On the Hopf algebra structure of the Lusztig quantum divided power algebras
aa r X i v : . [ m a t h . QA ] M a y ON THE HOPF ALGEBRA STRUCTURE OF THELUSZTIG QUANTUM DIVIDED POWER ALGEBRAS
NICOLÁS ANDRUSKIEWITSCH, IVÁN ANGIONO AND CRISTIAN VAY
Abstract.
We study the Hopf algebra structure of Lusztig’s quantumgroups. First we show that the zero part is the tensor product of thegroup algebra of a finite abelian group with the enveloping algebra of anabelian Lie algebra. Second we build them from the plus, minus and zeroparts by means of suitable actions and coactions within the formalismpresented by Sommerhauser to describe triangular decompositions. Introduction
There are two versions of quantum groups at roots of 1: the one introducedand studied by De Concini, Kac and Procesi [DK, DP] and the quantumdivided power algebra of Lusztig [Lu1, Lu2, Lu3, Lu4]. The small quantumgroups (aka Frobenius-Lusztig kernels) appear as quotients of the first andHopf subalgebras of the second; in both cases they fit into suitable exactsequences of Hopf algebras.The key actor in all these constructions is what we now call a Nicholsalgebra of diagonal type. Indeed all the Hopf algebras involved have trian-gular decompositions compatible with the mentioned exact sequences; thepositive part of the small quantum group is a Nichols algebra. The cele-brated classification of the finite-dimensional Nichols algebras of diagonaltype was achieved in [H]. The positive parts of the small quantum groupscorrespond to braidings of Cartan type, but there are also braidings of superand modular types in the list, see the survey [AA].The question of defining the versions of the quantum groups of De Concini,Kac and Procesi on one hand, and of Lusztig on the other, for every Nicholsalgebra in the classification arises unsurprisingly. The first was solved in[Ang2] introducing Hopf algebras also with triangular decompositions andwhose positive parts are now the distinguished pre-Nichols algebras of di-agonal type. These were introduced earlier in [Ang1], instrumental to thedescription of the defining relations of the Nichols algebras. A distinguishedpre-Nichols algebra projects onto the corresponding Nichols algebra and thekernel is a normal Hopf subalgebra that is even central under a mild technical
Mathematics Subject Classification. hypothesis, see [Ang2, AAR2]. The geometry behind these new Hopf alge-bras is studied in the forthcoming paper [AAY] for Nichols algebras comingin families.Towards the second goal, the graded duals of those distinguished pre-Nichols algebras were studied in [AAR1] under the name of Lusztig algebras;when the braiding is of Cartan type one recovers in this way the positive(and the negative) parts of Lusztig’s quantum groups. A Lusztig algebracontains the corresponding Nichols algebra as a normal Hopf subalgebra andthe cokernel is an enveloping algebra U ( n ) under the same mild technicalhypothesis mentioned above, see [AAR2]. In [AAR2, AAR3] it was shownthat n is either 0 or the positive part of a semisimple Lie algebra that wasdetermined explicitly in each case.In order to construct the analogues of Lusztig’s quantum groups at rootsof one for Nichols algebras of diagonal type, we still need to define the 0-partand the interactions with the positive and negative parts. This leads us tounderstand the Hopf algebra structure of a Lusztig’s quantum group whichis the objective of this Note.Let V be the Z [ v, v − ] -Hopf algebra as in [Lu3, 2.3]; the quantum group isdefined by specialization of V . Our first goal is to describe the specializationof the 0-part V , a commutative and cocommutative Hopf subalgebra of V .We show in Theorem 3.9 that it splits as the tensor product of the groupalgebra of a finite group and the enveloping algebra of the Cartan subalge-bra of the corresponding Lie algebra. For this we use some skew-primitiveelements h i,n ∈ V , cf. Definition 3.4, defined from the elements (cid:2) K i ;0 t (cid:3) and K ni of the original presentation of [Lu3] via elements p n,s ∈ Z [ v, v − ] . Theseare defined recursively in Lemma 3.3.In [S] it is explained that Hopf algebras U with a triangular decomposition U ≃ A ⊗ H ⊗ B , where H is a Hopf subalgebra, A is a Hopf algebra inthe category of left Yetter-Drinfeld modules and B is the same but right,plus natural compatibilities, can be described by some specific structurethat we call a TD-datum. Our second goal is to spell out the TD-datumcorresponding to the quantum group, see Theorem 4.4.The paper is organized as follows. In Section 2 we set up some notationand recall the formalism of [S]. Section 3 contains the analysis of the Hopfalgebra V from [Lu3]. In Section 4 we recall the definition of Lusztig’sversion of quantum groups at roots of 1, show that it fits into the settingof [S] and prove Theorem 4.4. For simplicity of the exposition we assumethat the underlying Dynkin diagram is simply-laced; in the last Subsectionwe discuss how one would extend the material to the general case.The Lusztig’s quantum groups enter into a cleft short exact sequence ofHopf algebras [A, 3.4.1,3.4.4] and contain an unrolled version of the finitequantum groups [Le] but as is apparent from the description here, they arenot unrolled Hopf algebras. We are not aware of other papers containinginformation on the matter of our interest. Other versions of triangular de-compositions similar to [S] appear in [Ma, L]. USZTIG QUANTUM DIVIDED POWER ALGEBRA 3 Preliminaries
Conventions.
We adhere to the notation in [Lu3, Lu4] as much aspossible. If t ∈ N , θ ∈ N and t < θ , then I t,θ := { t, t + 1 , . . . , θ } , I θ := I ,θ .Let A = Z [ v, v − ] , the ring of Laurent polynomials in the indeterminate v , A ′ = Q ( v ) , its field of fractions; later we also need A ′′ := Z [ v, v − , (1 − v ) − ] .The v -numbers are the polynomials [ s ] v = v s − v − s v − v − , [ N ] ! v = N Y s =1 [ s ] v , (cid:20) Ni (cid:21) v = [ N ] ! v [ N − i ] ! v [ i ] ! v ∈ A ,s, N ∈ N , i ∈ I ,N . We denote (cid:20) Ni (cid:21) v = 0 when i > N , i, N ∈ N .If B is a commutative ring and ν ∈ B is a unit, then B is an A -algebravia v ν ; the elements [ s ] v , [ N ] ! v , (cid:2) Ni (cid:3) v of A specialize to [ s ] ν , [ N ] ! ν , (cid:2) Ni (cid:3) ν of B . As in [Lu3, 5.1, pp. 287 ff], we fix ℓ ′ ∈ N and set ℓ = ℓ ′ if ℓ ′ is odd, ℓ ′ if ℓ ′ is even.Let φ ℓ ′ ∈ Z [ v ] be the ℓ ′ -th cyclotomic polynomial; let B be the field offractions of A / h φ ℓ ′ i and let ν be the image of v in B . We have in B φ ℓ ( ν ) = 0 , ν ℓ = ( − ℓ ′ +1 , ν ℓ = 1 , (2.1) (cid:20) N + MM (cid:21) ν = 0 , N, M ∈ I ,ℓ − , N + M ≥ ℓ. (2.2)We also have that [ mℓ ] ν [ nℓ ] ν = mn , [ mℓ + j ] ν [ nℓ + j ] ν = 1 , m, n ∈ N , j ∈ I ℓ − . (2.3)Hence for all m ≥ n ∈ N and j ∈ I ℓ − , we have (cid:20) mℓnℓ (cid:21) ν = (cid:18) mn (cid:19) , (cid:20) mℓ + jj (cid:21) ν = 1 , (cid:20) mℓ + j − j (cid:21) ν = 0 . (2.4)Let k be a field; all algebras, coalgebras, etc. below are over k unlessexplicitly stated otherwise. If A is an associative unital k -algebra, then weidentify k with a subalgebra of A .2.2. Hopf algebras with triangular decomposition.
Let H be a Hopfalgebra with multiplication m , comultiplication ∆ (with Sweedler notation ∆( h ) = h (1) ⊗ h (2) ), counit ε and bijective antipode S ; we add a subscript H when precision is desired. We denote by HH YD , respectively YD HH , thecategory of left-left, respectively right-right Yetter-Drinfeld modules over H .If M ∈ HH YD , then the action of H on M is denoted by ⊲ while the coactionis m m ( − ⊗ m (0) , whereas if N ∈ YD HH , the action is denoted by ⊳ andthe coaction is n n (0) ⊗ n (1) . For Hopf algebras either in HH YD or YD HH , we NICOLÁS ANDRUSKIEWITSCH, IVÁN ANGIONO AND CRISTIAN VAY use notations as above but with the variation ∆( r ) = r (1) ⊗ r (2) . Given Hopfalgebras R in HH YD and S in YD HH , their bosonizations are denoted R H , H S . If A π ⇄ ι H are morphisms of Hopf algebras, then R H ≃ A ≃ H S where R and S are the subalgebras of right and left coinvariants of π ; see[R, §11.6, §11.7].A TD-datum over H [S, Definition 3.2] is a collection ( A, B, ⇀, ↼, ♯ ) where(i) A is a Hopf algebra in HH YD ;(ii) B is a Hopf algebra in YD HH ;(iii) ⇀ : B ⊗ A → A is a left action so that A is a left B -module, and thefollowing identities hold for all a ∈ A , b ∈ B and h ∈ H : b ⇀ ( h ⊲ a ) = h (1) ⊲ (cid:0)(cid:0) b ⊳ h (2) (cid:1) ⇀ a (cid:1) , b ⇀ ε B ( b ) , ∆ A ( b ⇀ a ) = (cid:16) b (1)(0) ⇀ a (1) (cid:17) ⊗ (cid:16) b (1)(1) ⊲ (cid:16) b (2) ⇀ a (2) (cid:17)(cid:17) ; (2.5)(iv) ↼ : B ⊗ A → B is a right action so that B is a right A -module, andthe following identities hold for all a ∈ A , b ∈ B and h ∈ H : ( b ⊳ h ) ↼ a = (cid:0) b ↼ (cid:0) h (1) ⊲ a (cid:1)(cid:1) ⊳ h (2) , ↼ a = ε A ( a ) , ∆ B ( b ↼ a ) = (cid:16)(cid:16) b (1) ↼ a (1) (cid:17) ⊳ a (2)( − (cid:17) ⊗ (cid:16) b (2) ↼ a (2)(0) (cid:17) ; (2.6) both actions also satisfy for all a ∈ A and b ∈ B : (cid:16) b (1) ⇀ a (1) (cid:17) ⊗ (cid:16) b (2) ↼ a (2) (cid:17) = (cid:16) b (1)(1) ⊲ (cid:16) b (2) ⇀ a (2)(0) (cid:17)(cid:17) ⊗ (cid:16)(cid:16) b (1)(0) ↼ a (1) (cid:17) ⊳ a (2)( − (cid:17) (2.7)(v) ♯ : B ⊗ A → H is a linear map, b ⊗ a b♯a , satisfying the followingidentities for all a, c ∈ A , b, d ∈ B , h ∈ H . Compatibility of ♯ with the structure of H : (cid:0) b♯ (cid:0) h (1) ⊲ a (cid:1)(cid:1) h (2) = h (1) (cid:0)(cid:0) b ⊳ h (2) (cid:1) ♯a (cid:1) , ∆ H ( b♯a ) = (cid:16) b (1)(0) ♯a (1) (cid:17) a (2)( − ⊗ b (1)(1) (cid:16) b (2) ♯a (2)(0) (cid:17) ,ε H ( b♯a ) = ε B ( b ) ε A ( a ) , (2.8) Compatibility of ♯ with the products of A and B : b♯ ( ac ) = (cid:16) b (1) ♯a (1) (cid:17) a (2)( − (cid:16)(cid:16) b (2) ↼ a (2)(0) (cid:17) ♯c (cid:17) , ( bd ) ♯a = (cid:16) b♯ (cid:16) d (1)(0) ⇀ a (1) (cid:17)(cid:17) d (1)(1) (cid:16) d (2) ♯a (2) (cid:17) ,b♯ ε B ( b ) , ♯a = ε A ( a ) , (2.9) USZTIG QUANTUM DIVIDED POWER ALGEBRA 5
Compatibility of the actions with the multiplications via ♯ : b ⇀ ( ac ) = (cid:16) b (1)(0) ⇀ a (1) (cid:17) ×× (cid:0) b (1)(1) ( b (2) ♯a (2) ) a (3)( − ⊲ (cid:2)(cid:0) b (3) ↼ a (3)(0) (cid:1) ⇀ c (cid:3)(cid:1) , ( bd ) ↼ a = (cid:16)h b ↼ (cid:16) d (1)(0) ⇀ a (1) (cid:17)i ⊳ d (1)(1) (cid:0) d (2) ♯a (2) (cid:1) a (3)( − (cid:17) ×× (cid:16) d (3) ↼ a (3)(0) (cid:17) ; (2.10) Compatibility of the coactions with the comultiplications via ♯ : (cid:16) b (1)(0) ⇀ a (1) (cid:17) ( − b (1)(1) (cid:16) b (2) ♯a (2) (cid:17) ⊗ (cid:16) b (1)(0) ⇀ a (1) (cid:17) (0) = (cid:16) b (1)(0) ♯a (1) (cid:17) a (2)( − ⊗ (cid:16) b (1)(1) ⊲ (cid:16) b (2) ⇀ a (2)(0) (cid:17)(cid:17) ; (cid:16) b (2) ↼ a (2)(0) (cid:17) (0) ⊗ (cid:16) b (1) ♯a (1) (cid:17) a (2)( − (cid:16) b (2) ↼ a (2)(0) (cid:17) (1) = (cid:16)(cid:16) b (1)(0) ↼ a (1) (cid:17) ⊳ a (2)( − (cid:17) ⊗ b (1)(1) (cid:16) b (2) ♯a (2)(0) (cid:17) . (2.11) Proposition 2.1. (a) [S, 3.3, 3.4]
Let ( A, B, ⇀, ↼, ♯ ) be a TD-datum over H . Then U := A ⊗ H ⊗ B is a Hopf algebra with multiplication, comul-tiplication and antipode: ( a ⊗ h ⊗ b )( c ⊗ k ⊗ d ) = a (cid:16) h (1) ⊲ (cid:16) b (1)(0) ⇀ c (1) (cid:17)(cid:17) ⊗ h (2) b (1)(1) (cid:16) b (2) c (2) (cid:17) c (3)(1) k (1) ⊗ (cid:16)(cid:16) b (3) ↼ c (3)(2) (cid:17) ⊳ k (2) (cid:17) d, ∆( a ⊗ h ⊗ b ) = (cid:16) a (1) ⊗ a (2)( − h (1) ⊗ b (1)(0) (cid:17) ⊗ (cid:16) a (2)(0) ⊗ h (2) b (1)(1) ⊗ b (2) (cid:17) , S ( a ⊗ h ⊗ b ) = (1 ⊗ ⊗ S B ( b (0) ))(1 ⊗ S H ( a ( − hb (1) ) ⊗ S A ( a (0) ) ⊗ ⊗ . (b) [S, 3.5] Let U be a Hopf algebra. Let A and B be Hopf algebras in HH YD and YD HH respectively, provided with injective algebra maps ι A : A ֒ → U , ι H : H ֒ → U , ι B : B ֒ → U . Assume that (i)
The map A ⊗ H ⊗ B m U ( ι A ⊗ ι H ⊗ ι B ) / / U is a linear isomorphism. (ii) The induced maps A H → U , H B → U are Hopf algebra maps.Then there exists a TD-datum ( A, B, ⇀, ↼, ♯ ) over H such that U ≃ U . (cid:3) Clearly these constructions are mutually inverse. In the setting of theProposition, we say that U ≃ A ⊗ H ⊗ B is a triangular decomposition .As observed in [S], the verification of the conditions in the definition ofTD-datum is easier when H is commutative and cocommutative. NICOLÁS ANDRUSKIEWITSCH, IVÁN ANGIONO AND CRISTIAN VAY The algebra V Basic definitions.
We fix θ ∈ N . For simplicity, set I := I θ . Following[Lu3, 2.3, pp. 268 ff] we consider the A -algebra V presented by generators K i , K − i , (cid:20) K i ; ct (cid:21) , i ∈ I , c ∈ Z , t ∈ N (3.1)and relations for all i ∈ I , tagged as in loc. cit. , ( v − v − ) (cid:20) K i ; 01 (cid:21) = K i − K − i , (g5) the generators (3.1) commute with each other,(g6) K i K − i = 1 , (cid:20) K i ; 00 (cid:21) = 1 , (g7) (cid:20) t + t ′ t (cid:21) v (cid:20) K i ; 0 t + t ′ (cid:21) = X ≤ j ≤ t ′ ( − j v t ( t ′ − j ) (cid:20) t + j − j (cid:21) v K ji (cid:20) K i ; 0 t (cid:21)(cid:20) K i ; 0 t ′ − j (cid:21) , (g8) t ≥ , t ′ ≥ , (cid:20) K i ; − ct (cid:21) = X ≤ j ≤ t ( − j v c ( t − j ) (cid:20) c + j − j (cid:21) v K ji (cid:20) K i ; 0 t − j (cid:21) , t ≥ , c ≥ , (g9) (cid:20) K i ; ct (cid:21) = X ≤ j ≤ t v c ( t − j ) (cid:20) cj (cid:21) v K − ji (cid:20) K i ; 0 t − j (cid:21) , t ≥ , c ≥ . (g10)Observe that (g9) and (g10) actually define the elements (cid:20) K i ; ct (cid:21) , c ∈ Z − ,in terms of K ± i and k i,t := (cid:20) K i ; 0 t (cid:21) , t ∈ N , i ∈ I . (3.2)See also §4.4 for an equivalent formulation. Set a i,t = v − t K i − v t K − i v − v − i ∈ I , t ∈ Z . (3.3)Thus S ( a i,t ) = − a i, − t . Taking t ′ = 1 in (g8) we have [ t + 1] v k i,t +1 = k i,t ( v t k i, − [ t ] v K i ) = k i,t a i,t , (3.4)hence [ t ] ! v k i,t = Y ≤ s Proposition 3.1. (a) [Lu3, Lemma 2.21] The A -module V is free withbasis K δ · · · K δ θ θ k ,t · · · k θ,t θ , δ i ∈ { , } , t i ∈ N , i ∈ I . (3.7)(b) [Lu3, 2.22] V ⊗ A A ′ ≃ A ′ [ Z I ] as A ′ -algebras. (cid:3) Thus V is an A -form of the group algebra A ′ [ Z I ] ; actually it is a form ofthe Hopf algebra structure as we see next. Lemma 3.2. The A -algebra V is a Hopf algebra with comultiplication de-termined by ∆( K ± i ) = K ± i ⊗ K ± i i ∈ I . (3.8) Proof. Since V is a subalgebra of the Hopf algebra A ′ [ Z ] , we need to seethat ∆( V ) ⊂ V ⊗ A V . By (g9) and (g10), it is enough to show that ∆( k i,t ) = X ≤ s ≤ t k i,t − s K − si ⊗ k i,s K t − si (3.9)for all i ∈ I and t ∈ N . We proceed by induction on t . If t = 1 , then ∆( k i, ) = 1 v − v − (cid:0) K i ⊗ K i − K − i ⊗ K − i (cid:1) = k i, ⊗ K i + K − i ⊗ k i, . If (3.9) is valid for t , then ∆( k i,t +1 ) = 1[ t + 1] v ∆( k i,t )∆( a i,t ) = 1[ t + 1] v (cid:16) X ≤ s ≤ t k i,t − s K − si ⊗ k i,s K t − si (cid:17) × (cid:18) v − t K i ⊗ K i − v t K − i ⊗ K − i v − v − (cid:19) = X ≤ s ≤ t v − t k i,t − s K − s ⊗ k i,s K t +1 − s − v t k i,t − s K − − si ⊗ k i,s K t − − si [ t + 1] v ( v − v − )= X ≤ s ≤ t v − s [ t + 1] v (cid:18) [ t − s + 1] v k t +1 − s + v t − s k i,t − s K − i v − v − (cid:19) K − si ⊗ k i,s K t +1 − si + 1[ t + 1] v X ≤ s ≤ t v t − s k i,t − s K − − si ⊗ (cid:18) [ s + 1] v k i,s +1 − v − s k i,s K i v − v − (cid:19) K t − si = k i,t +1 ⊗ K t +1 i + 1[ t + 1] v X ≤ s ≤ t v − s [ t − s + 1] v k i,t +1 − s K − si ⊗ k i,s K t +1 − si + X ≤ s ≤ t − v t − s [ s + 1] v [ t + 1] v k i,t − s K − − si ⊗ k i,s +1 K t − si + K − − ti ⊗ k i,t +1 = k i,t +1 ⊗ K t +1 i + K − − ti ⊗ k i,t +1 + X ≤ j ≤ t v − j [ t − j + 1] v + v t +1 − j [ j ] v [ t + 1] v k i,t +1 − j K − ji ⊗ k i,j K t +1 − ji NICOLÁS ANDRUSKIEWITSCH, IVÁN ANGIONO AND CRISTIAN VAY = X ≤ s ≤ t +1 k i,t +1 − s K − si ⊗ k i,s K t +1 − si , which completes the inductive step. (cid:3) Some skew-primitive elements. We introduce some notation: • For n ∈ N , φ n : N → { , } is the map given by φ n ( j ) = 0 if n − j is evenand φ n ( j ) = 1 if n − j is odd. • Φ : V → V is the algebra automorphism determined by K i 7→ − K i , K − i 7→ − K − i , (cid:20) K i ; ct (cid:21) ( − t (cid:20) K i ; ct (cid:21) . (3.10)It is easy to see that (3.10) defines an algebra map. Notice that Φ( k i,n ) = ( − n k i,n , Φ( K ± ni ) = ( − n K ± ni , n ∈ N , i ∈ I . (3.11) Lemma 3.3. Let n ∈ N . We define p n,s ∈ Z [ v, v − ] , s ∈ I n , recursively on s by p n, = v − φ n (1) , p n,s = v ns − v − ns v φ n ( s ) s ( v n − v − n ) − X t ∈ I s − p n,t (cid:20) st (cid:21) v v ( φ n ( t ) − φ n ( s )) s , s > . Then K ni − K − ni = ( v n − v − n ) X s ∈ I n p n,s k i,s K φ n ( s ) i . (3.12) Proof. Fix i ∈ I . By Proposition 3.1 (a), K ni − K − ni is a linear combinationof k i,t , k i,t K i , t ∈ N . Indeed, it can be shown by induction on n that K ± ni belongs to the A -submodule spanned by k i,t , k i,t K i , t ≤ n . Using theinvolution Φ , we see by (3.11) that there are a n,t ∈ A , t ∈ I n such that K ni − K − ni = X t ∈ I ,n a n,t k i,t K φ n ( t ) i . (3.13)We extend scalars as in Proposition 3.1 (b) and consider the algebra maps Ξ i,j : A ′ [ Z I ] → A ′ , K i v j , K p , p = i, (3.14) j ∈ N . Notice that, with the convention (cid:2) Nn (cid:3) v = 0 when n > N , Ξ i,j ( k i,t ) = 1[ t ] ! v Y ≤ s Let n ∈ N . We set h i,n := K ni − K − ni n ( v n − v − n ) K ni = 1 n (cid:16) X s ∈ I n p n,s k i,s K φ n ( s ) i (cid:17) K ni ∈ V . (3.16)Then h i,n = K ni − n ( v n − v − n ) is (1 , K ni ) -skew primitive. Lemma 3.5. Let n ∈ N . Then p n,n = v − ( n )( − n − ( v − v − ) n − [ n − ! v . (3.17) Proof. We compute k i,t in V ⊗ A A ′ ≃ A ′ [ Z I ] : k i,t = 1[ t ] ! v t − Y j =0 v − j K i − v j K − i v − v − = t X s = − t f t,s K si , for some f t,s ∈ A ′ . In particular, f t, − t = ( − t v ( t )[ t ] ! v ( v − v − ) t . Looking at the equality (3.12), K − ni appears only in one summand, p n,n k i,n , on the right hand side. Hence − v n − v − n ) f n, − n p n,n = [ n ] v ( v − v − ) ( − n v ( n )[ n ] ! v ( v − v − ) n p n,n , and the claimed equality follows. (cid:3) Given ℓ ∈ N we consider the lower triangular matrix P i,ℓ = p . . . . . . p K i p . . . . . . p p K i p . . . . . . ... ... ... ... ... ... p ℓ K φ ℓ (1) i p ℓ K φ ℓ (2) i p ℓ K φ ℓ (3) i . . . p ℓℓ − K i p ℓℓ , and the column vectors k i,ℓ = k i, ... k i,ℓ , e h i,ℓ = e h i, ... e h i,ℓ , where e h i,n := nh i,n K − ni . Then (3.16) says that P i,ℓ k i,ℓ = e h i,ℓ . Recall A ′′ = Z [ v, v − , (1 − v ) − ] sothat the matrix P i,ℓ becomes invertible in V ⊗ A A ′′ by (3.17). Let P − i,ℓ = q . . . . . . q q . . . . . . q q q . . . . . . ... ... ... ... ... ... q ℓ q ℓ q ℓ . . . q ℓℓ − q ℓℓ . Then k i,n = X s ∈ I n q n,s e h i,s = X s ∈ I n q n,s s h i,s K − si , n ∈ N , i ∈ I . Example 3.6. We compute p n,s for small values of n . For n = 2 , p , = v − , p , = v − v − v − v − − p , (cid:20) (cid:21) v v = v + v − − v ( v + v − ) = v − − . This agrees with (3.17). Thus, h i, = (cid:16) v − − k i, + v − k i, K i (cid:17) K i . For n = 3 , we have that p , = 1 , p , = v − v − v ( v − v − ) − p , (cid:20) (cid:21) v v − = ( v − v − ) [2] v v ,p , = v − v − v − v − − X t ∈ I p ,t (cid:20) t (cid:21) v v φ ( t ) = v − v − v − v − − p , (cid:20) (cid:21) v − p , (cid:20) (cid:21) v v = 1 − v − − v − + v − = ( v − v − ) [2] v v . Again this agrees with (3.17). Thus, h i, = (cid:0) ( v − v − ) [2] v v k i, + ( v − v − ) [2] v v k i, K i + k i, (cid:1) K i . Remark . From the preceding formulas we conclude: k i, = h i, K − i , (3.18) k i, = 2 v − − h i, K − i − v − v − − h i, , (3.19) k i, = 3 v ( v − v − ) [2] v h i, K − i − vv − − h i, K − i + 1 v − − h i, K i (3.20) − v ( v − v − ) [2] v h i, K − i . USZTIG QUANTUM DIVIDED POWER ALGEBRA 11 Specializations of V . Recall that ℓ ′ ∈ N is defined in Section 2.1 andthat B is the field of fractions of A / h φ ℓ ′ i . We study now V B := V ⊗ A B .We also assume that ν = 1 . Thus the map A → B factorizes through A ′′ . Lemma 3.8. The algebra V B is generated by K i and k i,ℓ = (cid:20) K i ; 0 ℓ (cid:21) , i ∈ I .Furthermore, K ℓi = 1 . (3.21) Proof. We first prove (3.21). Taking t = ℓ − and t ′ = 1 in (g8) we have: (2.2) = (cid:20) ℓℓ − (cid:21) ν (cid:20) K i ; 0 ℓ (cid:21) = ν − (cid:20) K i ; 0 ℓ − (cid:21)(cid:20) K i ; 01 (cid:21) − (cid:20) ℓ − (cid:21) ν K i (cid:20) K i ; 0 ℓ − (cid:21) = (cid:20) K i ; 0 ℓ − (cid:21) νK i − ν − K − i ν − ν − (3.5) = Q ≤ s<ℓ ν − s K i − ν s K − i [ ℓ − ! ν ( ν − ν − ) ℓ = K ℓi − K − ℓi [ ℓ − ! ν ( ν − ν − ) ℓ . Hence (3.21) holds.Let t ∈ I ,ℓ − . Then [ t ] ! ν = 0 , so k i,t = t ] ! ν Q ≤ s Let Γ = ( Z / ℓ ) I , with g i ∈ Γ being generators of the corresponding copiesof the cyclic group Z / ℓ . Let h be the abelian Lie algebra with basis ( t i ) i ∈ I ,so that U ( h ) ≃ B [ t i : i ∈ I ] . Theorem 3.9. The assignement Ψ( g i ) = K i , Ψ( t i ) = h i,ℓ , i ∈ I , (3.23) determines an isomorphism of Hopf algebras Ψ : B Γ ⊗ U ( h ) → V B .Proof. That (3.23) defines an algebra map follows by (g6) and (3.21); thatis a surjective Hopf algebra map, by (3.8), Definition 3.4 and Lemma 3.8.It remains to prove that Ψ is injective. By [M, 5.3.1] it reduces to provethat Ψ is injective on the first term of the coradical filtration, i.e. that theset { K pi h jr,ℓ : i, r ∈ I θ , p ∈ I , ℓ − , j ∈ I , } is linearly independent. By theassumption ν = 1 , we have p ℓ,ℓ ( ν ) = 0 , so (3.16) implies that h i,ℓ ∈ B × k i,ℓ K ℓi + Bh K i i , (3.24)see the line before (3.22). We need then to prove that the set { K pi k jr,ℓ : i, r ∈ I θ , p ∈ I , ℓ − , j ∈ I , } is linearly independent. Indeed, suppose that X i,p e i,p K pi + b i,p K pi k i,ℓ , (3.25)where e i,p , b i,p ∈ B . Fix i ∈ I . The A ′ -algebra maps Ξ i,j : A ′ [ Z ] → A ′ asin (3.14) satisfy Ξ i,j ( V ) ⊆ A by (3.15). We restrict to A -algebra maps Ξ i,j : V → A and tensorize to get B -algebra maps Ξ i,j : V B → B such that Ξ i,j ( K i ) = ν j , Ξ i,j ( k i,ℓ ) = (cid:20) jℓ (cid:21) ν , Ξ i,j ( K r ) = 1 , Ξ i,j ( k r,ℓ ) = 0 if r = i. Applying Ξ i,j to (3.25), we get X p ∈ I , ℓ − e i,p ν pj , ≤ j < ℓ ; (3.26) X p ∈ I , ℓ − e i,p ν pj + b i,p ν pj , ℓ ≤ j < ℓ. (3.27)If ℓ ′ is even, then ℓ ′ = 2 ℓ and from (3.26) we deduce that e i,p = 0 for all p ∈ I , ℓ − . Hence P p ∈ I , ℓ − b i,p ν pj for all ≤ j < ℓ by (3.27), and thesame argument shows that b i,p = 0 for all p ∈ I , ℓ − .If ℓ ′ = ℓ is odd, then e i,p + e i,p + ℓ ⋆ = 0 for all p ∈ I ,ℓ − by (3.26). Similarly asabove, we consider the algebra maps e Ξ i,j : A ′ [ Z j ] → A ′ such that K i 7→ − v j and K r for r = i ; we get algebra maps e Ξ i,j : V B → B such that e Ξ i,j ( K i ) = − ν j , e Ξ i,j ( k i,ℓ ) = − (cid:20) jℓ (cid:21) ν , Ξ i,j ( K r ) = 1 , Ξ i,j ( k r,ℓ ) = 0 , r = i. Applying e Ξ i,j to the previous equality (3.25), we see that X p ∈ I ,ℓ − ( − i ( e i,p − e i,p + ℓ ) ν pj , ≤ j < ℓ. USZTIG QUANTUM DIVIDED POWER ALGEBRA 13 Hence e i,p − e i,p + ℓ = 0 for all p ∈ I ,ℓ − , so e i,p = 0 for all p ∈ I , ℓ − .Analogously b i,p = 0 for all p ∈ I , ℓ − . (cid:3) The algebra V , simply-laced diagram Definitions and first properties. As in [Lu3], we fix a finite Cartanmatrix A = ( a ij ) i,j ∈ I whose Dynkin diagram is connected and simply-laced,that is, of type A, D or E.Following [Lu3, 2.3, pp. 268 ff] we consider the A -algebra V presented bygenerators (3.1), E ( N ) i , F ( N ) i , i ∈ I , N ∈ N with relations (g5), . . . (g10),together with the following, tagged again as in loc. cit. , E ( N ) i E ( M ) i = (cid:20) N + MM (cid:21) v E ( N + M ) i , E (0) i = 1; (d1) F ( N ) i F ( M ) i = (cid:20) N + MM (cid:21) v F ( N + M ) i , F (0) i = 1; (f1)if i = j ∈ I , a ij = 0 : E ( N ) i E ( M ) j = E ( M ) j E ( N ) i , (d2) F ( N ) i F ( M ) j = F ( M ) j E ( N ) i , (f2)if i = j ∈ I , a ij = − , i < j : E ( N ) i E ( M ) j = min { M,N } X t =0 v t +( N − t )( M − t ) E ( M − t ) j E ( t ) ij E ( N − t ) i , (d3) v NM E ( N ) i E ( M ) ij = E ( M ) ij E ( N ) i , (d4) v NM E ( M ) ij E ( N ) j = E ( N ) j E ( M ) ij , (d5) F ( N ) i F ( M ) j = min { M,N } X t =0 v − t − ( N − t )( M − t ) F ( M − t ) j F ( t ) ij F ( N − t ) i , (f3) v NM F ( N ) i F ( M ) ij = F ( M ) ij F ( N ) i , (f4) v NM F ( M ) ij F ( N ) j = F ( N ) j F ( M ) ij , (f5)where E ( N ) ij = N P k =0 ( − N − k v − k E ( k ) i E ( N ) j E ( N − k ) i (cf. [Lu3, Lemma 2.5 (d)]), F ( N ) ij = N P k =0 ( − N − k v − k F ( k ) i F ( N ) j F ( N − k ) i ; E ( N ) i F ( M ) j = F ( M ) j E ( N ) i , i = j, (h1) E ( N ) i F ( M ) i = X ≤ t ≤ min { N,M } F ( M − t ) i (cid:20) K i ; 2 t − N − Mt (cid:21) E ( N − t ) i , (h2) K ± i E ( N ) j = v ± Na ij E ( N ) j K ± i , (h3) K ± i F ( N ) j = v ∓ Na ij F ( N ) j K ± i , (h4) (cid:20) K i ; ct (cid:21) E ( N ) j = E ( N ) j (cid:20) K i ; c + N a ij t (cid:21) , (h5) (cid:20) K i ; ct (cid:21) F ( N ) j = F ( N ) j (cid:20) K i ; c − N a ij t (cid:21) . (h6)Let V + , respectively V − , be the subalgebra of V generated by E ( N ) i , respec-tively F ( N ) i , i ∈ I , N ∈ N . Let (cid:20) K − i ; ct (cid:21) = S (cid:18)(cid:20) K i ; ct (cid:21)(cid:19) . (4.1)The following formula is analogous to (h2), cf. [Lu5, Corollary 3.19]: F ( N ) i E ( M ) i = X ≤ t ≤ min { N,M } E ( M − t ) i (cid:20) K − i ; 2 t − N − Mt (cid:21) F ( N − t ) i . (4.2)By [Lu3, Proposition 4.8, p. 287], we know that V has a unique Hopfalgebra structure determined by (3.8) and ∆( E ( N ) i ) = X ≤ b ≤ N v b ( N − b ) E ( N − b ) i K bi ⊗ E ( b ) i , ∆( F ( N ) i ) = X ≤ a ≤ N v − a ( N − a ) F ( a ) i ⊗ K − ai F ( N − a ) i , i ∈ I , N ∈ N . (4.3)4.2. Specializations of V . We define next V + B = V + ⊗ A B , V −B = V − ⊗ A B , V B = V ⊗ A B . By [Lu2, Proposition 3.2 (b)], V + B is generated by E i and E ( ℓ ) i , i ∈ I ; V −B isgenerated by F i and F ( ℓ ) i , i ∈ I . From now on, we abbreviate k i,N = (cid:20) K i ; 0 N (cid:21) , N ∈ N , E i = E (1) i , F i = F (1) i . Lemma 4.1. Let i, j ∈ I . We have in V B : k i,ℓ E i = E i (cid:0) k i,ℓ + ν − [2] ν K − i k i,ℓ − + ν − K − i k i,ℓ − (cid:1) , (4.4) k i,ℓ F i = F i (cid:16) k i,ℓ + X j ∈ I ℓ − ( − j ν − j [ j + 1] ν K ji k i,ℓ − j − K ℓi (cid:17) , (4.5) k i,ℓ E j = E j k i,ℓ , i = j, a ij = 0 , (4.6) k i,ℓ F j = F j k i,ℓ , i = j, a ij = 0 , (4.7) k i,ℓ E j = E j (cid:16) ℓ X s =0 ( − ν ) − s K si k i,ℓ − s (cid:17) , i = j, a ij = − , (4.8) k i,ℓ F j = F j (cid:0) k i,ℓ + ν − K − i k i,ℓ − (cid:1) , i = j, a ij = − , (4.9) k i,ℓ E ( ℓ ) j = E ( ℓ ) j (cid:16) k i,ℓ + a ij K ℓi (cid:17) , (4.10) USZTIG QUANTUM DIVIDED POWER ALGEBRA 15 k i,ℓ F ( ℓ ) j = F ( ℓ ) j (cid:16) k i,ℓ − a ij K ℓi (cid:17) . (4.11) Proof. We consider first the case j = i . We take t = ℓ , c = 0 and N = 1 in(h6) and use (g9) to obtain (4.5): k i,ℓ F i = F i (cid:20) K i ; − ℓ (cid:21) = F i (cid:16) X ≤ j ≤ ℓ ( − j ν ℓ − j ) (cid:20) j + 1 j (cid:21) ν K ji k i,ℓ − j (cid:17) = F i (cid:16) k i,ℓ + X j ∈ I ℓ − ( − j ν − j [ j + 1] ν K ji k i,ℓ − j − K ℓi (cid:17) . For (4.11), we take t = ℓ , c = 0 and N = 1 in (h6) and use (g9): k i,ℓ F ( ℓ ) i = F ( ℓ ) i (cid:20) K i ; − ℓℓ (cid:21) = F ( ℓ ) i (cid:16) X ≤ j ≤ ℓ ( − j ν ℓ ( ℓ − j ) (cid:20) ℓ + j − j (cid:21) ν K ji k i,ℓ − j (cid:17) = F ( ℓ ) i (cid:16) k i,ℓ − K ℓi (cid:17) . Now we take j = i and a ij = 0 . From (h5), k i,ℓ F ( N ) j = F ( N ) j k i,ℓ for all N ∈ N , hence we obtain (4.7) when N = 1 , and (4.11) when N = ℓ .Next we take j = i and a ij = − . From (h5) and (g10) we derive (4.9)when N = 1 , and (4.11) when N = ℓ : k i,ℓ F j = F j (cid:20) K i ; 1 ℓ (cid:21) = F j ℓ X s =0 ν ℓ − s (cid:20) s (cid:21) ν K − si (cid:20) K i ; 0 ℓ − s (cid:21)! = F j (cid:0) k i,ℓ + ν − K − i k i,ℓ − (cid:1) ,k i,ℓ F ( ℓ ) j = F ( ℓ ) j (cid:20) K i ; ℓℓ (cid:21) = F j (cid:0) ℓ X s =0 ν ℓ ( ℓ − s ) (cid:20) ℓs (cid:21) ν K − si (cid:20) K i ; 0 ℓ − s (cid:21)(cid:1) = F j (cid:16) k i,ℓ + K − ℓi (cid:17) . Finally we get (4.4), (4.6), (4.8) and (4.10) similarly but from (h5). (cid:3) Now we compute relations involving h i,ℓ . Lemma 4.2. Let i, j ∈ I . We have in V B : h i,ℓ E j = E j h i,ℓ , (4.12) h i,ℓ E ( ℓ ) j = E ( ℓ ) j h i,ℓ + a ij E ( ℓ ) j , (4.13) h i,ℓ F j = F j h i,ℓ , (4.14) h i,ℓ F ( ℓ ) j = F ( ℓ ) j h i,ℓ − a ij F ( ℓ ) j . (4.15) Proof. From (3.24) we have that h i,ℓ = k i,ℓ K ℓi + P ℓ − t =0 b t K ti for some b t ∈ B .Using (h3), (4.10) and (3.21), h i,ℓ E ( ℓ ) j = ( k i,ℓ K ℓi + ℓ − X t =0 b t K ti ) E ( ℓ ) j = E ( ℓ ) j (cid:16) k i,ℓ + a ij K ℓi (cid:17) K ℓi + ℓ − X t =0 b t E ( ℓ ) j K ti = E ( ℓ ) j h i,ℓ + a ij E ( ℓ ) j . The proof of (4.15) is similar. Next we check (4.12). By a direct computation, ∆([ h i,ℓ , E j ]) = [ h i,ℓ , E j ] ⊗ K j ⊗ [ h i,ℓ , E j ] . Thus [ h i,ℓ , E j ] is (1 , K j ) -primitive and belongs to the subalgebra generatedby K j , h i,ℓ and E j , so [ h i,ℓ , E j ] = c ij E j + d ij (1 − K j ) for some c ij , d ij ∈ B .As E ℓj = 0 , we have that h i,ℓ , E ℓj ] = X k ∈ I ℓ E k − [ h i,ℓ , E j ] E ℓ − kj = X k ∈ I ℓ E k − j ( c ij E j + d ij (1 − K j )) E ℓ − kj = d ij X k ∈ I ℓ E k − j (1 − K j ) E ℓ − kj = d ij (cid:16) ℓE ℓ − j − (cid:0) X k ∈ I ℓ ν a ij k (cid:1) E ℓ − j K j (cid:17) = d ij ℓE ℓ − j − d ij δ ,a ij ℓE ℓ − j K. Hence d ij = 0 . Analogously [ h i,ℓ , F j ] = c ′ ij F j for some c ′ ij ∈ B . Now h h i,ℓ , K j − K − j i = ( ν − ν − ) [ h i,ℓ , [ E j , F j ]]= ( ν − ν − )( c ij + c ′ ij ) (cid:16) K j − K − j (cid:17) , so c ′ ij = − c ij . We consider three cases:(1) j = i , a ij = 0 . Then [ h i,ℓ , E j ] = 0 by (4.6) and (h3). Thus c ij = 0 .(2) j = i , a ij = − . Let b t ∈ B such that ν − K − i k i,ℓ − = X ≤ t< ℓ b t K ti . Using the algebra maps such that K i ν s and K i 7→ − ν s , ≤ s < ℓ ,we conclude from the previous equality that δ ℓ − ,s = X ≤ t<ℓ ν st ( b t + b ℓ + t ) , − δ ℓ − ,s = X ≤ t<ℓ ( − t ν st ( b t − b ℓ + t ) . Hence b = 0 . From (4.9) and (h4): − c ij F j = [ h i,ℓ , F j ] = [ k i,ℓ , F j ] + X ≤ t< ℓ b t [ K ti , F j ]= X ≤ t< ℓ b t F j K ti + b t ( ν t − F j K ti . Thus c ij = − b = 0 .(3) j = i . The proof is analogous to the previous case, using (4.4).Hence c ij = 0 in all the cases, so (4.12) and (4.14) follow. (cid:3) USZTIG QUANTUM DIVIDED POWER ALGEBRA 17 The Hopf algebra structure of V B . Recall that by Theorem 3.9, V B = B [ K i , h i,ℓ : i ∈ I ] ≃ B Γ ⊗ U ( h ) . Remark . The counit on the elements (cid:20) K ± i ; ct (cid:21) takes the following values. ε (cid:18)(cid:20) K ± i ; ct (cid:21)(cid:19) = if c = t = 0 , if c = 0 and t = 0 , (cid:2) ct (cid:3) ν if c > , ( − t (cid:2) − c + t − t (cid:3) ν if c < .(4.16)In fact, we first note that ε (cid:18)(cid:20) K − i ; ct (cid:21)(cid:19) = ε S (cid:18)(cid:20) K i ; ct (cid:21)(cid:19) = ε (cid:18)(cid:20) K i ; ct (cid:21)(cid:19) by(4.1). The formula for c = 0 holds by (3.5). Then, for c > , we use (g10): ε (cid:18)(cid:20) K i ; ct (cid:21)(cid:19) = X ≤ j ≤ t v c ( t − j ) (cid:20) cj (cid:21) v ε ( K − ji ) ε (cid:18)(cid:20) K i ; 0 t − j (cid:21)(cid:19) = (cid:20) ct (cid:21) v . While for c < , we use (g9): ε (cid:18)(cid:20) K i ; ct (cid:21)(cid:19) = X ≤ j ≤ t ( − j v − c ( t − j ) (cid:20) − c + j − j (cid:21) v ε ( K ji ) ε (cid:18)(cid:20) K i ; 0 t − j (cid:21)(cid:19) = ( − t (cid:20) − c + t − t (cid:21) v Theorem 4.4. The Hopf algebra V B has a triangular decomposition givenby a TD-datum ( V + B , V −B , ⇀, ↼, ♯ ) over V B . The left action ⇀ of V −B on V + B , the right action ↼ of V + B on V −B and the map ♯ : V −B ⊗ V + B → V B aredetermined as follows: F ( N ) i ⇀ E ( M ) j = δ ij ( − N (cid:20) M − N (cid:21) ν E ( M − N ) i F ( N ) i ↼ E ( M ) j = δ ij ( − M (cid:20) N − M (cid:21) ν F ( N − M ) i F ( N ) i ♯E ( M ) j = δ M,N δ ij (cid:20) K − i ; 0 N (cid:21) , (4.17) cf. (4.1) , where E ( n ) i = 0 = F ( n ) i if n < .Proof. For the first claim, we just need to verify that the conditions of Propo-sition 2.1 (b) hold.Let V ≥ B := V + B V B and V ≤ B := V B V −B ; these are Hopf subalgebras of V B by definition. It is easy to see that the inclusions V B ֒ → V ≥ B and V B ֒ → V ≤ B admit Hopf algebra sections π + and π − respectively and that V + B = (cid:16) V ≥ B (cid:17) co π + , V −B = co π − (cid:16) V ≥ B (cid:17) . Thus V + B is a Hopf algebra in V B V B YD and V ≥ B ≃ V + B V B , respectively V −B is a Hopf algebra in YD V B V B and V ≤ B ≃ V B V −B . Also, by [Lu3, Theorem 4.5(a)], the multiplication induces a linear isomorphism V + B ⊗ V B ⊗ V −B ≃ V B .Thus we may apply Proposition 2.1 (b).The verification of (4.17) is direct using the formulas in the proof of [S,Theorem 3.5] and the natural projections ̟ ⋆ : V B → V ⋆ B , for ⋆ ∈ { + , , −} .In fact, F ( N ) i ⇀ E ( M ) j = ̟ + ( F ( N ) i E ( M ) j ) . If i = j , this zero by (h1). Other-wise, we use (4.2): F ( N ) i ⇀ E ( M ) i = ̟ + ( F ( N ) i E ( M ) i )= X ≤ t ≤ min { N,M } E ( M − t ) i ε (cid:18)(cid:20) K − i ; 2 t − N − Mt (cid:21)(cid:19) ε (cid:16) F ( N − t ) i (cid:17) which is zero for N ≥ M by Remark 4.3. If N < M , then F ( N ) i ⇀ E ( M ) i = E ( M − N ) i ε (cid:18)(cid:20) K − i ; N − MN (cid:21)(cid:19) = ( − N (cid:20) M − N (cid:21) E ( M − N ) i . We can verify the formulas for ↼ and ♯ in a similar way.We next show by induction that (4.17) completely determines ⇀ , ↼ and ♯ .We will use that the comultiplication of V ±B in the respective Yetter-Drinfeldcategory is given by ∆( E ( N ) i ) = X ≤ b ≤ N v b ( N − b ) E ( N − b ) i ⊗ E ( b ) i , ∆( F ( N ) i ) = X ≤ a ≤ N v − a ( N − a ) F ( a ) i ⊗ F ( N − a ) i , i ∈ I , N ∈ N . This follows from (4.3).Let E = E ( M ) j · · · E ( M r ) j r and F = F ( N ) i · · · F ( N s ) i s . First, we assume that F ( N ) i ⇀ E, F ↼ E ( M ) j , F ( N ) i ♯E and F ♯E ( M ) j are determined by (4.17) for all r, s ≤ n and prove the same claim for s = r = n + 1 . By (2.10), we have that F ( N ) i ⇀ (cid:16) E ( M ) j E (cid:17) = (cid:18)(cid:16) F ( N ) i (cid:17) (1)(0) ⇀ (cid:16) E ( M ) j (cid:17) (1) (cid:19) ×× (cid:18)(cid:16) F ( N ) i (cid:17) (1)(1) (cid:18)(cid:16) F ( N ) i (cid:17) (2) ♯ (cid:16) E ( M ) j (cid:17) (2) (cid:19) (cid:16) E ( M ) j (cid:17) (3)( − (cid:19) ⊲ (4.18) (cid:18)(cid:18)(cid:16) F ( N ) i (cid:17) (3) ↼ (cid:16) E ( M ) j (cid:17) (3)(0) (cid:19) ⇀ E (cid:19) . USZTIG QUANTUM DIVIDED POWER ALGEBRA 19 Hence, (4.18) is determined by (4.17) because of the inductive hypothesis.The same holds for F ( N ) i ♯ (cid:16) E ( M ) j E (cid:17) since F ( N ) i ♯ (cid:16) E ( M ) j E (cid:17) = (cid:18)(cid:16) F ( N ) i (cid:17) (1) ♯ (cid:16) E ( M ) j (cid:17) (1) (cid:19) × (cid:16) E ( M ) j (cid:17) (2)( − (cid:18)(cid:18)(cid:16) F ( N ) i (cid:17) (2) ↼ (cid:16) E ( M ) j (cid:17) (2)(0) (cid:19) ♯E (cid:19) by (2.9). A similar argument works for (cid:16) F F ( N ) i (cid:17) ↼ E ( M ) j and (cid:16) F F ( N ) i (cid:17) ♯E ( M ) j .Second, we prove that F ⇀ E, F ↼ E and F ♯E are determined by (4.17) for all r, s ≥ . This is true for F ⇀ E and F ↼ E because ⇀ and ↼ are actions. For the others we proceed again by inductionon r (or on s ) using (2.9); notice that the initial inductive step r = 1 wasproved above. (cid:3) Remark . Here are some particular instances of the first line in (4.17): F i ⇀ E ( M ) j = F ( ℓ ) i ⇀ E ( M ) j = 0 , if i = j,F i ⇀ E ( M ) i = ( − M − [ M − ν E ( M − i ,F ( ℓ ) i ⇀ E ( M ) i = 0 if ℓ does not divide M,F ( ℓ ) i ⇀ E ( ℓn ) i = ( − n − ( n − E ( ℓn − ℓ ) i . Remark . The structure of V + B as an object in V B V B YD is as follows: the(left) action of V B on V + B is given by (h3), (4.12) and (4.13), while thecoaction λ : V + B → V B ⊗ V + B is determined by λ ( E ( N ) i ) = K Ni ⊗ E ( N ) i , i ∈ I , N ∈ N . Analogously, the structure of V −B as an object in YD V B V B is as follows: the(right) action of V B on V −B is given by (h4), (4.14) and (4.15); meanwhilethe coaction ρ : V −B → V −B ⊗ V B is determined by ρ ( F ( N ) i ) = F ( N ) i ⊗ K − Ni , i ∈ I , N ∈ N . The multiply-laced diagrams. The arguments above can be ex-tended to the diagrams of types B, C, F, G. We just discuss the torus parthere.Let d = ( d i ) i ∈ I ∈ N I . Following [Lu4, 6.4] we consider the A -algebra V that is a (multiply-laced!) variation of the V studied so far. For the agilityof the exposition we do not stress d in the notation. This V is presentedby the generators analogous to those (3.1) of V : K i , K − i , (cid:20) K i ; ct (cid:21) , i ∈ I , c ∈ Z , t ∈ N (4.19) with slightly modified relations. Tagging them as in [Lu4], these are:the generators (4.19) commute with each other,(b1) K i K − i = 1 , (cid:20) K i ; c (cid:21) = 1 , (b2) (cid:20) K i ; 0 t (cid:21)(cid:20) K i ; − tt ′ (cid:21) = (cid:20) t + t ′ t (cid:21) v di (cid:20) K i ; 0 t + t ′ (cid:21) , t, t ′ ≥ , (b3) (cid:20) K i ; ct (cid:21) − v − d i t (cid:20) K i ; c + 1 t (cid:21) = − v − d i ( c +1) K − i (cid:20) K i ; ct − (cid:21) , t ≥ , (b4) ( v d i − v − d i ) (cid:20) K i ; 01 (cid:21) = K i − K − i . (b5)The algebra V is related to V in the following way. For i ∈ I , let V i , respectively V i , be the subalgebra of V , respectively V , generated by K ± i and (cid:2) K i ; ct (cid:3) , respectively K ± i and (cid:2) K i ; ct (cid:3) , c ∈ Z , t ∈ N . Then (g6) andProposition 3.1, respectively (b1) and [Lu4, Theorem 6.7] imply that thereare algebra isomorphisms V ≃ V ⊗ V ⊗ . . . ⊗ V θ , V ≃ V ⊗ V ⊗ . . . ⊗ V θ . (4.20) Lemma 4.7. Let e A = A regarded as A -algebra via v v d i . Then V i ≃ V i ⊗ A e A as algebras.Proof. First we claim that there is an algebra map ψ i : V i → V i ⊗ A e A givenby K ± i K ± i ⊗ and (cid:2) K i ; ct (cid:3) (cid:2) K i ; ct (cid:3) ⊗ . Indeed, the images satisfy (b2)by (g7) and (b5) by (g5). Taking c = t and y = t ′ in (g9) and inserting theright hand side in (g9) we get (b3), while (b4) follows applying (g9) to bothsides. The claim is proved and implies in turn that ψ i is an isomorphism, asit sends a basis to a basis by Proposition 3.1 and [Lu4, Theorem 6.7]. (cid:3) From Lemma 3.2, (4.20) and Lemma 4.7 we see that V is a Hopf algebraover A with comultiplication determined by the K i ’s being group-likes. Let k i,t := (cid:20) K i ; 0 t (cid:21) , h i,n := K ni − K − ni n ( v d i n − v − d i n ) K ni = 1 n (cid:16) X s ∈ I n p n,s ( v d i ) k i,s K φ n ( s ) i (cid:17) K ni ∈ V ,t, n ∈ N , i ∈ I . Let U ( h ) ≃ B [ t i : i ∈ I ] as above and let Γ = ( Z / ℓ ) I withgenerators ( g i ) i ∈ I . 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