Contemporary Mathematics Volume 506, 2010
Multi-parameter quantum groups and quantum shuffles, (I) Yufeng Pei† , Naihong Hu , and Marc Rosso Abstract. We study the multi-parameter quantum groups defined by the generators and relations associated with symmetrizable generalized Cartan matrices, together with their representations in the category O. We present two explicit descriptions here: as a Hopf 2-cocycle deformation, and as the multi-parameter quantum shuffle realization of the positive part.
1. Introduction In the early 90s of last century, much work has been done on the multiparameter deformations of the coordinate algebra of the general linear algebraic group. These deformations were firstly described in [5] and independently in [51]. These implied that multi-parameter deformations can be obtained by twisting the coalgebra structure [51] in the spirit of Drinfeld [9] or by twisting the algebra structure via a 2-cocycle on a free abelian group [5]. In fact, the original work of Drinfeld and Reshetikhin concerned only with quasitriangular Hopf algebras, but their constructions can be dualised to the case of co-quasitriangular Hopf algebras by Hopf 2-cocycle deformations [16, 48]. Benkart-Witherspoon [11, 12] investigated a class of two-parameter quantum groups of type A arising from the work on down-up algebras [10], which were early defined by Takeuchi [56]. Bergeron-Gao-Hu [7, 8] developed the corresponding theory for two-parameter quantum orthogonal and symplectic groups, in particular, they studied the distinguished Lusztig’s symmetries property for the two-parameter quantum groups of classical type. Recently, this fact has been generalized to the cases of Drinfeld doubles of bosonizations of Nichols algebras of diagonal type by Heckenberger in [26], that is, the study of Lusztig isomorphisms (only existed among a family of different objects) in the multi-parameter setting finds a beautiful realization model for his important notion of Weyl groupoid defined in [27]. It should be pointed out that this is also a remarkable feature for the multi-parameter quantum groups in question that are distinct from the well-known one-parameter ones (see [47]). 1991 Mathematics Subject Classification. Primary 17B37, 81R50; Secondary 17B35. † Y.Pei, supported in part by the NNSFC (Grant 10571119), the ZJNSF (Grant Y607136) and the Leading Academic Discipline Project of Shanghai Normal University (Grant DZL803). N.H., corresponding author, supported in part by the NNSFC (Grants 10971065, 10728102), the PCSIRT and the RFDP from the MOE, the National/Shanghai Leading Academic Discipline Projects (Project Number: B407). 1 145
c 2010 American Mathematical Society
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Y. PEI, HU,AND ANDROSSO M. ROSSO PEI,N.HU,
Hu-Shi [34], Bai-Hu [6] did contributions to exceptional types G2 , E, respectively; Hu-Wang [35, 36], Bai-Hu and Chen-Hu-Wang further investigated the structure theory of two-parameter restricted quantum groups for types B, G2 , D and C at roots of unity, including giving the explicit constructions of convex PBW-type Lyndon bases with detailed information on commutation relations, determining the isomorphisms as Hopf algebras and integrals, as well as necessary and sufficient conditions for them to be ribbon Hopf algebras. Another new interesting development is the work of Hu-Rosso-Zhang and Hu(1) Zhang [33, 37, 38] achieved for affine types X , where X = A, B, C, D, E, F4 , G2 . Of most importance among them are the following: (1) Drinfeld realizations in the two-parameter setting were worked out; (2) Axiomatic definition for Drinfeld realizations was achieved in terms of inventing τ -invariant generating function; (3) Quantum affine Lyndon bases were put forwarded and constructed for the first (1) time; (4) Constructions of two-parameter vertex representations of level 1 for X were obtained. Using the Euler form, the first two authors [31] introduced a unified definition for a class of two-parameter quantum groups for all types and studied their structure. Shortly after, this definition was quoted in [13]. On the other hand, (multi)two-parameter quantum groups have been deeply related to many interesting work. For instance, Krob and Thibon [44] on noncommutative symmetric functions; Reineke [50] on generic extensions and degenerate two-parameter quantum groups of simply-laced cases, and the classifications of Artin-Shelt regular algebras [46]. In [52, 53], the third author found a realization of Uq+ , the positive part of the standard quantized enveloping algebra associated with a Cartan matrix by quantizing the shuffle algebra (see also [17, 23, 45]). It was mentioned that the supersymmetric and multi-parameter versions of Uq+ (for a suitable choice of the Hopf bimodule) also can be treated in this uniform principle. From a more recent point of view, Andruskiewich and Schneider obtained remarkable results on the structure of pointed Hopf algebras arising from Nichols algebras (or say, quantum symmetric algebras as in [53]) and their lifting method [2, 3, 4]. In this paper, we study a class of multi-parameter quantum groups Uq (gA ) defined by generators and relations associated with symmetrizable generalized Cartan matrices A, together with their representations in the category O. In section 2, we show that Uq (g) can be realized as Drinfeld doubles of certain Hopf subalgebras with respect to a Hopf skew-pairing , q , and as a consequence, it has a natural triangular decomposition. Partially motivated by Doi-Takeuchi [16], Majid [48] and also Westreich [57] on Hopf 2-cocycle deformation theory, we construct an explicit Hopf 2-cocycle on Uq,q−1 (gA ) and use it to twist its multiplication to get the required multi-parameter quantum group Uq (gA ). In section 3, the representation theory of Uq (gA ) under the assumption that qii (i ∈ I) are not roots of unity is described, which is the generalization of the corresponding one for two-parameter quantum groups of types A, B, C, D developed in [8] and [12]. We show that the Hopf skew-pairing , q is non-degenerate when restricted to each grading component. In section 4, using a non-degenerate τ -sesquilinear form on Uq+ (where τ is an involution automorphism of the ground field K ⊃ Q(qij | i,j∈ I ) such that τ (qij ) = qji , i, j ∈ I), we prove that the positive part Uq+ of Uq (gA ) can be embedded into the multi-parameter quantum shuffle algebra (F, ). It turns out that
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MULTI-PARAMETER QUANTUM GROUPS
this realization plays a key role both in the study of PBW-bases of Uq (gA ) and the construction of multi-parameter Ringel-Hall algebras (see [49] for more details). Throughout the paper, we denote by Z, Z+ , N, C and Q the set of integers, the set of non-negative integers, the set of positive integers, the set of complex numbers and the set of rational numbers, respectively. 2. Multi-parameter quantum group and Hopf 2-cocycle deformation 2.1. Let us start with some notations. For n > 0, define vn − 1 (n)v = . v−1 (n)v ! = (n)v · · · (2)v (1)v , and (0)v ! = 1. n (n)v ! . = k v (k)v !(n − k)v ! The following identities are well-known. (1) (2) (3) (4) (5)
(m + n)v = (m)v + v m (n)v , m m (m−k)v = (k+1)v , k+1 v k v r−m−n k r−k m+n r r = , k−m v m v m+n v k v m v n v n−1 n−1 n k n−1 n−k n−1 =v + = +v , k−1 v k v k v k−1 v k v n n−1 k(k−1) n (−1)k v 2 an−k z k = (a − vz k ), ∀ scalar a. k v
k=0
k=0
2.2. Assume that R is a field (charR = 2) with an automorphism τ . Let V be a R-vector space. A τ -linear map f on V is a function: V → R such that f (av) = τ (a)f (v),
for any a ∈ R, v ∈ V.
A τ -sesquilinear form f on V is a function: V × V → R, subject to the conditions: f (x + y, z) = f (x, z) + f (y, z), f (x, y + z) = f (x, y) + f (x, z), f (ax, y) = τ (a)f (x, y) = f (x, τ (a)y),
∀ a ∈ R,
for any x, y, z ∈ V . If τ is the identity, f is an ordinary bilinear form on V . A τ -sesquilinear form f with τ 2 = id is called τ -Hermitian form if τ (f (x, y)) = f (y, x) for any x, y ∈ V . If τ = id, f is a symmetric bilinear form on V . 2.3. Let gA be a symmetrizable Kac-Moody algebra over Q and A = (aij )i,j∈I be an associated generalized Cartan matrix. Let di be relatively prime positive integers such that di aij = dj aji for i, j ∈ I. Let Φ be the root system, Π = {αi | i ∈ I} a set of simple roots, Q = i∈I Zαi the root lattice, and then with respect to Π, we have Φ+ the system of positive roots, Q+ = i∈I Z+ αi the positive root lattice, Λ the weight lattice, and Λ+ the set of dominant weights. Let qij be indeterminates over Q and Q(qij | i, j ∈ I) be the fraction field of polynomial ring Q[qij | i, j ∈ I] such that (6)
a
qij qji = qiiij .
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Y. PEI, HU,AND ANDROSSO M. ROSSO PEI,N.HU, 1
Let K ⊃ Q(qij | i,j∈ I ) be a field such that qiim ∈ K for m ∈ Z+ . Assume that there exists an involution Q-automorphism τ of K such that τ (qij ) = qji . Denote q := (qij )i,j∈I . Definition 7. The multi-parameter quantum group Uq (gA ) is an associative algebra over K with 1 generated by the elements ei , fi , ωi±1 , ωi±1 (i ∈ I), subject to the relations: (R1)
ωi±1 ωj±1 = ωj±1 ωi±1 ,
(R2)
ωi±1 ωj±1 = ωj±1 ωi±1 ,
(R3)
ωi ej ωi−1 = qij ej ,
(R4) (R5) (R6)
ωi±1 ωj±1 = ωj±1 ωi±1 , −1 ωi ej ωi−1 = qji ej ,
−1 ωi fj ωi−1 = qij fj , ωi fj ωi−1 = qji fj , qii (ωi − ωi ), [ ei , fj ] = δi,j qii − 1 1−aij k(k−1) k 1 − aij k 1−aij −k (−1) qii 2 qij ei ej eki = 0 k qii k=0
1−aij
(R7)
ωi±1 ωi∓1 = ωi±1 ωi∓1 = 1,
k=0
k(k−1) 1 − aij 1−a −k k k (−1) qii 2 qij fi fj fi ij = 0 k qii k
(i = j), (i = j).
Proposition 8. The associative algebra Uq (g) has a Hopf algebra structure with the comultiplication, the counit and the antipode given by: ∆(ωi±1 ) = ωi±1 ⊗ ωi±1 , ∆(ei ) = ei ⊗ 1 + ωi ⊗ ei , ε(ωi±1 ) = ε(ωi S(ωi±1 ) S(ei )
±1
= ωi∓1 , = −ωi−1 ei ,
) = 1,
±1
±1
±1
∆(ωi ) = ωi ⊗ ωi , ∆(fi ) = 1 ⊗ fi + fi ⊗ ωi , ε(ei ) = ε(fi ) = 0, S(ωi
±1
) = ωi
S(fi ) =
∓1
,
−1 −fi ωi .
Remark 9. (1) Assume that qij = q di aij (i, j ∈ I). In this case, we denote Uq,q−1 (gA ) := Uq (gA ), and Uq (gA )/(ωi − ωi−1 ) Uq (gA ), where Uq (gA ) is the one-parameter quantum group of Drinfeld-Jimbo type [40]. (2) Assume that qij = r j,i s−i,j , where ⎧ ⎪ ⎨ di aij i < j, i, j := i = j, di ⎪ ⎩ 0 i > j. Uq (gA ) is one of a class of two-parameter quantum groups introduced uniformly by Hu-Pei [31], which, owing to nonuniqueness of definitions for two-parameter quantum groups, have some overlaps with the former examples defined in such as [7, 8, 6, 9, 10, 11, 12, 31, 34, 33] and references therein. (3) Assume that gA is of finite type and qij = q −u(αi ,αj )−di aij , where u is a skew Z-bilinear form on root lattice Q. Then Uq (gA ) is the multi-parameter quantum group Uq,Q introduced by Hodge et al [28, 29]. Note that the Hopf dual
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MULTI-PARAMETER QUANTUM GROUPS
objects of these quantum groups are isomorphic to those quantum groups discussed by Reshetikhin [51] (also see [15]). (4) Assume that qij = q di aij pij where P = (pij )i,j∈I such that pij pji = 1, pii = 1. Then Uq (g) are the multi-parameter quantum groups Uq,P introduced by Hayashi in [25]. (5) Assume that g = An , Uq (g) is the multi-parameter quantum groups or their dual object studied by many authors (see [1], [5], [14], and references therein). Remark 10. The definition of Uq (gA ) has appeared in [18, 19]. The positive part of Uq (gA ) has appeared in [53]. The Borel part of Uq (gA ) has appeared in [43]. From now on, we always assume that qii are not roots of unity. 2.4. Note that τ : K → K that is defined by τ (qij ) = qji for i, j ∈ I is a Q-automorphism of K. Lemma 11. (1) There is a τ -linear Q-algebra automorphism Φ of Uq (g) defined by ei → fi ,
(12)
fi → ei ,
ωi → ωi ,
ωi → ωi .
(2) There is a K-algebra anti-automorphism Ψ of Uq (g) defined by ei → fi ,
(13)
fi → ei ,
ωi → ωi ,
ωi → ωi .
Proof. (2) is clear. (1) is due to the fact: The q-Serre relation 1−aij k(k−1) 1 − aij k 1−aij −k (−1)k qii 2 qij ei ej eki = 0, k qii k=0
is equivalent to
1−aij
k=0
k(k−1) 1 − aij 1−a −k k k (−1) qii 2 qji ei ej ei ij = 0. k qii k
This completes the proof.
˜q (g) defined by the same 2.5. It will be convenient to work with the algebra U ±1 ±1 generators ei , fi , ωi , ωi for i ∈ I, and subject to relations (R1)—(R5) only ˜q (g) Uq (g). (without Serre relations). We have the canonical homomorphism U ˜q (g) and Uq (g), We abuse the notations both for the corresponding elements in U which will be clear from the context. For any i, j ∈ I with i = j, set 1−aij k(k−1) k 1 − aij k 1−aij −k u+ (14) := (−1) qii 2 qij ei ej eki , ij k qii k=0
1−aij
(15)
u− ij :=
(−1)k
k=0
k(k−1) 1 − aij 1−a −k k k qii 2 qij fi fj fi ij . k qii
Lemma 16. Let i, j ∈ I with i = j. Then 1−aij
+ ∆(u+ ij ) = uij ⊗ 1 + ωi
Proof. See Appendix A.
ωj ⊗ u+ ij ,
1−aij
− ∆(u− ij ) = uij ⊗ ωi
ωj + 1 ⊗ u− ij .
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150 6
Y. PEI, HU,AND ANDROSSO M. ROSSO PEI,N.HU,
2.6. Let Uq+ (respectively, Uq− ) be the subalgebra of Uq generated by the elements ei (respectively, fi ) for i ∈ I, Uq+0 (respectively, Uq−0 ) the subalgebra of Uq generated by ωi±1 (respectively, ωi±1 ) for i ∈ I. Let Uq0 be the subalgebra of Uq generated by ωi±1 , ωi±1 for i ∈ I. Moreover, Let Uq≤0 (respectively, Uq≥0 ) be the subalgebra of Uq generated by the elements ei , ωi±1 for i ∈ I (respectively, fi , ωi±1 for i ∈ I). It is clear that Uq0 , Uq±0 are commutative algebras. Similarly, we can ˜q− , U ˜q0 , etc. For each µ ∈ Q, we can define the elements ωµ and ωµ by ˜q+ , U define U µ µi ωµ = ωi i , ωµ = ωi if µ =
if µ =
i∈I
i∈I
i∈I
i∈I
µi αi ∈ Q. For any µ, ν ∈ Q, we denote µν qµν := qiji j µi αi and ν = deg ei = αi ,
i,j∈I
j∈I
νj αj . Let
deg fi = −αi ,
Then
Uq± =
deg ωi±1 = deg ωi±1 = 0. (Uq± )±β ,
β∈Q+
where
−1 (Uq± )±β = x ∈ Uq± ωµ xω−µ = qµβ x, ωµ xω−µ = qβµ x, ∀ µ ∈ Q .
2.7. (Skew) Hopf pairings. For i ∈ I, we define a linear form τi on Uq≥0 by τi (ei ωµ ) =
qii , 1 − qii
for all µ ∈ Q,
and τi (Uν≥0 ) = 0,
for all ν ∈ Q with ν = αi .
For each sequence J = (β1 , . . . , βl ) of simple roots, let τJ = τβ1 · · · τβl , and for J = ∅, τJ = 1. Then τi (eJ ωµ ) =
deg J = β1 + · · · + βl .
qii 1−qii ,
0,
if J = (αi ), otherwise.
For any µ ∈ Q, let kµ : Uq≥0 → K be the algebra homomorphism with kµ (xKν ) = ε(x)qνµ
for all ν ∈ Q and x ∈ Uq+ .
Then we have for all sequences J of simple roots and all µ ∈ Q, qνµ , if J = ∅; kµ (eJ ων ) = 0, otherwise.
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MULTI-PARAMETER QUANTUM GROUPS
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Lemma 17. (1) For all sequences J, J of simple roots and all µ ∈ Q, we have τJ (eJ ωµ ) = τJ (eJ ) and if deg(J) = deg(J ), then τJ = 0. (2) For all µ, ν ∈ Q and all sequences J of simple roots, we have kµ kν = kµ+ν ,
kµ τJ = q|J|µ τJ kµ .
Elements fJ ωµ with all finite sequences J of simple roots and µ ∈ Q form q≤0 −→ (Uq≥0 )∗ with q≤0 . Then there is a unique linear map ψ : U a basis of U ψ(fJ ωµ ) = τJ kµ for all J and µ. Since ) = q|J |µ τJ+J kµ+ν , ψ(fJ ωµ fJ ων ) = q|J |µ ψ(fJ+J ωµ+ν
ψ(fJ ωµ )ψ(fJ ων ) = τJ kµ τJ kν = q|J |µ τJ+J kµ+ν . We have, for all J, J and µ, ν, ψ(fJ ωµ fJ ων ) = ψ(fJ ωµ )ψ(fJ ων ), which implies that ψ is in fact an algebra homomorphism. Now we define a bilinear q≤0 × Uq≥0 −→ K by pairing , : U q≤0 , x ∈ Uq≥0 . y, x = ψ(y)(x) for all y ∈ U Then we have for all J, J , µ and ν, fJ ωµ , eJ ων = τJ kµ (eJ ων ). Moreover, we have yωµ , xων = qνµ y, x. and if µ, ν ∈ Q with µ = ν, then y, x = 0,
q− )−ν . for all x ∈ (Uq+ )µ , y ∈ (U
q≤0 , we have Lemma 18. For all x, x1 , x2 ∈ Uq≥0 and all y, y1 , y2 ∈ U y1 y2 , x = y1 ⊗ y2 , ∆(x),
y, x1 x2 = ∆(y), x2 ⊗ x1 .
Lemma 19. For all x ∈ Uq≥0 and i = j ∈ I, we have u− ij , x = 0. Proof. It suffices to prove u− ij , eJ = 0 with |J| = (1 − aij )αi + αj . We have J = (γ, J ) with γ ∈ {αi , αj } where J is the sequence with |J | = |J| − γ. Hence |J| = |J | and |J | = 0. Then, by Lemma 16, − u− ij , eJ = ∆(uij ), eJ ⊗ eγ 1−aij
= u− ij ⊗ ωi
ωj + 1 ⊗ u− ij , eJ ⊗ eγ
1−aij
= u− ij , eJ ωi = 0. This completes the proof.
ωj , eγ + 1, eJ u− ij , eγ
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Y. PEI, HU,AND ANDROSSO M. ROSSO PEI,N.HU,
Theorem 20. There exists a unique bilinear pairing , q : Uq≤0 × Uq≥0 → K such that for all x, x ∈ Uq≥0 , y, y ∈ Uq≤0 , µ, ν ∈ Q, and i, j ∈ I y, xx q = ∆(y), x ⊗ xq , yy , xq = y ⊗ y , ∆(x)q , qii , fi , ej q = δij 1 − qii ωµ , ων q = qνµ , ωµ , ei q = 0, fi , ωµ q = 0. q≤0 modulo the ideal generated by u− for Proof. Since is isomorphic to U ij any i = j, and by Lemma 19, we have a homomorphism ψ¯ : Uq≤0 −→ (Uq≥0 )∗ . ¯ It is easy to Then we get a bilinear pairing of Uq≤0 and Uq≥0 via y, xq = ψ(y)(x). see that the pairing satisfies all the properties as desired. Uq≤0
For any two Hopf algebras A and B paired by a skew-dual pairing , , one may consider the Drinfeld double construction D(A, B, , ), which is a Hopf algebra whose underlying vector space is A ⊗ B with the tensor product coalgebra structure and the algebra structure defined by SB (b(1) ), a(1) b(3) , a(3) aa(2) ⊗ b(2) b , (a ⊗ b)(a ⊗ b ) = for a, a ∈ A and b, b ∈ B, and whose antipode S is given by S(a ⊗ b) = (1 ⊗ SB (b))(SA (a) ⊗ 1). Therefore we have Corollary 21. Uq (g) is isomorphic to the Drinfeld double D(Uq≥0 , Uq≤0 , , q ). 2.8. Triangular decomposition. By the same argument as Coro. 2.6 in [7], we have Corollary 22. Uq (g) has a triangular decomposition: Uq (g) Uq− ⊗ Uq0 ⊗ Uq+ . 2.9. Hopf 2-cocycle deformation. Let (H, m, ∆, 1, ε, S) be a Hopf algebra over a field F . The bilinear form σ : H × H → F is called a (left) Hopf 2-cocycle of H if (23) (24)
σ(a, 1) = σ(1, a) = ε(a), ∀ a ∈ H, σ(b1 , c1 )σ(a, b2 c2 ), σ(a1 , b1 )σ(a2 b2 , c) =
∀ a, b, c ∈ H.
Let σ be a Hopf 2-cocycle on (H, m, ∆, 1, ε, S), σ −1 the inverse of σ under the convolution product. So, by [16], we can construct a new Hopf algebra (H σ , mσ , ∆, 1, ε, S σ ), where H = H σ as coalgebras, and mσ (a ⊗ b) = (25) σ(a1 , b1 )a2 b2 σ −1 (a3 , b3 ), ∀ a, b ∈ H, (26) σ −1 (a1 , S(a2 ))S(a3 )σ(S(a4 ), a5 ), ∀ a ∈ H. S σ (a) = H and H σ are called twisted-equivalent. Consider the (standard) one-parameter quantum group Uq,q−1 (gA ) generated by Ei , Fi , Ki±1 and Ki±1 (i ∈ I) and satisfying the same relations as those in
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MULTI-PARAMETER QUANTUM GROUPS
153 9
Definition 7 except that ei , fi , ωi±1 , ωi±1 and qij are replaced by Ei , Fi , Ki±1 , Ki±1 and q di aij , respectively. Assume qii = q 2di (i ∈ I). Next we shall show that Uq,q−1 (gA ) is twistedequivalent to Uq (gA ). Proposition 27. Let σ : Uq,q−1 (gA ) × Uq,q−1 (gA ) → K be a bilinear form on Uq,q−1 (gA ) defined by 1 2 qµν , x = Kµ or Kµ , y = Kν or Kν , σ(x, y) = 0, otherwise. Then σ is a Hopf 2-cocycle of Uq,q−1 (gA ). Proof. Let x, y, z be any homogenous elements in Uq,q−1 (gA ). If x, y, z ∈ 0 Uq,q / −1 , it is easy to check that the cocycle conditions (23) and (24) hold. If x ∈ 0 0 0 / Uq,q−1 ⊗ Uq,q−1 . Uq,q−1 , then we can assume ∆(x) = a ⊗ b + · · · such that a ⊗ b ∈ 0 0 and by ∈ / U , Since a ∈ / Uq,q −1 2 q,q −1 σ(a, y1 )σ(b, y2 z) = 0.
0 Hence, σ(x1 , y1 )σ(x2 y2 , z) = 0. Since x ∈ / Uq,q −1 , σ(y1 , z1 )σ(x, y2 z2 ) = 0. Therefore, σ also satisfies the cocycle conditions (23) and (24). Similarly, if y or 0 z∈ / Uq,q −1 , we can show that σ satisfies the cocycle conditions. Theorem 28. Let σ be the Hopf 2-cocycle defined in Proposition 27. Then we have the following Hopf algebra isomorphism: σ Uq (gA ) Uq,q −1 (gA ), σ where Uq,q −1 (gA ) is the deformed Hopf algebra via the Hopf 2-cocycle deformation of Uq,q−1 (gA ).
Proof. Denote a ∗ b := mσ (a, b) for a, b ∈ Uq,q−1 (gA ). It suffices to check the relations: (R∗ 1)
Ki±1 ∗ Kj±1 = Kj±1 ∗ Ki±1 ,
(R∗ 2)
Ki±1 ∗ Kj±1 = Kj±1 ∗ Ki±1 ,
(R∗ 3)
Ki ∗ Ej ∗ Ki−1 = qij Ej ,
−1 Ki ∗ Ej ∗ Ki−1 = qji Ej ,
(R∗ 4)
−1 Ki ∗ Fj ∗ Ki−1 = qij Fj ,
Ki ∗ Fj ∗ Ki−1 = qji Fj ,
(R∗ 5)
Ei ∗ Fj − Fj ∗ Ei = δi,j
1−aij ∗
(R 6)
k=0
1−aij ∗
(R 7)
k=0
Ki±1 ∗ Ki∓1 = Ki±1 Ki∓1 = 1, Ki±1 ∗ Kj±1 = Kj±1 ∗ Ki±1 ,
qii (Ki − Ki ), qii − 1
k(k−1) 1 − aij ∗(1−aij −k) k (−1) qii 2 qij Ei ∗ Ej ∗ Ei∗k = 0 (i = j), k qii k
k(k−1) 1 − aij ∗(1−aij −k) k k (−1) qii 2 qij Fi ∗ Fj ∗ Fi = 0 (i = j). k qii k
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Y. PEI, HU,AND ANDROSSO M. ROSSO PEI,N.HU,
Since ∆2 (Ki ) = Ki ⊗ Ki ⊗ Ki , ∆2 (Ki ) = Ki ⊗ Ki ⊗ Ki , ∆2 (Ei ) = Ei ⊗ 1 ⊗ 1 + Ki ⊗ Ei ⊗ 1 + Ki ⊗ Ki ⊗ Ei , ∆2 (Fi ) = 1 ⊗ 1 ⊗ Fi + 1 ⊗ Fi ⊗ Ki + Fi ⊗ Ki ⊗ Ki . It is straightforward to check (R∗ 1) and (R∗ 2). For (R∗ 3) and (R∗ 4): Ki ∗ Ej = σ(Ki , Kj )Ki Ej = σ(Ki , Kj )q di aij Ej Ki = σ(Ki , Kj )q di aij σ(Kj , Ki )−1 Ej ∗ Ki −1
1
= qij2 q di aij qji 2 Ej ∗ Ki = qij (qij qji )− 2 q di aij Ej ∗ Ki 1
aij 2
= qij (qii )−
q di aij Ej ∗ Ki
= qij q −di aij q di aij Ej ∗ Ki = qij Ej ∗ Ki , Ki ∗ Ej = σ(Ki , Kj )Ki Ej = σ(Ki , Kj )q −di aij Ej Ki = σ(Ki , Kj )q −di aij σ(Kj , Ki )−1 Ej ∗ Ki −1
1
= qij2 q −di aij qji 2 Ej ∗ Ki −1 = qji (qij qji ) 2 q −di aij Ej ∗ Ki 1
−1 = qji (qii )
aij 2
q −di aij Ej ∗ Ki
−1 di aij −di aij = qji q q Ej ∗ Ki −1 = qji Ej ∗ Ki ,
Ki ∗ Fj = σ(Ki , Kj )−1 Ki Fj = σ(Ki , Kj )−1 q −di aij Fj Ki = σ(Ki , Kj )−1 q −di aij σ(Kj , Ki )Fj ∗ Ki −1
1
= qij 2 q −di aij qji2 Fj ∗ Ki −1 = qij (qij qji ) 2 q −di aij Fj ∗ Ki 1
−1 = qij (qii )
aij 2
q −di aij Fj ∗ Ki
−1 di aij −di aij = qij q q F j ∗ Ki −1 = qij F j ∗ Ki ,
Ki ∗ Fj = σ(Ki , Kj )−1 Ki Fj = σ(Ki , Kj )−1 q di aij Fj Ki = σ(Ki , Kj )−1 q di aij σ(Kj , Ki )Fj ∗ Ki −1
1
= qij 2 q di aij qji2 Fj ∗ Ki = qji (qij qji )− 2 q di aij Fj ∗ Ki 1
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MULTI-PARAMETER QUANTUM GROUPS
= qji (qii )−
aij 2
q di aij Fj ∗ Ki
= qji q −di aij q di aij Fj ∗ Ki = qji Fj ∗ Ki . For (R∗ 5): Ei ∗ Fj − Fj ∗ Ei = Ei Fj − Fj Ei = δi,j
qii (Ki − Ki ). qii − 1
For (R∗ 6): ∗(1−aij −k)
Ei
(aij −1)aij 4
∗ Ej ∗ Ei∗k = qii
1−aij −k 2
qij
k
1−aij −k
qji2 Ei
Ej Eik .
Hence
1−aij
k=0
1−aij
=
k=0
k(k−1) 1 − aij k ∗(1−aij −k) (−1) qii 2 qij Ei ∗ Ej ∗ Ei∗k k qii k
(aij −1)aij 1−aij −k k(k−1) k 1 − aij 1−a −k k (−1) qii 2 qij qii 4 qij 2 qji2 Ei ij Ej Eik k qii k
(aij −1)aij 4
= qii
(aij −1)aij 4
= qii
(aij −1)aij 4
= qii
1−aij 2
qij
1−aij 2
qij
1−aij 2
qij
1−aij
k=0 1−aij
k=0
k(k−1) 1 − aij k 1−a −k (−1) qii 2 (qij qji ) 2 Ei ij Ej Eik k qii k
k=0 1−aij
k(k−1) k k 1 − aij 1−a −k (−1) qii 2 qij2 qji2 Ei ij Ej Eik k qii k
1 − aij 1−a −k (−1) q di k(k−1+aij ) Ei ij Ej Eik k 2d q i k
= 0. For (R∗ 7): Since ∗(1−aij −k)
Fi∗k ∗ Fj ∗ Fi
(1−aij )aij 4
= qii
−
qji
1−aij −k 2
−k
1−aij −k
qij 2 Fik Fj Fi
.
Therefore,
1−aij
k=0 1−aij
=
k=0
k(k−1) 1 − aij ∗(1−aij −k) k (−1) qii 2 qij Fi∗k ∗ Fj ∗ Fi k qii k
(1−aij )aij 1−aij −k k(k−1) 1 − aij − −k 1−a −k k (−1) qii 2 qij qii 4 qji 2 qij 2 Fik Fj Fi ij k qii
(1−aij )aij 4
= qii
k
aij −1 2
qji
1−aij
k=0
k(k−1) k k 1 − aij 1−a −k (−1) qii 2 qij2 qji2 Fik Fj Fi ij k qii k
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156 12
Y. PEI, HU,AND ANDROSSO M. ROSSO PEI,N.HU,
=
=
(1−aij )aij 4
qii
(1−aij )aij 4
qii
(1−aij )aij 4
=
qii
=
0.
aij −1
aij −1
k(k−1) k 1 − aij 1−a −k qii 2 (qij qji ) 2 Fik Fj Fi ij k qii
(−1)k
aij k k(k−1) 1 − aij 1−a −k qii 2 qii 2 Fik Fj Fi ij k qii
(−1)k
1 − aij 1−a −k q di k(k−1+aij ) Fik Fj Fi ij k 2d q i
k=0 1−aij
qji 2 aij −1
(−1)k
1−aij
qji 2
k=0 1−aij
qji 2
k=0
The proof is complete. 3. Representation Theory When gA is of finite type, we denote qµν =
µν qij
i,j∈I
for µ = ν = i∈I µi αi ,
i∈I νi αi ∈ Λ. When gA is of affine type, let I = {0, 1, · · · , l} and Λ = i∈I ZΛi such that Λi (hj ) = δi,j for i, j ∈ I, where Λi is the ith fundamental weight of gA . Let qΛ0 αi , qαi Λ0 ∈ K (i ∈ I) such that δ
∀ i ∈ I.
qΛ0 αi qαi Λ0 = qiii,0 ,
(29)
Now we can define qµν for µ, ν ∈ Λ as above. q 3.1. Category Oint . q consists of Uq (gA )-modules V q with the Definition 30. The category Oint following conditions satisfied: (1) V q has a weight space decomposition V q = λ∈Λ Vλq , where −1 Vλq = {v ∈ V q | ωi v = qαi λ v, ωi v = qλα v, ∀ i ∈ I} i
and dim Vλq < ∞ for all λ ∈ Λ. (2) There exist a finite number of elements λ1 , . . . , λt ∈ Λ such that wt(V q ) ⊂ D(λ1 ) ∪ · · · ∪ D(λt ), where D(λi ) := {µ ∈ Λ | µ < λi }. (3) ei and fi are locally nilpotent on V q . The morphisms are taken to be usual Uq (gA )-module homomorphisms. Lemma 31. For any λ ∈ Λ, and i ∈ I, we have λ(hi )
(32)
qαi λ qλαi = qii
.
Proof. It suffices to prove Λ (hi )
By (29), let λj =
qαi Λj qΛj αi = qiij
δ
∀ i, j ∈ I.
= qiiij ,
xkj αk . Then x x qikkj qkikj = (qik qki )xkj qαi λj qλj αi = k∈I
k∈I
= (qii ) This completes the proof.
k∈I
k∈I
aik xkj
=
k∈I δ qiiij .
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MULTI-PARAMETER QUANTUM GROUPS
Lemma 33. For any i ∈ I, m ∈ Z and m ≥ 1, we have qii (34) fim−1 (m)q−1 ωi − (m)qii ωi , ei fim = fim ei + ii qii − 1 q ii m−1 m −1 em f = f e + ω − (m) ω (35) (m) e i i qii i i i i . qii qii − 1 i Proof. For m = 1, it is the relation (R6). For m > 1, we have qii fim−1 (m)q−1 ωi − (m)qii ωi . ei fim = fim ei + ii qii − 1 Then
qii fim−1 (m)q−1 ωi − (m)qii ωi fi ei fim+1 = fim ei fi + ii qii − 1 qii qii −1 (ωi − ωi ) + fim qii = fim fi ei + (m)q−1 ωi − qii (m)qii ωi ii qii − 1 qii − 1 q ii f m (m + 1)q−1 ωi − (m + 1)qii ωi . = fim+1 ei + ii qii − 1 i
Similarly, the second equation holds.
For each i ∈ I, let Ui be a subalgebra of Uq (gA ) generated by ei , fi , ωi±1 , ωi±1 . Proposition 36. Let φ : Ui0 → K be a homomorphism of algebras. Denote φi := φ(ωi ),
φi := φ(ωi ),
vj := f j ⊗ vφ ∈ M (φ), j ≥ 0.
Then −j φi = 0, ∀ j ≥ 0. (i) M (φ) is a simple Ui -module if and only if φi − qii −m (ii) If φi = φi qii for m ≥ 0, then M (φ) has a unique maximal submodule N = SpanK { vj | j ≥ m + 1 } ∼ = M (φ − (m + 1)αi ). (iii) The simple Ui -module L(φ) is (m+1)-dimensional. Moreover, it is spanned by v0 , v1 , · · · , vm such that −j ωi .vj = φi qii vj , j−m ωi .vj = φi qii vj ,
fi .vj = vj+1 , (vm+1 = 0), ei .vj =
−m+1 φi qii (m
− j + 1)qii (j)qii vj−1 ,
(v−1 = 0).
(iv) Any (m + 1)-dimensional simple Ui -module is isomorphic to L(φ) for some φ.
−νi and Ui -module L(νi Λi ) (v) Let ν = i∈I νi Λi ∈ Λ+ . Then νˆ(ωi ) = νˆ(ωi )qii 0 is (νi + 1)-dimensional and φi = νˆ(ωi ). Here νˆ : U → K is the algebra homomor−1 , ∀ i ∈ I. phism such that νˆ(ωi ) = qαi µ , νˆ(ωi ) = qµα i
Proof. Similar to the argument of two-parameter cases (see [8]), in particular, for (v), by Lemma 31, we have νˆ(ωi ) ν i λi (ωi ) −ν(h ) −νi −1 −1 = qΛ , q = qii i = qii = i ν νΛi νˆ(ωi ) ν i λi (ωi )
The proof is complete.
∀ i ∈ I.
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158 14
Y. PEI, HU,AND ANDROSSO M. ROSSO PEI,N.HU,
Proposition 37. Let λ ∈ Λ+ . Let V q (λ) be an irreducible highest module with highest weight vector vλ . Then λ(hi )+1
fi
vλ = 0,
∀ i ∈ I.
Proof. By Lemma 33, −1 )fim−1 .vλ ei fim .vλ = ((m)q−1 qαi λ − (m)qii qλα i ii
−1 −m+1 = (m)qii qλα (qii qαi λ qλαi − 1)fim−1 .vλ . i
By Lemma 31, λ(hi )+1
ei fi
.vλ = 0.
By Lemma 33, λ(hi )+1
ej fi
.vλ = 0,
∀j = i.
λ(h )+1 fi i .vλ
If 0, then there exists a nontrivial submodule, contradicting the = irreducibility of V q (λ). Let V q (λ) be an irreducible highest module with Corollary 38. Let λ ∈ Λ+ .
highest weight vector vλ . Let β = i∈I mi αi ∈ Q+ such that λ(hi ) ≥ mi , ∀ i ∈ I. Then for any x ∈ (Uq− )−β , the map x → x.vλ is injective. Proposition 39. Let V q (λ) be an irreducible highest module with highest weight vector vλ . Then V q (λ) is integrable if and only if for every i ∈ I, there exists some Ni such that fiNi .vλ = 0. Proof. It is clear that ei (i ∈ I) are locally nilpotent on any highest weight module. It suffices to show that fi (i ∈ I) are locally nilpotent on V q (λ). Let j = i. We shall show that for N ≥ 1 − aij , (40) fiN fj ∈ Kfim fj fin . m+n=−aij ,N +aij ≤n≤N
For N = 1 − aij , it is just q-Serre relation (R7). Assume for N ≥ 1 − aij , the claim holds. For N + 1, by induction, Kfim+1 fj fin . fiN +1 fj ∈ m+n=−aij ,N +aij ≤n≤N
By q-Serre relation (R7), 1−aij
fi
N +aij
fj fi
∈
t+N +aij
Kfis fj fi
.
s+t=1−aij ,1≤t≤1−aij
Then (40) holds. For a sufficiently large N , fiN y ∈ Uq− fiNi , y ∈ Uq− . Note that every element of V q can be written in the form yvλ , y ∈ Uq− . This completes the proof. Proposition 41. Let V q (λ) be an irreducible highest module with highest q if and only if λ ∈ Λ+ . weight vector vλ . Then V q (λ) belongs to category Oint Proof. By Propositions 37 and 39, we get the “if” part. Now we shall show the “only if” part. It suffices to prove (λ, αi∨ ) ≥ 0 for any i ∈ I. Since fi is locally
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159 15
MULTI-PARAMETER QUANTUM GROUPS
nilpotent on V q (λ), there exists some mi ≥ 0 such that fimi +1 .v = 0 and fimi .v = 0 for i ∈ I. By Lemma 33 and ei .v = 0, we have qii fimi (mi + 1)q−1 ωi − (mi + 1)qii ωi v 0 = ei fimi +1 .v = fimi +1 ei .v + ii qii − 1 qii −1 = . fimi .v (mi + 1)q−1 qαi λ − (mi + 1)qii qλα i ii qii − 1 λ(hi )
mi mi Hence qii = qαi λ qλαi . With the help of Lemma 31, we have qii = qii qii (i ∈ I) are not roots of unity, λ(hi ) = mi .
. Since
Lemma 42. (1) Let y ∈ (Uq− )−β such that [ei , y] = 0 for all i ∈ I. Then x = 0. (2) Let x ∈ (Uq+ )β such that [fi , x] = 0 for all i ∈ I. Then x = 0. Proof. Let y ∈ (Uq− )−β such that [ei , y] = 0 for all i ∈ I. By Corollary 38, we can choose a sufficiently large λ ∈ Λ+ such that (Uq− )−β −→ V q (λ),
u → u.vλ
q
is injective. Here V (λ) is an irreducible highest module with highest weight vector vλ . yvλ generates a submodule of V q (λ). Since V q (λ) is irreducible, yvλ = 0, which implies y = 0. Using the anti-automorphism Ψ of Uq (g) in Lemma 11, we can prove (2) directly. 3.2. Skew derivations. By coproduct, we have ∆(x) ∈ (Uq+ )β−ν ων ⊗ (Uq+ )ν , for all x ∈ (Uq+ )β , 0≤ν≤β
For i ∈ I and β ∈ Q , we can define the skew-derivations ∂ˆi , i ∂ˆ : (U + )β −→ (U + )β−α +
q
such that ∆(x) = x ⊗ 1 +
q
i
∂ˆi (x) ωi ⊗ ei + the rest,
i∈I
∆(x) = ωβ ⊗ x +
ˆ ei ωβ−αi ⊗ i ∂(x) + the rest,
i∈I
where in each case “the rest” refers to terms involving products of more than one ej in the second (resp. first) factor. Let qii ˆ qii ˆ ∂i , i ∂ := ∂i := i∂ 1 − qii 1 − qii Lemma 43. For all x ∈ (Uq+ )β , x ∈ (Uq+ )β , and y ∈ Uq− , we have the following relations: (i) ∂i (xx ) = qαi β ∂i (x) x + x ∂i (x ), (ii) i ∂(xx ) = i ∂(x) x + qβαi x i ∂(x ), (iii) fi y, xq = y, i ∂(x)q , (iv) yfi , xq = y, ∂i (x)q , (v) fi x − xfi = ∂i (x) ωi − ωi i ∂(x). Proof. It is straightforward to check.
Proposition 44. For each β ∈ Q+ , the restriction of pairing , q to (Uq− )−β × is nondegenerate.
(Uq+ )β
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160 16
Y. PEI, HU,AND ANDROSSO M. ROSSO PEI,N.HU,
Proof. We use induction on β with respect to the usual partial order: β ≤ β if β − β ∈ Q+ . The claim holds for β = 0, since 1, 1q = 1. Assume that β ≥ 0, and the claim holds for all α with 0 ≤ α < β. Let x ∈ (Uq+ )β with y, xq = 0 for all y ∈ (Uq− )β . In particular, we have for all y ∈ (Uq− )−β+αi that fi y, xq = 0,
yfi , xq = 0,
i ∈ I.
It follows from Lemma 43 (iii) and (iv) that fi y, xq = y, i ∂(x)q = 0,
yfi , xq = y, ∂i (x)q = 0.
By the induction hypothesis, we have i ∂(x) = ∂i (x) = 0, and it follows from Lemma 43 (v) that fi x = xfi for all i. Now Lemma 42 applies to give x = 0, as desired. d
β 3.3. By Proposition 44, we can take a basis {uβk }k=1 , (dβ = dim (Uq+ )β ) of
dβ {vkβ }k=1
(Uq+ )β , and the dual basis y ∈ (Uq− )−β , (45)
x=
dβ
of (Uq− )−β . Then, for any x ∈ (Uq+ )β and
vkβ , xq uβk ,
y=
k=1
dβ
y, uβk q vkβ .
k=1
For β ∈ Q , let +
(46)
Θβ =
dβ
vkβ ⊗ uβk .
k=1
Set Θβ = 0 if β ∈ Q . +
(47)
Θ=
Θβ .
β∈Q+
Lemma 48. For i ∈ I, β ∈ Q+ , (i) (ωi ⊗ ωi ) Θβ = Θβ (ωi ⊗ ωi ), (ωi ⊗ ωi ) Θβ = Θβ (ωi ⊗ ωi ), (ii) (ei ⊗ 1) Θβ + (ωi ⊗ ei ) Θβ−αi = Θβ (ei ⊗ 1) + Θβ−αi (ωi ⊗ ei ), (iii) (1 ⊗ fi ) Θβ + (fi ⊗ ωi ) Θβ−αi = Θβ (1 ⊗ fi ) + Θβ−αi (fi ⊗ ωi ).
Let Ωqβ = k S(vkβ )uβk , where S is the antipode. The quantum Casimir operator Ωq can be defined q (49) Ωq := Ωβ = S(vkβ )uβk . β∈Q+
β∈Q+
k
q
Note that Ω is well-defined. Lemma 50. Let ψ be the automorphism of Uq (gA ) defined by ψ(ωi ) = ωi ,
ψ(ωi ) = ωi ,
ψ(ei ) = ωi ωi−1 ei ,
ψ(fi ) = fi ωi
−1
ωi .
Then ψ(x)Ωq = Ωq x,
∀ x ∈ Uq .
Proof. It is straightforward to check. q ) and v ∈ Vλ , we have Corollary 51. For any V ∈ Ob(Oint
(52)
−(λ+αi )(hi )
Ωq ei v = qii
ei Ωq v,
λ(hi )
Ωq fi v = qii
fi Ωq v.
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161 17
MULTI-PARAMETER QUANTUM GROUPS
Proof. For any v ∈ Vλ and i ∈ I, by Lemma 50, −(λ+αi )(hi )
q −1 ei Ωq .m = qii ψ(ei ) Ωq .v = ωi ωi−1 ei Ωq .v = qα−1 i ,λ+αi λ+αi ,αi ψ(fi ) Ωq .v = fi ωi
−1
λ(hi )
ωi Ωq .v = qαi λ qλαi fi Ωq .v = qii
ei Ωq .v,
fi Ωq .v.
This completes the proof. Note that the following fact: a
a
∀ i, j ∈ I,
qij qji = qiiij = qjjji = qji qij , and
∀ i, j ∈ I.
di aij = dj aji , Then 1 d
1 d
∀ i, j ∈ I.
qiii = qjjj ,
(53) 1 di
q ), we can define Let t = qii , ∀ i ∈ I. For V q ∈ Ob(Oint
Ξq : V q −→ V q such that for vµ ∈ Vµq , i ∈ I,
Ξq vµ = g(µ)vµ ,
(54) where g(µ) = t
(µ+ρ,µ+ρ) 2
.
q ), then the action of Ωq · Ξq : V q −→ V q Proposition 55. For V ∈ Ob(Oint commutes with the action of Uq on V .
Proof. It suffices to check the result on generators. Then for v ∈ Vµq and i ∈ I, we have Ωq · Ξq (ei .v)
= g(µ + αi )Ωq ei .v −(µ+αi )(hi )
= g(µ + αi )qii
ei Ωq .v
−(µ+αi )(hi )
= g(µ + αi )g(µ)−1 qii = t
(µ+αi +ρ,µ+αi +ρ) 2
t−
(µ+ρ,µ+ρ) 2
ei Ωq Ξq .v
−
qii
(µ+αi ,αi ) di
ei Ωq Ξq .v
= t(µ+ρ,αi )+di t−(µ+αi ,αi ) ei Ωq · Ξq .v = t(µ,αi )+2di t−(µ,αi )−2di ei Ωq · Ξq .v = ei Ωq · Ξq .v. Moreover, Ωq · Ξq (fi .v)
= g(µ − αi )Ωq fi .v µ(hi )
= g(µ − αi )qii
fi Ωq .v
= g(µ − αi )g(µ)−1 qii
µ(hi )
= t
(µ−αi +ρ,µ−αi +ρ) 2
t−
fi Ωq · Ξq .v
(µ+ρ,µ+ρ) 2
(µ,αi ) di
qii
fi Ωq · Ξq .v
= t−(µ,αi ) t(µ,αi ) fi Ωq · Ξq .v = fi Ωq · Ξq .v.
We complete the proof. Lemma 56. Let λ, µ ∈ Λ . If λ ≥ µ and g(λ) = g(µ), then λ = µ. +
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Y. PEI, HU,AND ANDROSSO M. ROSSO PEI,N.HU,
Proof. Since λ ≥ µ, we can assume that λ = µ + β for some β ∈ Q+ . Then g(λ) = g(µ), t t
(λ+ρ,λ+ρ) 2
(µ+β+ρ,µ+β+ρ) 2
t
(µ+β,β)
= t = t
(µ+ρ,µ+ρ) 2 (µ+ρ,µ+ρ) 2
, ,
= 1.
Hence, (µ + β, β) = 0.
Because of µ ∈ Λ , β = 0. +
q ). Then the action of Ωq · Ξq is the scalar Lemma 57. Let V q (λ) ∈ Ob(Oint
g(λ) = t
(λ+ρ,λ+ρ) 2
.
Proof. Let vλ be the highest weight vector of V q (λ). Then Ωq · Ξq .vλ = g(λ)vλ . By Lemma 55, we have Ωq · Ξq .v = g(λ)v,
∀ v ∈ V q (λ).
By all above lemmas, similar to Lusztig [47] for the one-parameter ones, we have q ). Then V q is completely reducible. Theorem 58. Let V q ∈ Ob(Oint
q 3.4. R-matrix. Let M, M ∈ Ob(Oint ). The map
pM,M : M ⊗ M −→ M ⊗ M is defined by −1 (m ⊗ m ), ∀ m ∈ Mµ , m ∈ Mν . pM,M (m ⊗ m ) = qµν
(59)
d
β We can take a basis {uβk }k=1 , (dβ = dim (Uq+ )β ) of (Uq+ )β , and the dual basis
d
β of (Uq− )−β . Then, for any x ∈ (Uq+ )β and y ∈ (Uq− )−β , {vkβ }k=1
(60)
x=
dβ
vkβ , xq k=1
Lemma 61. Let x ∈ (62)
uβk ,
y, uβk q vkβ .
k=1
(Uq− )−β (β
∈
(Uq+ )β , y
∆(x) =
y=
dβ
∈ Q+ ). Then
viβ−γ vjγ , xq uβ−γ ωγ ⊗ uγj , i
0≤γ≤β i,j
(63)
∆(y) =
y, uβ−γ uγj q vjγ ⊗ viβ−γ ωγ . i
0≤γ≤β i,j + Denote Θβ = Θ− β ⊗ Θβ . By a direct computation, we have the following lemma
Lemma 64. For any η ∈ Q+ , (∆ ⊗ 1)Θη
=
(Θη−γ )23 (1 ⊗ ωγ ⊗ 1)(Θγ )13
0≤γ≤η
=
− + + Θ− β ⊗ Θγ ωβ ⊗ Θγ Θβ ,
β+γ=η
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163 19
MULTI-PARAMETER QUANTUM GROUPS
and (1 ⊗ ∆)Θη
=
(Θη−γ )12 (1 ⊗ ωγ ⊗ 1)(Θγ )13
0≤γ≤η
=
+ + − Θ− β Θγ ⊗ Θβ ωγ ⊗ Θβ .
β+γ=η
Let M and M ∈
q Ob(Oint ).
Define
ΘqM,M : M ⊗ M −→ M ⊗ M , and Θβ : Mλ ⊗Mµ −→ Mλ−β ⊗Mµ+β , ∀ λ, µ ∈ Λ. Note that ΘqM,M is well-defined. q Theorem 65. Let M and M ∈ Ob(Oint ). Then q q RM,M := ΘM,M ◦ pM ,M ◦ P : M ⊗ M −→ M ⊗ M
(66)
is an isomorphism of Uq -modules, where P : M ⊗ M −→ M ⊗ M is the flip map such that P (m ⊗ m ) = m ⊗ m,
(67)
∀ m ∈ M, m ∈ M .
q Proof. It is clear that RM,M is invertible. We shall show that q q ∆(x)RM,M (m ⊗ m ) = RM,M ∆(x)(m ⊗ m )
for any x ∈ Uq , m ∈ Mλ and m ∈ Mµ . In fact, it suffices to check it for generators ei , fi , ωi , ωi (i ∈ I). Here we only check this for fi , i ∈ I, similarly for ei , ωi , ωi . By Lemma 48 (iii), q −1 ∆(fi )RM,M (m ⊗ m ) = qµλ ∆(fi )Θ(m ⊗ m) −1 −1 (fi ⊗ ωi ) Θβ−αi )(m ⊗ m) + qµ,λ (1 ⊗ fi )( Θβ )(m ⊗ m) = qµλ −1 ( = qµλ
β∈Q+
−1 qαi λ ( = qµλ
β∈Q+ −1 Θβ−αi )(fi ⊗ ωi )(m ⊗ m) + qµλ (
β∈Q+
Θβ )(1 ⊗ fi )(m ⊗ m)
β∈Q+
−1 Θβ−αi )(fi m ⊗ m) + qµλ (
β∈Q+
Θβ )(m ⊗ fi m).
β∈Q+
On the other hand, q q RM,M ∆(fi )(m⊗m ) = RM,M (m ⊗ fi m + fi m ⊗ ωi m ) −1 −1 Θ(fi m ⊗ m) + qµλ−α Θ(ωi m ⊗ fi m) = qµ−α iλ i −1 −1 = qµλ qαi λ ( Θβ−αi )(fi m ⊗ m) + qµλ ( Θβ )(m ⊗ fi m). β∈Q+
β∈Q+
So the proof is complete.
q Corollary 68. For any M, M , M ∈ Ob(Oint ), we have the following quantum Yang-Baxter equation: q q q q q q R23 R12 = R23 R12 R23 . R12 q q The category Oint is a braided tensor category with the braiding RM,M .
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164 20
Y. PEI, HU,AND ANDROSSO M. ROSSO PEI,N.HU,
4. Quantum Shuffle Realization 4.1. τ -sesquilinear form on Uq+ . Proposition 69. Let τ be an involution automorphism of K such that τ (qij ) = qji , ∀ i, j ∈ I. Then there exists a unique nondegenerate τ -bilinear form ( , ) : Uq+ × Uq+ → K such that, for any i ∈ I and x, y ∈ Uq+ , (70)
(1, 1) = 1,
(xei , y) = (x, ∂i y),
(ei x, y) = (x, i ∂y).
Proof. Let ( , ) : Uq+ × Uq+ −→ K defined by (x, y) := Φ(x), yq ,
∀ x, y ∈ Uq+ ,
where , q is the skew Hopf pairing defined in Proposition 20 and Φ is the τ linear automorphism of Uq (g) defined in Lemma 11. Since Φ is τ -linear, ( , ) is τ -sesquilinear. By Lemma 43 (iii) and (iv), the condition (70) is satisfied. It is clear that ( , ) is unique and nondegenerate. Corollary 71. Let x ∈ Uq+ . If ∂i x = 0 for any i ∈ I, then x ∈ K. 4.2. Quantum shuffle algebra. Let (F, ·) be the
free associative K-algebra with 1 with generators wi (i ∈ I). For any ν = i νi αi ∈ Q, we denote by Fν the K-subspace of F spanned by the monomials wi1 · · · wir such that for any i ∈ I, the number of occurrences of i in the sequence i1 , · · · , ir is equal to νi . Then F = ⊕ν∈Q Fν with Fν is a finite dimensional K-vector space. We have Fµ Fν ⊂ Fµ+ν , 1 ∈ F0 and wi ∈ Fαi . An element x of F is said to be homogeneous if it belongs to Fν for some ν. Let |x| = ν. w[i1 , · · · , ik ] := wi1 · · · wik . Definition 72. The quantum shuffle product on F is defined by 1 x = x 1 = x,
for x ∈ F,
xwi ywj = (xwi y)wj + qαi ,ν+αj (x ywj )wi , for i, j ∈ I and x ∈ F, y ∈ Fν , µ ∈ Q+ . Lemma 73. For any i = j ∈ I and m, l ∈ Z+ , we have wi m wj wi l =
l m
k(l−t)
k l−t qij qji qii
k=0 t=0
m l (m−k+l−t)qii !(k+t)qii !× k qii t qii × wim−k+l−t wj wik+t .
Proof. See Appendix B. Proposition 74. For any i = j ∈ I, we have
1−aij
(75)
k=0
(−1)k
1 − aij k
Proof. See Appendix C.
k(k−1) 2
qii
(1−aij −k)
k qij wi
wj wi k = 0.
qii
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MULTI-PARAMETER QUANTUM GROUPS
165 21
4.3. Embedding. We will adopt a similar treatment due to Leclerc [45] used in the one-parameter setting. For w = w[i1 , · · · , ik ], let ∂w := ∂i1 · · · ∂ik and ∂w = Id for w = 1. Next we introduce a K-linear map Γ : Uq+ −→ (F, ) defined by
Γ(x) =
∂w (x)w,
∀ x ∈ (Uq+ )µ .
w∈F |w|=µ
Lemma 76. Γ is injective. Proof. Assume Γ(x) = 0 for x ∈ (Uq+ )µ . Then ∂w (x) = 0 for all |w| = µ. By Corollary 71, we have x = 0, which implies Φ is injective. Let Di ∈ End(F ) (i ∈ I) defined as Di (w[i1 , · · · , ik ]) = δi,ik w[i1 , · · · , ik−1 ].
Di (1) = 0,
Lemma 77. Each Di (i ∈ I) satisfies the relations Di (wj ) = δi,j , Di (x y) = qαi ν Di (x) y + x Di (y) for any y ∈ Fµ and x ∈ F. Proof. Let x = x wk , y = y wl . Then Di (x y) = Di (x wk y wl ) = Di ((x wk y )wl + qαk µ (x y wl )wk ) = δi,l (x wk y ) + δi,k qαk µ (x y wl ) = δi,k qαi µ (x y wl ) + (x wk Di (y)) = qαi µ Di (x) y + x Di (y).
This completes the proof. Theorem 78. For any x, y ∈ Uq+ , we have Γ(xy) = Γ(x) Γ(y).
Proof. By Proposition 74, there exists a linear map Γ : Uq+ −→ (F, ) such that Γ (ei ) = wi ,
Γ (xy) = Γ (x) Γ (y)
for i ∈ I and x, y ∈ Uq+ . By Lemmas 20 and 77, Γ ∂i = Di Γ , ∀ i ∈ I. For x ∈ Uµ+ , µ ∈ Q+ and w = w[ii , · · · , ik ] ∈ Fµ , let γw (x) be the coefficient of w in Γ (x). Then γw (x) = Di1 · · · Dik Γ (x) = Γ ∂i1 · · · ∂ik (x) = ∂w (x). Hence Γ(x) = Γ (x).
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166 22
Y. PEI, HU,AND ANDROSSO M. ROSSO PEI,N.HU,
5. Appendix 5.1. Appendix A: The proof of Lemma 16.
1−aij
∆(u+ ij ) =
(−1)k
k=0 1−aij 1−aij −k
=
k k(k−1) 1 − aij 1 − aij − k k k (−1)k qii 2 qij n k m qii qii qii n=0
m=0 m 1−aij −k−m (ei ωi
k=0
×
k(k−1) 1 − aij k qii 2 qij ∆(ei )1−aij −k ∆(ej )∆(ei )k k qii
1−a −k−m
⊗ ei ij )(ej ⊗ 1 + ωj ⊗ ej )(eni ωik−n ⊗ ek−n ) i 1−aij 1−aij −k k 1 − aij 1 − aij − k k = k m qii qii n qii m=0 n=0 k=0
k(k−1) 2
× (−1)k qii
1−a −k−m n(1−a −k−m)
1−aij −m−n
ij k n qij qij ij qii (em i ej ei ωi 1−aij 1−aij −k k 1 − aij 1 − aij − k k + k m qii qii n qii m=0 n=0
1−aij −m−n
⊗ ei
)
k=0
k(k−1) 2
× (−1)k qii
n(1−a −k−m)
1−a −m−n
1−aij −k−m
ij k n qij qji qii (em+n ωi ij ωj ⊗ ei i 1−aij 1−aij −k k 1 − aij 1 − aij − k k = m k qii qii n qii m=0 n=0
ej ek−n ) i
k=0
k(k−1) +n(1−aij −k−m) 2
× (−1)k qii
1−a −m
1−a −m−n
ij n qij ij (em i ej ei ωi 1−aij 1−aij −k k 1 − aij 1 − aij − k k + k m qii qii n qii m=0 n=0
1−aij −m−n
⊗ ei
)
k=0
k(k−1) +n(1−aij −k−m) 2
1−a −m−n
× (−1)k qii
1−a −k−m
k n m+n qij qji (ei ωi ij ωj ⊗ ei ij 1−aij 1−aij −k k 1 − aij − m − n 1 − aij m + n = n k−n qii m + n qii qii m=0 n=0
ej ek−n ) i
k=0
k(k−1) +n(1−aij −k−m) 2
× (−1)k qii
1−aij −m
1−a −m−n
1−aij −m−n
ij n (em ⊗ ei i ej ei ωi 1−aij 1−aij −k k 1 − aij − m − n 1 − aij m + n + n k−n qii m + n qii qii m=0 n=0
qij
)
k=0
k(k−1) +n(1−aij −k−m) 2
× (−1)k qii
1−a −m−n
1−aij −k−m
ej ek−n ) i
1−aij −t
1−aij −t
k n m+n qij qji (ei ωi ij ωj ⊗ ei 1−aij 1−aij −t t 1 − aij − t 1 − aij t = n t u qii qii qii t=0 u=0 n=0 (n+u)(n+u−1) +n(1−aij −u−t) 2
× (−1)u+n qii
1−a −t+n
qij ij (et−n ej eni ωi i 1−aij 1−aij −t t 1 − aij − t 1 − aij t + t u qii qii n qii t=0 u=0 n=0 (n+u)(n+u−1) +n(1−aij −u−t) 2
× (−1)u+n qii
1−aij −t
u+n n t qij qji (ei ωi
⊗ ei
1−aij −u−t
ωj ⊗ ei
)
ej eui )
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167 23
MULTI-PARAMETER QUANTUM GROUPS
1−aij
=
1−a −t qij ij
t=0
t
1−aij
+
(−1) qii
1 − aij − t u
u=0
1 − aij − t u
u=0
1−aij
=
u(u−1) 2
(−1)u qii
1−aij −t
qij
t
1−aij −t
1 − aij t
u(u−1) 2
(−1)u qii
+
t
n
1−aij −t
⊗ ei
)
t n qii qii 1−aij −t
u qij (eti ωi
1−aij −u−t
ωj ⊗ ei
ej eui )
qii n(n−1) +n(1−aij −t) 2
(−1)n qii
n=0 1−a −t t−n δt,1−aij (ei ej eni ωi ij 1−aij
1 − aij t n t qii qii
(et−n ej eni ωi i
t=0
×
n qij
qii
n(n−1) +n(1−t) 2
t=0 n=0
(−1)n qii
1−aij −t
×
n(n−1) +n(1−aij −t) 2
n
n=0
1−aij −t
×
t
n(n−1) +n(1−t) 2
(−1) qii
t=0 n=0 1−aij −t
1−aij −t
⊗ ei
n qij
1 − aij t t qii n qii
)
1 − aij t
t n qii qii
u(u−1) 1 − aij − t 1−a −t 1−a −u−t u (−1)u qii 2 qij (eti ωi ij ωj ⊗ ei ij ej eui ) u qii u=0 1−aij 1−aij 1 − aij n(n−1) 1−a −n n 1 − aij (−1)n qii 2 qij (ei ij ej eni ⊗ 1) + δt,0 = n t qii qii n=0 t=0 1−aij −t u(u−1) 1 − aij − t 1−a −t 1−a −u−t u × (−1)u qii 2 qij (eti ωi ij ωj ⊗ ei ij ej eui ) u qii u=0 1−aij 1 − aij u(u−1) 1−a 1−a −u u ⊗ 1 + (−1)u qii 2 qij (ωi ij ωj ⊗ ei ij ej eui ) = u+ ij u qii u=0 ×
1−aij
= u+ ij ⊗ 1 + ωi
ωj ⊗ u+ ij .
5.2. Appendix B: The proof of Lemma 73. If l = 0, we have wi m = (m)qii !wim . Assume that Lemma 73 holds for l. Then for l+1, we have m k m wi m wj wi l+1 = qij × k qii k=0 l l−t k(l−t) l m−k+l−t k+t × qji qii (m−k+l−t)qii !(k+t)qii !(wi wj wi ) wi . t qii t=0 Then l t=0
l−t k(l−t) qji qii
l (m−k+l−t)qii !(k+t)qii !(wim−k+l−t wj wik+t ) wi t qii
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168 24
Y. PEI, HU,AND ANDROSSO M. ROSSO PEI,N.HU,
=
l
k(l−t)
l−t qji qii
t=0
l (m−k+l−t)qii !(k+t)qii !× t qii
k+t m−k+l−t+1 wi wj wik+t × (m−k+l−t+1)qii qji qii + (k+t+1)qii wim−k+l−t wj wik+t+1 ) l l−t+1 k(l−t)+k+t l = qji qii (m−k+l−t+1)qii !(k+t)qii !wim−k+l−t+1 wj wik+t t q ii t=0 l l−t k(l−t) l + qji qii (m−k+l−t)qii !(k+t+1)qii !wim−k+l−t wj wik+t+1 t qii t=0 l l l−t k(l−t)+t+1 = qji qii (m−k+l−t)qii !(k+t+1)qii !wim−k+l−t wj wik+t+1 t+1 qii t=−1 l l−t k(l−t) l + qji qii (m−k+l−t)qii !(k+t+1)qii !wim−k+l−t wj wik+t+1 t qii t=−1 l l l l−t k(l−t) t+1 = qji qii + qii (m−k+l−t)qii !(k+t+1)qii !× t qii t+1 qii t=−1 × wim−k+l−t wj wik+t+1 l l−t k(l−t) l+1 = qji qii (m−k+l−t)qii !(k+t+1)qii !wim−k+l−t wj wik+t+1 t+1 qii t=−1 l+1 l−t+1 k(l−t+1) l+1 = qji qii (m−k+l−t+1)qii !(k+t)qii !wim−k+l−t+1 wj wik+t . t qii t=0
5.3. Appendix C: The proof of Proposition 74. By Lemma 73, we have
1−aij
k=0
k(k−1) 1 − aij k (1−aij −k) (−1) qii 2 qij wi wj wi k k qii k
1−aij 1−aij −k
=
m=0
k=0
k 1−aij 1−aij −k k k m qii qii n qii n=0
k(k−1) +m(k−n) 2
1−aij −n−m
wj wim+n
k(k−1) +m(k−n) 2
1−aij −n−m
wj wim+n
× (−1) qii
k+m k−n qij qji (1−aij −n−m)qii !(m+n)qii !wi 1−aij 1−aij −k k 1−aij −m−n 1−aij m+n = m qii k−n qii m+n qii m=0 n=0 k
k=0
× (−1)k qii
k+m k−n qij qji (1−aij −n−m)qii !(m+n)qii !wi 1−aij 1−aij −t t 1−aij −t 1−aij t = t u qii qii n qii t=0 u=0 n=0 (n+u)(n+u−1) +(t−n)u 2
× (−1)n+u qii
1−aij −t
u+t u qij qji (1−aij −t)qii !(t)qii !wi
wj wit
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MULTI-PARAMETER QUANTUM GROUPS
169 25
t 1−aij n(n−1) t t n 2 = qij (−1) qii n t qii qii t=0 n=0 1−aij −t u(u−1) 1−aij −t 1−a −t t × (−1)u qii 2 (qii qij qji )u (1−aij −t)qii !(t)qii !wi ij wj wit u qii u=0 1−aij 1−aij −t 1 − aij u(u−1) 1−aij −t t t = δt,0 qij (−1)u qii 2 (qii qij qji )u t u qii qii t=0 u=0 1−aij
1−a −t
× (1−aij −t)qii !(t)qii !wi ij wj wit 1−aij 1−aij u(u−1) 1−a = (−1)u qii 2 (qij qji )u (1 − aij )qii !wi ij wj u qii u=0 −aij
= (1 −
1−a aij )qii !wi ij wj
n (1 − qii qij qji )
n=0
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[email protected] Department of Mathematics, East China Normal University, Minhang Campus, Dong Chuan Road 500, Shanghai 200241, PR China E-mail address:
[email protected] ´matiques, Universit´ UFR de Mathe e Denis Diderot - Paris 7, 2 Place Jussieu, 75251 Paris Cedex 05, France E-mail address:
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