Braid group action and quantum affine algebras Jonathan Beck
arXiv:hep-th/9404165v1 27 Apr 1994
Massachusetts Institute of Technology Room 2-130 77 Massachusetts Avenue Cambridge, MA 02139 email:
[email protected] July 22, 1993 Abstract. We lift the lattice of translations in the extended affine Weyl group to a braid group action on the quantum affine algebra. This action fixes the Heisenberg subalgebra pointwise. Loop like generators are found for the algebra which satisfy the relations of Drinfel′ d’s new realization. Coproduct formulas are given and a PBW type basis is constructed. §0. Introduction. The purpose of this paper is to explicitly establish the isomorphism between the quantum enveloping algebra Uq (b g) of Drinfel′ d and Jimbo (b g an untwisted ′ affine Kac–Moody algebra) and the “new realization” [D2] of Drinfel d. This is done using the braid group action defined on Uq (b g) by Lusztig. In particular, we consider a group of operators P arising from the lattice of translations in the extended affine Weyl group. Drinfel′ d found that the study of finite dimensional representations of Uq (b g) is made easier by the use of a “new realization” on a set of loop algebra–like generators over C[[h]]. He gives (the proof is unpublished) an isomorphism to the usual presentation, although by his construction there is no explicit correspondence between the two sets of generators. Here we find the loop–like generators in Uq (b g) and prove a version of [D2] which sits inside the Lusztig −1 form over Q[q, q ]. We also give formulas for the coproduct of the Drinfel′ d generators. The method is to show that Uq (b g) contains n (= rank g) “vertex” subalgebras Ui , each c2 ). Applying work of Damiani [Da], it follows that Uq (b isomorphic to Uq (sl g) contains a Heisenberg subalgebra pointwise fixed by the group of translations P. This subalgebra contains the purely imaginary Drinfel′ d generators. We find the remaining generators as P translations of the usual Drinfel′ d–Jimbo generators. Having found expressions for imaginary root vectors in the usual presentation of Uq (b g), it is a straightforward application to define a basis of Poincar´e–Birkhoff–Witt type (with the method of [L5]). Acknowledgments. I would like to thank Ian Grojnowski, Victor Kac and George Lusztig for helpful conversations. §1. Notation. 1.1 We review the following standard notation (see [K]). Let (aij ), i, j ∈ I = {0, . . . , n} be the (n + 1) × (n + 1) Cartan matrix of b g so that (aij ), 1 ≤ i, j ≤ n is the Cartan matrix of the simple Lie algebra g. Let di be relatively prime positive integers such that (di aij ) is a symmetric matrix. Let P ∨ be a lattice over Z with basis ωi∨ , 1 ≤ i ≤ n. Let
Pn P ∨ ∨ ∨ α∨ ⊂ P ∨ . Then P ∨ , Q∨ are called respectively j = i=1 aji ωi , 1 ≤ j ≤ n and Q = i Zαi P P ∨ ∨ ∨ the coweight and coroot lattices of g. Let Q∨ Z α , P = + + = + i i i Z+ ωi . ∨ Define the root lattice Q = Hom(P , Z) with basis given by αi such that hαi , ωj∨ i = δij . For 1 ≤ i ≤ n define the reflection si acting on P ∨ by si (x) = x − hαi , xiα∨ i . Additionally, si ∨ acts on Q by si (y) = y − hy, αi iαi for y ∈ Q. Let W0 be the subgroup of Aut(P ∨ ) generated ∨ ∨ by s1 , . . . , sn . Let Π = {α1 , α2 , . . . αn }, Π∨ = {α∨ 1 , α2 , . . . αn }. Define the root system (resp. coroot system) R = W0 Π (resp. R∨ = W0 Π∨ ), then the correspondence αi ↔ α∨ i extends to R ↔ R∨ and for α ∈ R, hα, α∨ i = 2. 1.2 Using the W0 action on P ∨ define W = W0 ⋉ P ∨ where the product is given by −1 (s, x)(s′ , y) = (ss′ , s′ (x) + y). P ∨ is characterized as the subgroup of W consisting of elements with finitely many conjugates. For s ∈ W0 write s for (s, 0). Similarly for x ∈ P ∨ write x for (1, x). Let θ be the highest root of R. Then writing s0 for (sθ , θ ∨ ), the set {s0 , . . . , sn } generates ˜ of W with defining relations determined by (aij ). T = W/W ˜ a normal Coxeter subgroup W is a finite group in correspondence with a certain subgroup of diagram automorphisms of the Dynkin diagram of b g (see [B]). Identifying τ ∈ T with such an automorphism, τ acts −1 ˜ ˜ . The length function of W ˜ on W by τ si τ = sτ (i) , for 0 ≤ i ≤ n. We have W ∼ = T ⋉W ∨ ˜ extends to W by setting lW (τ w) = lW ˜ (w), for τ ∈ T , w ∈ W . The semigroup P+ has the properties: l(si x) = l(x) + 1, 1 ≤ i ≤ n, l(xy) = l(x) + l(y), x, y ∈ P+∨ . ˜ = Zα0 ⊕Q and set δ = α0 +θ. Then W acts as an affine Extend Q to the affine root lattice Q ˜ In particular, for x ∈ P ∨ , 1 ≤ j ≤ n, x(αj ) = αj − hαj , xiδ. transformation group on Q. ˜×Q ˜ → Z determined by (αi |αj ) = di aij . Introduce the symmetric bilinear form (.|.) : Q di Let qi = q . Introduce the q–integer notation in C(q) by: q n − qi−n [n]i = i , qi − qi−1
[n]i ! =
n Y
[k]i .
k=1
1.3 One defines the quantum affine algebra Uq (b g)(= Uq ) of Drinfel′ d and Jimbo as an ˜ C ±1/2 , D±1 subject to the algebra over C(q) on generators Ei , Fi (i ∈ I), Kα (α ∈ Q), following relations: [Kα , Kβ ] = [Kα , D] = 0, Kα Kβ = Kα+β , K0 = 1, C ±1/2 is central, (C ±1/2 )2 = Kδ±1 , Kα Ej Kα−1 = q (α|αj ) Ej , DEj D−1 = q δ0j Ej , Kα Fj Kα−1 = q −(α|αj ) Fj , DFj D−1 = q −δ0j Fj , Ki − Ki−1 , [Ei , Fj ] = δij qi − qi−1 1−aij
X
1−aij
(−1)
s
(1−aij −s) (s) Ej Ei Ei
= 0,
s=0
X s=0
2
s
(1−aij −s)
(−1) Fi
(s)
Fj Fi
= 0.
(s)
Here Ki = Kαi and Ei = Ei /[s]i !. We have added the square root of the canonical central element Kδ for later notational convenience. Introduce the C–algebra automorphism Φ, and anti-automorphism Ω of Uq , defined by: Φ(Ei ) = Fi , Φ(Fi ) = Ei , Φ(Kα ) = Kα , Φ(D) = D, Φ(q) = q −1 , Ω(Ei ) = Fi , Ω(Fi ) = Ei , Ω(Kα ) = K−α , Ω(D) = D−1 , Ω(q) = q −1 , As usual, let Uq+ (resp. Uq− ) denote the span of monomials in Ei (resp. Fi ) and T the span of monomials in Kα , C ±1/2 and D±1 . Then Uq = Uq− ⊗ T ⊗ Uq+ [L, Ro]. Uq+ is graded ˜ + in the usual way and U + = ⊕ν (U + )ν where ν ∈ Q ˜ + . An element x ∈ U + is called by Q q q q homogeneous if x ∈ (Uq+ )ν for some ν. In this case let [c.f. L 1.1.1] |x| = ν. Note that |0| = ν for all ν. For i ∈ I introduce the twisted derivations ri , i r of Uq+ [cf. L 1.2.13] defined uniquely as linear maps over C(q) with the properties: ri (1) = i r(1) = 0, ri (Ej ) = i r(Ej ) = δij , i r(xy)
= i r(x)y + q (|x|,αi ) x i r(y),
ri (xy) = q (|y|,αi ) ri (x)y + xri (y), x, y homogeneous. The Braid group B associated to W is the group on generators Tw (w ∈ W ) with the relation Tw Tw′ = Tww′ if l(w) + l(w′ ) = l(ww′ ). A reduced presentation of w ∈ W is an expression w = τ si1 . . . sin where l(w) = n, τ ∈ T . ˜ , whose canonical generators one denotes by Recall that the braid group associated to W Ti = Tsi , i ∈ I, acts as a group of automorphisms of the algebra Uq ([L]): −aij
Ti Ei = −Fi Ki , Ti Ej =
X
(−aij −s)
(s)
if i 6= j,
(−aij −s)
if i 6= j,
(−1)s−aij qi−s Ei
Ej Ei
s=0 −aij
Ti Fi = −Ki−1 Ei , Ti Fj =
X
(s)
(−1)s−aij qis Fi Fj Fi
s=0
Ti Kβ = Ksi β ,
˜ β ∈ Q,
Ti (D) = DKi−δi0 .
Then ΩTi = Ti Ω, and ΦTi = Ti−1 Φ. We extend this action to W by defining Tτ by Tτ (Ei ) = Eτ (i) , Tτ (Fi ) = Fτ (i) , Tτ (Ki ) = Kτ (i) . Write τ for Tτ . Denote by P the group generated by the operators Tωi∨ (1 ≤ i ≤ n) and their inverses. From now on, for notational convenience refer to ωi∨ by ωi . §2. Some preliminary material. 2.1 We review the following method (c.f. [DC–K], [L4], [L5]) of recovering the usual affine algebra through specialization at 1. Let A be the ring C[q, q −1 ] localized at (q − 1). Let UA be the A subalgebra of Uq generated by the elements Ei , Fi , Ki±1 , D±1 , C ±1/2 , and: Hi =
C − C −1 Ki − Ki−1 D − D−1 , c = , d = . q − q −1 q − q −1 qi − qi−1 3
UA includes the elements: [Ki ; di n] = Hi qi−n + Ki [n]i and [D; n] = dq −n + D[n]. Note the identities: Ej Hi = [Ki ; −di aij ]Ej , Fj Hi = [Ki ; di aij ]Fj E0 d = [D; −1]E0 , F0 d = [D; 1]F0 ˆ1 , the Let (q − 1)UA be the left ideal generated by (q − 1) in UA . Define the algebra U ˆ1 = UA /(q − 1)UA . We obtain the following: specialization of Uq at 1, by U ˆ1 is an associative algebra over C on the above generators Proposition [c.f. DC–K 1.5]. U with relations: c is central [Ei , Fj ] = δij Hi , [Hi , Ej ] = aij Ki Ej , [Hi , Fj ] = −aij Ki Fj , [d, Ej ] = δj0 DEj , [d, Fj ] = δj0 DFj , Ki2 = 1, D2 = 1, C 2 = 1, ad(1−aij ) Ei (Ej ) = 0, ad(1−aij ) Fi (Fj ) = 0, i 6= j. ˆ1 /(Ki − 1, D − 1, C 1/2 − 1) is isomorphic to the universal enveloping In particular, U1 = U algebra of the affine Kac–Moody Lie algebra. The following is due to Iwahori, Matsumoto and Tits. Proposition. Let w ∈ W and let τ si1 si2 . . . sin be a reduced expression of w. Then the automorphism Tw = τ Ti1 Ti2 . . . Tin of Uq depends only w and not on the reduced expression chosen. In particular, one reduced expression can be transformed to another by a finite sequence of braid relations. We recall the following from [L]. The notation is adapted to this paper. ˜ + , ν 6= 0. Let x ∈ (Uq )ν : Lemma [L 1.2.15]. Let ν ∈ Q (a) If ri (x) = 0 for all i ∈ I then x = 0. (b) If i r(x) = 0 for all i ∈ I then x = 0. Proposition [L 3.1.6]. Let x ∈ Uq+ , then: ri (x)Ki − Ki−1 i r(x) [x, Fi ] = , qi − qi−1 Proposition [L 38.1.6]. (a) {x ∈ Uq+ | i r(x) = 0} = {x ∈ Uq+ |Ti (x) ∈ Uq+ }. (b) {x ∈ Uq+ | ri (x) = 0} = {x ∈ Uq+ |Ti−1 (x) ∈ Uq+ }. Proposition [L 40.1.2]. Let w ∈ W, i ∈ I be such that l(wsi ) = l(w) + 1. If w = si1 . . . sir is a reduced presentation then Ti1 . . . Tir (Ei ) ∈ Uq+ . Lemma [L2 2.7]. Let x ∈ P ∨ , i = 1 . . . n, si ∈ S. (a) If si x = xsi then Ti TQ x = Tx Ti . Q a aj −1 −1 −1 = j Tωjj , in particular Ti−1 Tωi Ti−1 = x = = α (b) If si xs−1 i i j ωj then Ti Tx Ti Q −aij Tω−1 . j6=i Tωj i Remark: ΦTω−1 = Tω−1 Φ. i i
4
§3. Subalgebras of Uq (b g). c2 ). In this section we find certain subalgebras of Uq (b g) which are isomorphic to Uq (sl 3.1 Lemma. Let ωi ∈ P ∨ , 1 ≤ i ≤ n. (a) Any reduced presentation of ωi starts with τ sj where τ ∈ T and τ sj = s0 τ. (b) Any reduced presentation of ωi ends with si .
Proof: For j 6= 0, l(sj ωi ) = l(ωi ) + 1 and this implies (a). For (b), if l(ωi sj ) < l(ωi ) then ωi (αj ) < 0 which is only the case when i = j. Definition. For 1 ≤ i ≤ n let ωi′ = ωi si . Then l(ωi′ ) = l(ωi ) − 1. Remark. Tωi′ = Tωi Ti−1 . Definition. For 1 ≤ i ≤ n let Ui ⊂ Uq (b g) be the subalgebra generated over C(qi ) by Ei , Fi , Ki±1 , Tωi′ (Ei ), Tωi′ (Fi ), Tωi′ (Ki±1 ), C ±1/2 , D±di . It is clear Tωi′ Ei ∈ Uq+ , Tωi′ Fi ∈ Uq− since l(ωi′ si ) = l(ωi ) = l(ωi′ ) + 1. The following is proved as in [L5 1.8]: 3.2 Lemma. Let i, j ∈ I and let w ∈ W be such that w(αi ) = αj . Then Tw (Ei ) = Ej . Corollary. Let 1 ≤ i 6= j ≤ n. Then for x ∈ Uj , Tωi (x) = x. Proof. ωi (αj ) = αj . Definition. For 1 ≤ i 6= j ≤ n, aij ≤ 0 introduce the elements: Fij = −Fj Fi + q −(αi |αj ) Fi Fj , Eij = −Ei Ej + q (αi |αj ) Ej Ei . 3.3 Lemma. Let 1 ≤ i 6= j ≤ n. (a) Tωi (Fji ) = Tωj (Fij ), (b) Tωi (Eji ) = Tωj (Eij ). Proof: For (a) if aij = 0 then both sides of the equation equal 0. Otherwise, since the statement is symmetric in i and j we may assume aji = −1. Then: Tωi (Fi ) Tωj (Fij ) = Tωj (Tj−1 Fi ) = Tj Tω−1 j = Tωi Tj (Fi ) = Tωi (Fji ) which implies (a). (b) follows by applying Ω. 3.4 Lemma. Let 1 ≤ i ≤ n, [Fi , Tωi′ (Ei )] = 0. Proof: By [L 3.1.6, 38.1.6] it suffices to check that both Ti Tωi′ (Ei ) ∈ Uq+ and Ti−1 Tωi′ (Ei ) ∈ Uq+ . Since ωi ∈ P+∨ , l(siωi′ si ) = l(ωi ) + 1 = l(si ωi′ ) + 1 so that Ti Tωi′ (Ei ) ∈ Uq+ . Now Ti−1 Tωi′ (Ei ) = Ti−1 Tωi Ti−1 (Ei ) = Tω−1 (Ei ). Since Φ ◦ Ω(Uq+ ) = Uq+ and Φ ◦ Ω(Tω−1 (Ei )) = i i + ∨ Tω−1 (Ei ) it is enough to check Tω−1 (Ei ) ∈ Uq . This follows because ωi ∈ P+ and l(ωi−1 si ) = i
l(ωi−1 ) + 1.
i
5
3.5 Lemma. Let 1 ≤ i ≤ n, j 6= i, 0, [Fj , Tωi′ (Ei )] = −CKi−1 Tωj (Fij ). Proof: [Fj , Tωi′ (Ei )] = [Fj , Tωi (−Ki−1 Fi )] = −Tωi ([Fj , Ki−1 Fi ]) = −Tωi (Ki−1 )Tωi (q −(αi |αj ) Fj Fi −Fi Fj ) = −CKi−1 Tωi (Fji ) = −CKi−1 Tωj (Fij ). c2 then r0 (Tω′ (Ei )) = 0. 3.6 Lemma. Let 1 ≤ i ≤ n, if b g 6= sl i
Proof: Since l(ωi ) > 1 and Tωi′ (Ei ) ∈ Uq+ it follows T0−1 Tωi′ (Ei ) ∈ Uq+ . 3.7 Proposition. Let 1 ≤ i ≤ n. (3)
(2)
(2)
(3)
(a) Ei Tωi′ (Ei ) − Ei Tωi′ (Ei )Ei + Ei Tωi′ (Ei )Ei − Tωi′ (Ei )Ei = 0, (b) Tωi′ (Ei )(3) (Ei ) − Tωi′ (Ei )(2) Ei Tωi′ (Ei ) + Tωi′ (Ei )Ei Tωi′ (Ei )(2) − Ei Tωi′ (Ei )(3) = 0. Proof: (b) follows from (a) by applying Tωi′ . Denote the expression in (a) by xi . To check xi = 0 it suffices [L 1.2.15] to check rj (xi ) = 0 for j ∈ I. For j = 0 this is by the preceding lemma. Since for x ∈ Uq+ [Fj , x] = 0 implies rj (x) = 0, we can check that [Fj , xi ] = 0. This is straightforward using the expressions for [Fj , Tωi′ (Ei )] in 3.4 and 3.5. c2 ) → Ui 3.8 Proposition. For each 1 ≤ i ≤ n there is a algebra isomorphism hi : Uq (sl ±1 ±1 ±1 ±1 given by hi (E1 ) = Ei , hi (E0 ) = Tωi′ (Ei ), hi (K1 ) = Ki , hi (K0 ) = Tωi′ (Ki ), hi (F1 ) = Fi , hi (F0 ) = Tωi′ (Fi ), hi (C ±1/2 ) = C ±1/2 , hi (D±1 ) = D±di , hi (q) = qi . c2 ). By the previous Proposition and some Proof: Consider the defining relations of Uq (sl simple checks they hold in Ui where q is replaced by qi . Therefore hi is surjective. For ˜ g), let U ± = Ui ∩ U ± , then hi − is homogeneous with respect to this grading. ν ∈ Q(b ν i,ν |Ui P Therefore if x ∈ Kerhi |U − , writing x = j bj xj in terms of homogeneous components i ˜ hi − (xj ) = 0 for each j. Fix some xj . By [L4 Prop. 2.6] (see also remark 4.14) for β ∈ Q |Ui
c2 ) with highest weight vector there is a unique irreducible highest weight module M of Uq (sl v such that Ki v = q (αi |β) v for i = 0, 1 and Dv = q dβ v. Further we can pick β so that xj c2 ) imbeds into that of Uq (b acts non–trivially on M . The root system of Uq (sl g) via hi and ′ ′ ˜ we can fix a β ∈ Q(b g) so that pulling back the highest weight module M with weight β ′ c2 ) module M ′ has an through hi we have K0 , K1 , and D acting as on M . Now as a Uq (sl irreducible quotient which is isomorphic to M . In particular, xj must act non–trivially in M ′ which is a contradiction. Therefore Ker hi |U − = 0. Since multiplication induces a vector i m c2 ) it follows that hi factors space isomorphism U − ⊗ T ⊗ U + −→ U both in Ui and Uq (sl through this decomposition. Therefore Ker hi = 0. Corollary. For 1 ≤ i ≤ n, (a) Ti |Ui = hi ◦ T1 ◦ h−1 i , (b) Tωi |Ui = hi ◦ Tω1 ◦ h−1 i . Proof: Let M be an integrable Uq module. Decompose M into weight spaces with respect to the action of Ki , M = ⊕j M j . Let u ∈ Ui , m ∈ M n for a particular n. From the defining properties of the braid group action it follows: 6
X
T1 (h−1 i (u)) · =
X
(a)
(b)
(c)
(−1)b q −ac+b E1 F1 E1 m
a,b,c;−a+b−c=n (a)
(b)
(c)
(−1)b q −ac+b E1 F1 E1 (h−1 i u)m
a,b,c
= h−1 i
X
a,b,c
(a) (b) (c) (−1)b qi−ac+b Ei Fi Ei um
Ti (u) = h−1 i
X
(a) (b) (c) (−1)b qi−ac+b Ei Fi Ei m =⇒ hi ◦ T1 ◦ h−1 = Ti|Ui i
a,b,c
(2)
(2)
This implies (a). Tω−1 (Ei ) = Ti−1 Tωi′ (Ei ) = qi−2 Ei Tωi′ (Ei )−qi−1 Ei Tωi′ (Ei )Ei +Tωi′ (Ei )Ei i and Tω−1 Tωi′ (Ei ) = −Ki−1 Fi so that Tωi |Ui acts on the generators of Ui as does hi ◦ Tω1 ◦ h−1 i . i (b) follows. 3.9 Definition. For 1 ≤ i ≤ n, k > 0, let ψ ik = C −k/2 (qi−2 Ei Tωki (Ki−1 Fi )−Tωki (Ki−1 Fi )Ei ). Note that ψ ik ∈ Ui . c2 ). Versions of the next two propositions appear in the work of [Da §4] for Uq (sl
3.10 Proposition 1. Let d = (qi2 C −1/2 ), r > 0, m ∈ Z then:
[2]i
r−1 X
(Fi ) + d(1−r) Tωm+r (Fi ) d(1−k) (qi − qi−1 )ψ i,r−k Tωm+k i i
[ψ ir , Tωmi (Ei )] = C −1/2 [2]i
r−1 X
(Ei ) d(k−1) (qi − qi−1 )Tωm−k (Ei )ψ i,r−k + d(r−1) Tωm−r i i
[ψ ir , Tωmi (Fi )]
= −C
1/2
k=1
k=1
Proposition 2. Let r > 0, 1 ≤ i ≤ n. (a) [ψ i1 , ψ ir ] = 0, (b) Tωi (ψ ir ) = ψ ir . c2 ). Here i = 1 and Proof: It is sufficient to prove the previous two statements for Uq (sl ω1 = τ s1 , where τ is the non–trivial Dynkin diagram automorphism. This follows because l(ω1 ) = 1, l(s1 ω1 ) = l(ω1 ) + 1 and ω1 has only finitely many conjugates in W . For the sake of exposition, we sketch a proof by induction on r which appears in [Da §4]. For r = 1 the statements are readily checked. A direct calculation shows [ψ 11 , ψ 1r ] = −1 [2] (τ T1 ) (ψ 1,r+1 )−ψ 1,r+1 . This implies that 2a)r is equivalent to 2b)r+1 . Here we denote by 2a)r′ the statement 2a) for all r ≤ r ′ . Proposition 2b)r implies 1)r . This follows from an inductive calculation using the identities: [ψ 1r , F1 ] = C 1/2 q −2 [ψ 1,r−1 , τ T1 (F1 )] − [2](q − q −1 )ψ 1,r−1 τ T1 (F1 ) [ψ 1r , E1 ] = C −1/2 q 2 [ψ 1,r−1 , T1−1 τ (E1 )] + [2](q − q −1 )T1−1 τ (E1 )ψ 1,r−1 7
To show 2a) it is sufficient to show rj (C (r+1)/2 [ψ 11 , ψ 1r ]) = 0 for j = 0, 1, r > 0. For j = 0 this is straightforward. For j = 1 this follows from [[ψ 11 , ψ 1r ], F1 ] = 0. This is shown by induction on r. Assuming 2a)r−1 , 2b)r , and 1)r , a direct calculation gives [[ψ11 , ψ 1r ], F1 ] = −C
1/2
[2]
r−1 X
d(1−k) [ψ 11 , ψ1s ]Tωk1 F1 = 0.
s=1
This implies 2a)r . As noted this now implies 2b)r+1 and 1)r+1 . This completes the proof of Propositions 1 and 2. Remark: Much of the calculation through the end of §3 is inspired by the work of [Da] for c2 ). The statements of Proposition 2 also appear for Uq (sl c2 ) in [LSS]. Uq (sl
3.11 Define ϕik = Ω(ψ ik ). Applying the anti–automorphism Ω to the above propositions gives similar identities with ψ ik replaced by ϕik and Fi (resp. Ei ) replaced by Ei (resp. Fi ). Here and in the future we omit writing these identities down although we implicitly assume them. Let H be the subalgebra of Uq generated by ψ ik , ϕik for 1 ≤ i ≤ n, then we have shown: 3.12 Proposition. The group of translations P fixes H pointwise. 3.13 Lemma. Let 1 ≤ i ≤ n, r ∈ Z. Tωri (Fi )Fi − qi−2 Fi Tωri (Fi ) = qi−2 Tωr−1 (Fi )Tωi (Fi ) − Tωi (Fi )Tωr−1 (Fi ). i i c2 ) directly. Proof: This is checked in Uq (sl 3.14 Lemma. Let aij ≤ 0, m ∈ Z.
(Fj ), (a) [ψ i1 , Tωmj (Fj )] = C 1/2 [aij ]i Tωm+1 j m −1/2 m−1 (b) [ψ i1 , Tωj (Ej )] = −C [aij ]i Tωj (Ej ). Proof: We check (a) for aij ≤ 0. Note that by previous lemmas [Tωi (Ki−1 Fi ), Fj ] = −Ki−1 CTωi (Fji ) and Tωi (Fji ) = Tωj (Fij ). Then: [ψ i1 , Fj ] = C −1/2 ([qi−2 Ei Tωi (Ki−1 Fi ), Fj ] − [Tωi (Ki−1 Fi )Ei , Fj ]) = −C 1/2 Ki−1 ([Ei , Tωi (Fji )]) = −C 1/2 Ki−1 Tωj ([Ei , Fij ]) = −C 1/2 Ki−1 Tωj ([Ei , −Fj Fi + q −(αi |αj ) Fi Fj ]) = C 1/2 [aij ]i Tωj (Fj ) Now (a) follows by applying Tωmj to the above equality. Using ψi1 = Tω−1 ψi1 = i −1 −1 −(1/2) −2 −1 −1 C qi Tωi (Ei )(Ki Fi ) − (Ki Fi )Tωi (Ei ) (b) follows similarly. 3.15 Lemma. Let a = aij ≤ 0, r > 0, m ∈ Z, and let d = (−qia C −1/2 ). [ψ ir , Tωmj (Fj )]
=C
1/2
[a]i
r−1 X
k=1
[ψ ir , Tωmj (Ej )]
= −C
−1/2
[a]i
(Fj ) + d(1−r) Tωm+r (Fj ) , d(1−k) (qi − qi−1 )ψ i,r−k Tωm+k j j r−1 X
k=1
(r−1) m−r + d T (E ) . d(k−1) (qi − qi−1 )Tωm−k (E )ψ j j i,r−k ω j j 8
Proof: We check the second equation.
[ψ ir , Ej ] = C −r/2 (qi−2 Tω−1 (Ei )Tωr−1 (Ki−1 Fi )Ej − Tωr−1 (Ki−1 Fi )Tω−1 (Ei )Ej i i i i − qi−2 Ej Tω−1 (Ei )Tωr−1 (Ki−1 Fi ) + Ej Tωr−1 (Ki−1 Fi )Tω−1 (Ei )) i i i i since:
Tωr−1 (Ki−1 Fi )Ej = qi−a Ej Tωr−1 (Ki−1 Fi ) i i
= C −r/2 (qi−2−a Tω−1 (Ei )Ej Tωr−1 (Ki−1 Fi ) − Tωr−1 (Ki−1 Fi )Tω−1 (Ei )Ej i i i i − qi−2 Ej Tω−1 (Ei )Tωr−1 (Ki−1 Fi ) + qia Tωr−1 (Ki−1 Fi )Ej Tω−1 (Ei )) i i i i −2−a −1 −1 −1 −r/2 r−1 r−1 −1 =C −qi Tωi (Eij )Tωi (Ki Fi ) + Tωi (Ki Fi )Tωi (Eij )
now use:
Tω−1 (Eij ) = Tω−1 (Eji ) = Tω−1 (−Ej Ei + qia Ei Ej ), i j j
= C −r/2 (qi−2−a Tω−1 (Ej )Ei Tωr−1 (Ki−1 Fi ) − qi−2+a Ei Tωr−1 (Ki−1 Fi )Tω−1 Ej j i i j − qi−a Tω−1 (Ej )Tωr−1 (Ki−1 Fi )Ei + qia Tωr−1 (Ki−1 Fi )Ei Tω−1 Ej ) j i i j (Ej )ψ i,r−1 ) (Ej )] − [a]i (qi − qi−1 )Tω−1 = C −1/2 (−qia [ψ i,r−1 , Tω−1 j j Now the second statement follows by induction and applying Tωmj . The first statement follows by a similar calculation. 3.16 Lemma. Let r ∈ Z, (αi |αj ) ≤ 0. −Tωri Fi Fj + q −(αi |αj ) Fj Tωri Fi = q −(αi |αj ) Tωr−1 Fi Tωj Fj − Tωj Fj Tωr−1 Fi . i i Proof: The left hand side equals Tωri (Fji ). The right hand side equals: q −(αi |αj ) Tωr−1 Fi Tωj Fj − Tωj Fj Tωr−1 Fi = Tωr−1 Tωj (Fij ) = Tωri (Fji ) i i i §4. The relations in Drinfel′ d’s realization. Let Γ be the the Dynkin diagram of g. Orient the vertices of Γ by defining o : V → {±1} so that for i and j are adjacent in Γ, o(i) = −o(j). Now define Tˆωi = o(i)Tωi , and modify all the definitions by replacing Tωi with Tˆωi . 4.1 Lemma. Let a = aij , r > 0, m ∈ Z, and let d = (qia C −1/2 ).
[ψ ir , Tˆωmj (Fj )] = −C 1/2 [a]i [ψ ir , Tˆωmj (Ej )] = C −1/2 [a]i
r−1 X
k=1 r−1 X
Fj + d(1−r) Tˆωm+r Fj d(1−k) (qi − qi−1 )ψ i,r−k Tˆωm+k j j
Ej Ej ψ i,r−k + d(r−1) Tˆωm−r d(k−1) (qi − qi−1 )Tˆωm−k j j
k=1
Proof: This follows directly from §3. 9
Now for k > 0 introduce generators hik ∈ H by the change of variables (c.f. [D2], [G]): X X ′ hik z k = log(1 + (qi − qi−1 ) (qi − qi−1 ) ψ i,k′ z k ), k′ >0
k>0
Differentiating both sides and considering the coefficient of z r gives: rhir = rψ ir − (qi −
(*)
qi−1 )
r−1 X
kψ i,r−k hik .
k=1
Similarly introduce hi,−k = Ω(hik ) so that: (**)
rhi,−r = rϕir −
(qi−1
− qi )
r−1 X
khi,−k ϕi,r−k .
k=1
4.2 Lemma. Let 1 ≤ i, j ≤ n, k > 0. Fj , (a) [hik , Tˆωmj Fj ] = − k1 [kaij ]i C k/2 Tˆωm+k j Ej , (b) [hik , Tˆωmj Ej ] = k1 [kaij ]i C −k/2 Tˆωm−k j Proof: Part (b) is an induction on k using the following identities: a [ψ ik , Tˆωmj Ej ] = C −1/2 −qi ij [ψ i,k−1 , Tˆωm−1 Ej ] − [aij ]i (qi − qi−1 )Tˆωm−1 Ej ψ i,k−1 , j j (k ′ − 1)[k ′ aij ]i [hi,k′ −1 , Tˆωm−1 Ej ], where k ′ < k, [hik′ , Tˆωmj Ej ] = C −1/2 ′ ′ j k [(k − 1)aij ]i
and (a) is similar. Remark. As before, we omit the identities obtained by applying Ω. For 1 ≤ i ≤ n, r > 0, introduce the elements ψir = (qi − qi−1 )Ki ψ ir , ϕir = Ω(ψir ). Then: ψir = (qi − q −1 )C r/2 [Ei , Tˆ r Fi ], ϕir = (qi −
ωi i −1 −r/2 [Fi , Tˆωri Ei ]. qi )C
Set ψi,0 = Ki , ϕi,0 = Ki−1 . 4.3 Lemma. Let k, l ≥ 1. Then [hik , ψjl ] = 0. Proof: 1 [hik , ψjl ] = C l/2 [hik , [Ej , Tˆωl j Fj ]] qj − qj−1 = C l/2 [[hik , Ej ], Tˆωl j Fj ] + [Ej , [hik , Tˆωl j Fj ]] [kaij ]i −k/2 ˆ −k [kaij ]i k/2 ˆ l+k C [Tωj Ej , Tˆωl j Fj ] + [Ej , − C Tωj Fj ] k k [kaij ]i −k ˆ−k = C k+l/2 C [Tωj Ej , Tˆωl j Fj ] − [Ej , Tˆωl+k Fj ] = 0, j k −1 since Tωj [Ej , Tωl j Fj ] = C[Ej , Tωl j Fj ].
= C l/2
Similarly: 10
4.4 Lemma. Let k, r > 0. Then
[kaij ]i k (C − C −k )ϕj,r−k [hik , ϕjr ] = k 0
if r ≥ k,
−
if r < k.
Rewriting (**) in terms of the ϕir we have:
(***)
r(qi−1
− qi )hi,−r = rKi ϕir + (qi −
qi−1 )Ki
r−1 X
kϕi,r−k hi,−k .
k=1
4.5 Lemma. Let k, l > 0. Then C k − C −k 1 . [hik , hjl ] = δk,−l [kaij ]i k qj − qj−1 Proof: Induction using (***). + ˆk ˆ−k 4.6 Definition. For 1 ≤ i ≤ n, k ∈ Z define x− ik = Tωi (Fi ), xik = Tωi (Ei ).
We can now prove: ± ±1 ±1/2 4.7 Theorem [c.f. D2]. Uq (b g) is generated over C(q) by the elements x± , D, ij , hik , Ki , C where 1 ≤ i ≤ n, j ∈ Z, and k ∈ Z \ {0}. The following are defining relations for Uq (b g) :
(1)
[C ±1/2 , hik ] = [C ±1/2 , x± ik ] = [Kj , hik ] = [Ki , Kj ] = 0, −1 ± −1 −1 K i x± = q ±(αi ,αj ) x± = q k x± = q k hik jk Ki jk , Dxjk D jk , Dhik D
(2) (3) (4) (5)
C k − C −k 1 [hik , hjl ] = δk,−l [kaij ]i , k qj − qj−1 1 ∓(|k|/2) ± xj,k+l , [hik , x± jl ] = ± [kaij ]i C k ± ± ± ± ±(αi |αj ) ± ± x± xjl xi,k+1 = q ±(αi |αj ) x± i,k+1 xjl − q ik xj,l+1 − xj,l+1 xik , 1 − k−l/2 [x+ ψi,k+l − C l−k/2 ϕi,k+l , ik , xjl ] = δij −1 C qi − qi For i 6= j, n = 1 − aij , 1−aij
(6)
Symk1 ,k2 ,...,kn
X r=0
(−1)r
n
r i
± ± ± ± x± i,k1 . . . xi,kr xjl xi,kr+1 . . . xi,kn = 0.
Sym denotes symmetrization with respect to the indices k1 , k2 , . . . kn . Here ψik and ϕik are defined by the following functional equations: 11
∞ X
X
ψik u = Ki exp (qi −
X
ϕik uk = Ki−1 exp (qi−1 − qi )
k
qi−1 )
hik uk ,
k=1 ∞ X
k≥0
k=1
k≥0
hi,−k u−k .
Proof: Relations (1)–(5) follow from the previous calculations. Relation (6) is obtained by applying Tˆωi , i = 1, . . . , n to the Chevalley relations and an induction on max{|kir − kis |}. Let R be the algebra over C(q) on the above generators with defining relations (1)–(6). By the previous consideration there exists an algebra surjection F : R → Uq . To check that F is an isomorphism we specialize at 1 as in §2. Let RA be the A subalgebra of R generated by: Ki±1 , C ±1/2 , D±1 , hi,0 = c=
Ki − Ki−1 , qi − qi−1
D − D−1 C − C −1 , d = , hik , x± ik q − q −1 q − q −1
ˆ 1 = RA /(q−1)RA . Then R ˆ 1 is an associative algebra over C on the above generators Define R with the defining relations: (1)
(2)
[Ki , Kj ] = [D, Ki ] = 0, C 2 = D2 = Ki2 = 1, ± [d, hik ] = khik , [d, x± jk ] = kDxjk , c [hik , hjl ] = δk,−l aij (C k−1 + · · · + C 1−k ), dj
(3)
± ∓|k|/2 ± [hik , x± xj,k+l , [hi0 , x± jl ] = ±aij C jl ] = ±aij Ki xjl ,
(4)
± ± ± ± ± ± ± x± i,k+1 xjl − xjl xi,k+1 = xik xj,l+1 − xj,l+1 xik ,
(5)
− (k−l)/2 [x+ hi,k+l , ik , xjl ] = δij Ki C
(6)
± ± ± [x± i,k1 , [xi,k2 , . . . , [xi,kn , xjl ] = 0,
n = 1 − aij .
It follows from the Gabber–Kac theorem [G–K] (see [G] for the relations in R1 below) that: ˆ 1 /(Ki − 1, C 1/2 − 1, D − 1) ∼ R1 = R = U (g ⊗ C[t, t−1 ] ⊕ Cc ⊕ Cd). Now specialize Uq (b g) to U1 (b g) as in §2. Then F induces the isomorphism: F : R1 ∼ g), = U1 (b Since specialization doesn’t change the root multiplicities, F : R → Uq is an isomorphism. Remark: Let sθi ∈ W0 so that sθi (αi ) = θ. By Lemma 3.2 it follows Tθi Tωi (−Ki−1 Fi ) = E0 . This gives the inverse to the isomorphism F : R → Uq . In particular, F −1 (E0 ) = −o(i)CKθ−1 Tθi x− i1 . 12
§ 5. The coproduct. Since the Drinfel′ d generators are now expressed in terms of the braid group, calculating their coproduct depends on how the coproduct commutes with the braid group. Define for 1 ≤ i ≤ n :
Ri =
X
−k(k−1) 2
(−1)k qi
(qi − qi−1 )k [k]i !Ti (Fi )(k) ⊗ Ti (Ei )(k)
k≥0
Ri = (Ti−1 ⊗ Ti−1 )Ri−1 =
X
k(k−1) 2
qi
(k)
(qi − qi−1 )k [k]i !Fi
(k)
⊗ Ei .
k≥0
The following proposition is due in the finite type case to [K–R], [L–S]. The Kac–Moody case is due to [L 37.3.2]. 5.1 Proposition. Let Si = Ti ⊗ Ti . Let 1 ≤ i ≤ n, x ∈ Uq . (a) ∆(Ti (x)) = Ri−1 · Si ∆(x) · Ri , −1 (b) ∆(Ti−1 (x)) = Ri · Si−1 ∆(x) · Ri . Let τ si1 . . . sir be a reduced presentation of w. Define Rw = τ Si1 Si2 . . . Sir−1 (Rir ) . . . Si1 (Ri2 )Ri1 ,
Rw = Si−1 . . . Si−1 (Ri1 ) . . . Si−1 (Rir−1 )Rir . r 2 r 5.2 Lemma. Let w ∈ W, Rw , Rw are well defined.
c2 any reduced presentation is unique. Otherwise, Proof: If W is the affine Weyl group of sl since any two reduced presentations differ by a finite sequence of braid relations it is enough to check the statement for the rank two case. Consider Rsi sj si , Rsj si sj in the simply laced case. They are certainly equal since both (up to a torus element) are expressions for the rank 2 universal R–matrix (see [K–R], [L–S]). 5.3 Proposition. Let 1 ≤ i ≤ n, k ≥ 0. Let w = kωi . − −1 − (a) ∆(x− ik ) = Rw (xik ⊗ K−αi +kδ + 1 ⊗ xik )Rw , −1 − − (b) ∆(x− i,−k ) = Rw (xi,−k ⊗ K−αi −kδ + 1 ⊗ xi,−k )Rw .
Proof: This follows inductively from the above formulas. − + To obtain the coproduct on x+ ik note that Ω(xi,−k ) = xik and use ∆ ◦ Ω = Ω ⊗ Ω ◦ σ ◦ ∆ on the above formulas.
§6. A PBW basis of Uq . + For 1 ≤ i ≤ n, C k/2 ψ ik ∈ UA . On specialization to q = 1 these elements form a basis of the root space kδ of b g. This follows from the previous section since ψ ik = hik mod (q − 1), which implies their linear independence on specialization. Note that if w(αi ) = β (αi simple, β positive, w ∈ W ) then Tw (Ei ) specializes to a root vector of b g of root β. 13
˜ so that wβ (αi ) = β for some iβ ∈ I. Define Eβ = For β ∈ ∆re g) choose wβ ∈ W + (b β im Twβ (Eiβ ). For κ : ∆re + → N, ι : {1, . . . , n} × ∆+ → N define E κ,ι =
Y
κ(β)
Eβ
ι(i,kδ)
(C k/2 ψ ik )
′
′
′
′
, F κ ,ι = Ω(E κ ,ι )
where the product is in a predetermined total order over the positive roots counted with multiplicity. 6.1 Proposition. The E κ,ι form a basis of Uq+ (b g) as a C(q)–vector space. The elements ′ ′ ′ ˜ r, r ′ ∈ Z, κ, ι as above) form a basis of Uq (b F κ ,ι Kα C r +1/2 Dr E κ,ι (α ∈ Q, g) as a C(q)– vector space. Proof: The proof can be repeated almost word for word as found in [L5 §1]. In the proof of linear independence of the E κ,ι , a dominant integral highest weight should be chosen so κ,ι that for κ, ι ∈ G (in the notation found there) the E form a linearly independent set in M. Remark: The above basis is called of Poincar´e–Birkhoff–Witt type because on specialization to 1 it degenerates to a PBW basis of the enveloping algebra U (b g). References [B] N. Bourbaki “Groupes et alg` ebres de Lie Ch. 4,5,6,” Hermann, Paris, 1968. [Da] I. Damiani, A basis of type Poincar´ e–Birkhoff–Witt for the quantum algebra of sc l2 , Journal of Algebra 161 (1993), 291–310. [D] V.G. Drinfel′ d, Quantum groups, Proc. ICM Berkeley 1 (1986), 789–820. [D2] V.G. Drinfel′ d, A new realization of Yangians and Quantized Affine Algebras, Soviet Math. Dokl. Vol 36 (1988). [DC–K] C. De Concini, V.G. Kac, Representations of quantum groups at roots of 1, Progress in Math 92 (1990), 471–506, Birkh¨ auser. [DC–K–P] C. De Concini, V.G. Kac, C. Procesi, Quantum coadjoint action, Journal of the AMS 5 (1992), 151–190. [DC–K–P2] C. De Concini, V.G. Kac, C. Procesi, Some remarkable degenerations of quantum groups, Comm. Math. Phys. (1993). [G–K] O. Gabber, V.G. Kac, On defining relations of certain infinite dimensional Lie algebras, Bulletin of the AMS 5 (1981), 185–189. [G] H. Garland, The Arithmetic Theory of Loop Algebras, J. of Algebra 53 (1978), 480–551. [H] J.E. Humphreys, “Reflection Groups and Coxeter Groups,” Cambridge University Press, 1990. [J] M. Jimbo, A q-difference analog of U(g) and the Yang–Baxter equation, Lett. Math. Physics 10(1985), 63–69. [K] V.G. Kac, “Infinite Dimensional Lie Algebras, Third Edition,” Cambridge University Press, 1990. [K–R] A.N. Kirillov, N.Reshetikhin, q-Weyl group and a multiplicative formula for universal R-matrices, Comm. Math. Phys. 134 (1990), 421–431. [L–S] S. Levendorskii, Y. Soibelman, Some applications of the quantum Weyl groups, J. Geom. Phys. 7 (1990,), 241–254. [LSS] S. Levendorskii, Y. Soibelman, V. Stukopin Quantum Weyl Group and Universal Quantum R-matrix (1) for Affine Lie Algebra A1 , Lett. in Math. Physics 27 (1993), 253–264. [L] G. Lusztig, “Introduction to Quantum Groups,” Birkh¨ auser, 1993. [L2] G. Lusztig, Affine Hecke algebras and their graded version, Journal of the AMS 2 (1989), 599–625. [L3] G. Lusztig, Some examples of square integrable representations of semisimple p–adic groups, Transactions of the AMS 277 (1983), 623–653.
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[L4] G. Lusztig, Quantum deformations of certain simple modules over enveloping algebras, Advances in Mathematics (1988), 237–249. [L5] G. Lusztig, Finite dimensional Hopf algebras arising from quantized universal enveloping algebras, Journal of the AMS 3 (1990), 257–296. [Ro] M. Rosso, Finite dimensional representations of the quantum analog of the enveloping algebra of a complex simple Lie algebra, Comm. Math. Phys. 117 (1988), 581–593.
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