ C f ] is isomorphic to Z [ X ] W , with the isomorphism given by[ V ] → ch V . Let R = [ C f ] ⊗ Z Q ( q ), then R ∼ = Q ( q )[ X ] W and it has a natural basis ofsimple characters { ch L λ , λ ∈ X + } .For V ∈ C f , define the trace of V as tr V = φ V ( P i v i ⊗ f i ) ∈ M ( V ), where { v i } isa basis of V and { f i } is the dual basis of V ∗ . Let tr ⊂ O q be the Q ( q )-linear spanof traces of modules from C f . If U ֒ → V ։ W is a short exact sequence of modulesfrom C f , we have tr V = tr U + tr W , therefore each tr V can be written as a linearcombination of traces of its composition factors. Note that tr λ (:= tr L λ ), λ ∈ X + , arelinearly independent, hence they form a basis of tr , and tr ∼ = R as a vector space.Since tr V ⊗ V ′ = tr V tr V ′ , it is in fact an isomorphism of algebras.Denote by Co the set of elements in O q that are cocommutative, it suffices toshow that Co = tr . It is obvious that tr ⊂ Co . To prove the inverse inclusion,define P λ = P µ ≤ λ,µ ∈ X + M ( T µ ) ⊂ O q . Since O q = S λ ∈ X + P λ , it suffices to provethat P λ ∩ Co ⊂ tr . If λ is minimal (for the ordering ≤ ) among the weights in X + ,then P λ = M ( T λ ) = M ( L λ ) ∼ = L λ ⊗ L ∗ λ . It follows from Lemma 3.4 that P λ ∩ Co = Q ( q ) tr λ ⊂ tr . Now assume that P µ ∩ Co ⊂ tr is true for any µ < λ, µ ∈ X + . Fromthe proof of Theorem 3.3, we have P λ / P µ<λ,µ ∈ X + M ( T µ ) ∼ = V ∗− ω λ ⊗ V ∗ λ . Suppose f ∈ P λ ∩ Co , then ρ ( u ) f = ρ ( S − u ) f for any u ∈ U q , hence the image of f in P λ / P µ<λ,µ ∈ X + M ( T µ ) belongs to the subspace Y defined in Lemma 3.4. Since Y isone-dimensional and is spanned by the image of the trace of L λ , there exists a scalar ζ such that f − ζtr λ ∈ P µ<λ,µ ∈ X + M ( T µ ) ∩ Co . By induction f − ζtr λ ∈ tr , hence f ∈ tr . (cid:3) It is well known that the category of finite dimensional representations of U q issemisimple when q is not a root of unity, in which case the quantum function alge-bra O q is the direct sum of matrix coefficients of irreducible modules, and all thecocommutative elements of O q come from the traces of finite dimensional modules.Proposition 3.5 says that the last statement is also true at roots of 1. Remark 3.6. For other types of simple Lie algebras, I am not sure if O q is linearlyspanned by the matrix coefficients of finite dimensional U q -modules. Nonetheless ifwe denote the latter by O ′ q , then obviously O q ⊂ O ′ q , and the results in this sectionhold for O ′ q . 4. the case of sl In this section we study the sl case more thoroughly. Let ℓ > q be aprimitive ℓ -th root of unity. The quantum function algebra O q is generated by a, b, c, d over Q ( q ) subject to the relations: ab = qba, ac = qca, EGULAR REPRESENTATIONS OF THE QUANTUM GROUPS AT ROOTS OF UNITY 11 bd = qdb, cd = qdc,bc = cb, ad − qbc = da − q − bc = 1 . The comultiplication △ , counit ε and antipode S are defined by △ ( a ) = a ⊗ a + b ⊗ c, △ ( b ) = a ⊗ b + b ⊗ d, △ ( c ) = c ⊗ a + d ⊗ c, △ ( d ) = c ⊗ b + d ⊗ d,ε ( a ) = ε ( d ) = 1 , ε ( b ) = ε ( c ) = 0 ,S ( a ) = d, S ( d ) = a, S ( b ) = − q − b, S ( c ) = − qc. The quantum group U q is generated by E ( i ) , F ( i ) , K ± , (cid:20) K ; ct (cid:21) subject to some re-lations.For g = sl , we have X = Z , X + = N . The Weyl module V n , for n ∈ N , is( n + 1)-dimensional, with a basis f , f , · · · , f n such that f i is of weight − n + 2 i and E ( j ) f i = (cid:20) i + ji (cid:21) q f i + j , F ( j ) f i = (cid:20) n − i + jj (cid:21) q f i − j . The dual representation V ∗ n is also ( n + 1)-dimensional, with a basis e , e , · · · , e n such that e i is of weight n − i and E ( j ) e i = (cid:20) ij (cid:21) q e i − j , F ( j ) e i = (cid:20) n − ij (cid:21) q e i + j . The Weyl modules V n and their duals V ∗ n are reducible in general, but their compo-sition series are well-known, so are the Weyl filtrations of the tilting modules T n . Lemma 4.1. Write n = n + ℓn with ≤ n ≤ ℓ − , n ≥ , then (1) if n = 0 or n = ℓ − , V n is irreduible, hence T n = V n = V ∗ n = L n ; (2) assume now that ≤ n ≤ ℓ − and n ≥ , set n ′ = ( ℓ − − n )+ ℓ ( n − , thenwe have the following exact sequences: L n ′ ֒ → V n ։ L n , L n ֒ → V ∗ n ։ L n ′ , V n ֒ → T n ։ V n ′ and V ∗ n ′ ֒ → T n ։ V ∗ n .Proof. See [L2, Proposition 9.2] or [APW, Corollary 4.6] for assertions about V n , V ∗ n .Since − ω n = − ( − n = n , we have L ∗ n ∼ = L n and T ∗ n = T n . By [A2, Proposition5.8], T n is the projective cover of L n ′ . Since Ext i C f ( V m , V ∗ k ) = Q ( q ) if i = 0 and m = k ; 0 otherwise, we have the following reciprocity of multiplicities: ( T n , V k ) =dim Q ( q ) Hom C f ( T n , V ∗ k ) = ( V ∗ k , L n ′ ). Hence the composition factors of the Weyl mod-ules imply the Weyl filtrations of the tilting modules. (cid:3) Let W ∼ = Z = { , − } be the Weyl group of sl , and let W ℓ ∼ = Z ⋉ Z be theaffine Weyl group. The shifted action of W ℓ on X = Z is defined by: (1 , m ) · n = n + 2 mℓ ; ( − , m ) · n = − n − mℓ . The fundamental domain for W ℓ is given by¯ C = {− , , · · · , ℓ − } , and the linkage principal yields the following decompositionof O q . Proposition 4.2. O q = ( ⊕ k ≥ V kℓ − ⊗ V kℓ − ) ⊕ ( ⊕ ℓ − m =0 Λ m ) as a U q × U q -module, where Λ m = P s ∈W ℓ · m,s ≥ M ( T s ) .Proof. See Proposition 3.1 and Lemma 4.1. (cid:3) To analyze the structure of Λ m , 0 ≤ m ≤ ℓ − M ( T s ).We say that n < · · · < n i < n i +1 < · · · is a sequence if n i = n ′ i +1 for any i ≥ n i ≥ , n i = − ℓ is assumed). We can form ℓ − , , · · · , ℓ − W ℓ -orbits of 0 , , · · · , ℓ − Lemma 4.3. Let n < n be a sequence, i.e. n = n ′ , then (1) M ( L n ) = L n ⊗ L n and M ( L n ) = L n ⊗ L n . (2) M ( V n ) (resp. M ( V ∗ n ) ) is rigid and the Loewy series is given by ⊂ M ( L n ) ⊕ M ( L n ) ⊂ M ( V n ) (resp. ⊂ M ( L n ) ⊕ M ( L n ) ⊂ M ( V ∗ n ) ) with layers de-picted by M ( V n ) ∼ L n ⊗ L n L n ⊗ L n L L n ⊗ L n (resp. M ( V ∗ n ) ∼ L n ⊗ L n L n ⊗ L n L L n ⊗ L n ) . Proof. (1) is obvious. Tensoring 0 ⊂ L n ⊂ V n together with 0 ⊂ L n = Ann ( L n ) ⊂ V ∗ n , we obtain a filtration of V n ⊗ V ∗ n : 0 ⊂ L n ⊗ Ann ( L n ) ⊂ L n ⊗ V ∗ n + V n ⊗ Ann ( L n ) ⊂ V n ⊗ V ∗ n . Recall the U q × U q -map φ V n : V n ⊗ V ∗ n ։ M ( V n ), itis easy to see that Ker φ V n = L n ⊗ Ann ( L n ); φ V n ( L n ⊗ V ∗ n ) = M ( L n ); and φ V n ( V n ⊗ Ann ( L n )) = M ( L n ). It follows that M ( V n ) admits the filtration asclaimed in (2). The constituent L n ⊗ L n is nontrivially linked with both L n ⊗ L n and L n ⊗ L n , since the exact sequence L n ֒ → V n ։ L n does not split. Similararguments apply to M ( V ∗ n ). (cid:3) Lemma 4.4. Let n < n be a sequence and ≤ n ≤ ℓ − , then (1) M ( T n ) is rigid and indecomposable as a U q × U q -module. The Loewy seriesis given by ⊂ M ( L n ) ⊕ M ( L n ) ⊂ M ( V n ) + M ( V ∗ n ) ⊂ M ( T n ) with layersdepicted by M ( T n ) ∼ L n ⊗ L n L n ⊗ L n L L n ⊗ L n L n ⊗ L n L L n ⊗ L n . (2) M ( T n ) ⊂ M ( T n ) and M ( T n ) / M ( T n ) ∼ = V ∗ n ⊗ V ∗ n .Proof. Tensoring 0 ⊂ L n ⊂ V n ⊂ T n together with 0 ⊂ Ann ( V n ) ⊂ Ann ( L n ) ⊂ T ∗ n gives a filtration of T n ⊗ T ∗ n ; applying φ T n : T n ⊗ T ∗ n ։ M ( T n ) to it, we EGULAR REPRESENTATIONS OF THE QUANTUM GROUPS AT ROOTS OF UNITY 13 obtain the desired filtration for M ( T n ). Choose a basis of T n so that the matrixrepresentations with respect to this basis look like M ( L n ) △ ⋆ M ( L n ) ▽ M ( L n ) . The diagonal blocks correspond to the simple layers of T n ; the matrix coefficients M ( L n ) and M ( L n ), together with △ (resp. ▽ ), span M ( V n ) (resp. M ( V ∗ n )); thecoefficients in ⋆ generate the whole M ( T n ). It is not hard to see that M ( T n ) isindeed rigid and indecomposable.It is obvious that M ( T n ) ⊂ M ( T n ) since T n ∼ = L n for 0 ≤ n ≤ ℓ − 2. More-over M ( T n ) / M ( T n ) ∼ = ( T n ⊗ T ∗ n ) /φ − T n M ( L n ) = ( T n ⊗ T ∗ n ) / ( L n ⊗ T ∗ n + T n ⊗ Ann ( V n )) ∼ = V ∗ n ⊗ V ∗ n . (cid:3) Lemma 4.5. Let n < n < n be a sequence, then (1) M ( T n ) is rigid and indecomposable. The Loewy series is given by ⊂ M ( L n ) ⊕ M ( L n ) ⊕ M ( L n ) ⊂ M ( V n )+ M ( V ∗ n )+ M ( V n )+ M ( V ∗ n ) ⊂ M ( T n ) with layersdepicted by M ( T n ) ∼ L n ⊗ L n L n ⊗ L n L L n ⊗ L n L L n ⊗ L n L L n ⊗ L n L n ⊗ L n L L n ⊗ L n L L n ⊗ L n . (2) M ( T n ) ∩ M ( T n ) = M ( V n ) + M ( V ∗ n ) . Moreover we have M ( T n ) / ( M ( T n ) ∩ M ( T n )) ∼ = V ∗ n ⊗ V ∗ n and M ( T n ) / ( M ( V n ) + M ( V ∗ n )) ∼ = V n ⊗ V n .Proof. The proof is parallel to the proof of the previous two lemmas. Since T n isrigid with layers L n , L n ⊕ L n and L n from the socle to the top, we can choose abasis of T n so that the matrix representations with respect to this basis look like M ( L n ) △ n ▽ n ⋆ M ( L n ) 0 ▽ n M ( L n ) △ n M ( L n ) . We have the matrix coefficients of the irreducibles on the diagonal; △ n (resp. ▽ n )together with M ( L n ), M ( L n ) span M ( V n ) (resp. M ( V ∗ n )); △ n (resp. ▽ n ) togetherwith M ( L n ), M ( L n ) span M ( V n ) (resp. M ( V ∗ n )); the top ⋆ generates the whole M ( T n ). Again it is not difficult to see that M ( T n ) is rigid and indecomposable, andthe nonzero blocks in the matrix correspond to the layers of the Loewy series.It’s clear that M ( T n ) ∩ M ( T n ) = M ( V n )+ M ( V ∗ n ). Since φ − T n ( M ( V n )+ M ( V ∗ n )) = V ∗ n ⊗ T ∗ n + T n ⊗ Ann ( V n ) and φ − T n ( M ( V n )+ M ( V ∗ n )) = V n ⊗ T ∗ n + T n ⊗ Ann ( V ∗ n ),the last two isomorphisms hold. (cid:3) Theorem 4.6. Let n = n < n < · · · < n i < · · · be the sequence of infinite lengthstarting at n for ≤ n ≤ ℓ − . (1) Λ n is rigid and indecomposable as a U q × U q -module. The Loewy series isgiven by ⊂ ⊕ i ≥ M ( L n i ) ⊂ X i ≥ M ( V n i ) + X i ≥ M ( V ∗ n i ) ⊂ X i ≥ M ( T n i ) = Λ n with layers ⊕ i ≥ L n i ⊗ L n i , ⊕ i ≥ ( L n i +1 ⊗ L n i ⊕ L n i ⊗ L n i +1 ) and ⊕ i ≥ L n i ⊗ L n i . (2) Λ n also admits an increasing filtration of U q × U q -submodules P ⊂ P ⊂ · · · ⊂ P i ⊂ · · · and a decreasing filtration of U q × U q -submodules · · · ⊂ Q i ⊂ · · · ⊂ Q ⊂ Q ⊂ Q = Λ n such that ∪ i P i = Λ n , P i /P i − ∼ = V ∗ n i ⊗ V ∗ n i , and ∩ i Q i = 0 , Q i − /Q i ∼ = V n i ⊗ V n i .Proof. It follows from the three lemmas. For (2), set P i = P j ≤ i M ( T n j ) and Q i = P j ≥ i +2 M ( T n j ). (cid:3) Finally let’s find out explicitly the cocommutative elements of O q . Let Y n ⊂ O q bethe linear span of monomials a m b k c h , b k c h d l of degree ≤ n , i.e. Y n = P i ≤ n M ( T i ) = P i ≤ n M ( T ) i . Lemma 4.7. Y ⊂ Y ⊂ · · · ⊂ Y n − ⊂ Y n ⊂ · · · is a filtration of U q × U q -submodulesof O q with subquotients Y n /Y n − ∼ = V ∗ n ⊗ V ∗ n .Proof. It follows from Lemma 4.4 (2) and Lemma 4.5 (2). (cid:3) Lemma 4.8. The subspace { x ∈ V ∗ n ⊗ V ∗ n : ρ ′ ( u ) x = ρ ′ ( S − u ) x, ∀ u ∈ U q } is one-dimensional where ρ ′ , ρ ′ denote the actions of U q on the two copies of V ∗ n .Proof. Of course it follows from Lemma 3.4, the general version of it. But here wecan compute more explicitly, which is actually the motivation behind the proof ofLemma 3.4.Recall that V ∗ n is ( n + 1)-dimensional and has a basis e , e , · · · , e n such that e i is of weight n − i and E ( j ) e i = (cid:20) ij (cid:21) q e i − j ; F ( j ) e i = (cid:20) n − ij (cid:21) q e i + j . For any x = P i,j x ij e i ⊗ e j ∈ V ∗ n ⊗ V ∗ n , if it satisfies that ρ ′ ( u ) x = ρ ′ ( S − u ) x for any u ∈ U q , we must have x ij = 0 except for i + j = n . Now let x = P i x i,n − i e i ⊗ e n − i ,since S − E ( j ) = ( − j q − j ( j − E ( j ) K − j , it follows that ρ ′ ( E ( j ) ) x = n X i = j x i,n − i (cid:20) ij (cid:21) q e i − j ⊗ e n − i and ρ ′ ( S − E ( j ) ) x = n − j X i =0 x i,n − i ( − j q − j ( j − q − j (2 i − n ) (cid:20) n − ij (cid:21) q e i ⊗ e n − i − j EGULAR REPRESENTATIONS OF THE QUANTUM GROUPS AT ROOTS OF UNITY 15 for any j ∈ N , j ≤ n . Hence if ρ ′ ( E ( j ) ) x = ρ ′ ( S − E ( j ) ) x , then x i,n − i (cid:20) ij (cid:21) q = x i − j,n − i + j ( − j q j ( j + n − i +1) (cid:20) n − i + jj (cid:21) q , in particular x i,n − i = x ,n ( − i q i ( n − i +1) (cid:20) ni (cid:21) q ,which implies that x = x ,n y with y = P ni =0 ( − i q i ( n − i +1) (cid:20) ni (cid:21) q e i ⊗ e n − i . On theother hand it is straightforward to check that ρ ′ ( u ) y = ρ ′ ( S − u ) y holds for any u ∈ U q . (cid:3) Proposition 4.9. HH ( O q , O q ) = Q ( q )[ a + d ] .Proof. Denote the set of cocommutative elements of O q by Co . We need to showthat Co consists of polynomials in a + d . Recall that O q = ∪ n ≥ Y n , and it is trivialthat Y ∩ Co = Q ( q ). Assume now that Y n ∩ Co is linearly spanned by polynomialsof degree ≤ n in a + d . Suppose f ∈ Y n +1 ∩ Co , then ρ ( u ) f = ρ ( S − u ) f for any u ∈ U q . Since the image of ( a + d ) n +1 in Y n +1 /Y n is nonzero, by Lemma 4.8 thereexists a scalar ζ such that f − ζ ( a + d ) n +1 ∈ Y n . Note that f − ζ ( a + d ) n +1 is alsococommutative, i.e. it belongs to Y n ∩ Co , by induction f − ζ ( a + d ) n +1 is a polynomialof degree ≤ n in a + d , therefore f is a polynomial of degree ≤ n + 1 in a + d . (cid:3) References [AGL] A. Alekseev, D. Glushchenkov, A. Lyakhovskaya, Regular representation of the quantumgroup sl q ( ) ( q is a root of unity). Algebra i Analiz 6 (1994), no. 5, 88-125.[APW] H. H. Andersen, P. Polo, Wen K., Representations of quantum algebras, Invent. math. 104,1-59(1991).[A1] H. H. 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