q-VERTEX OPERATORS FOR QUANTUM AFFINE ALGEBRAS

arXiv:math/9901026v1 [math.QA] 6 Jan 1999

q-VERTEX OPERATORS FOR QUANTUM AFFINE ALGEBRAS NAIHUAN JING AND KAILASH C. MISRA Abstract. q-vertex operators for quantum affine algebras have played important role in the theory of solvable lattice models and the quantum KnizhnikZamolodchikov equation. Explicit constructions of these vertex operators for (1) most level one modules are known for classical types except for type Cn , where the level −1/2 have been constructed. In this paper we survey these (1) (1) (1) (1) results for the quantum affine algebras of types An , Bn , Cn and Dn .

1. Introduction Algebraic conformal field theory can be simply described by the beautiful properties of the Virasoro algebra, vertex operators of affine Lie algebras, and the Knizhnik-Zamolodchikov equation. The vertex operators or intertwining operators between certain representations of affine Lie algebras stand at the core of the foundation since they also provide solutions to the KZ equation. Most advances in this part of Lie theory around 1980s circled around this interesting subject. Vertex operators also helped the introduction of the vertex algebra by Borcherds [Bo] (cf. [FLM]). In 1985 Drinfeld [D1] and Jimbo [Jb] introduced the notion of quantum group, a q-deformation of enveloping algebras of Kac-Moody algebras. This has inspired torrential activity to quantize existed phenomena in Lie theory. Lusztig first q-deformed the integrable modules of Kac-Moody algebras [L]. Then the q-deformed level one modules of quantum affine algebras were constructed by vertex representations [FJ, J1]. Quantum KZ equations were defined by FrenkelReshetikhin using the q-vertex operators [FR], and their matrix coefficients provide the solutions for the q-KZ equations. The Kyoto school used the q-vertex operators in solvable lattice models (see [JM]). They proposed that the half of the space of states H, on which Baxter’s corner transfer matrix is acting, can be identified with highest weight representations of quantum affine algebras. Then the correlation functions and the form factors can be computed in terms of the q-deformed vertex operators. Moveover their integral formulas are immediately given by explicit bosonization of the q-vertex operators. Because of its applications in statistical mechanics models and q-conformal field theory, the program for realizing various q-vertex operators attracted the attention of several researchers. It started with (1) the explicit bosonization of level one case for Uq (An ) first for n = 1 [JMMN], then for general n by Koyama [Ko1]. Kang, Koyama, and the first author [JKK] gener(1) alized the bosonization to type Dn . More recently we gave explicit bosonizations 1991 Mathematics Subject Classification. Primary: 17B. Key words and phrases. q-Vertex operators, quantum affine algebras, NJ: Research supported in part by NSA grant MDA904-97-1-0062. KCM: Research supported in part by NSA grant MDA904-96-1-0013. 1

2

NAIHUAN JING AND KAILASH C. MISRA

for other types of quantum affine algebras [JM1, JM2, JK]. Some lower rank cases of other modules are also considered in [I, Ko2]. In this work we will review the techniques involved in the realization of q-vertex operators and provide a unified description for the untwisted quantum affine algebras of classical types. To minimize the length of the paper we do not include twisted quantum affine algebras which have been considered in [JM2]. However the basic ingredients are similar in all cases. Currently two methods of deriving q-vertex operators are used: coproduct formula (cf. Theorem 3.1 ) and direct computation using commutation relations. We take the approach of coproduct to present the results, however this method is not (1) sufficient to determine the q-vertex operator in all the case (cf. type Cn in [JK]). We refer the reader to [JK] for more detailed information about the direct computation approach. Due to lack of space we do not include the intertwining operators for the Drinfeld comultiplication. Their bosonization agrees with our vertex operators in the special computable component. The rest of the components in [DI] are then derived using normal ordering and commutation relations. This paper is arranged as follows. In section two we review the basic notations. Drinfeld realization, the coproduct formula and the level zero action of the Drinfeld generators are computed in section three. In section four we discuss the construc(1) (1) (1) tions of level one modules for types An , Bn , Dn and level −1/2 modules for (1) type Cn . These constructions are used to construct the q-vertex operators in the last section. 2. Notation and Preliminaries Let A = (Aij ), i, j ∈ I = {0, 1, · · · , n} be the generalized Cartan matrix of type (1) (1) (1) (1) An , Bn , Cn , or Dn . Let b g=b g(A) be the associated Kac-Moody algebra , and hi (i ∈ I) and d be the b We define the affine root lattice of b generators [K] of the Cartan subalgebra h. g to be ˆ = Zα0 ⊕ · · · ⊕ Zαn , Q b ∗ such that < αi , hj >= Aji and < αi , d >= δi0 . The affine weight where αi ∈ h ˆ lattice P is defined to be Pˆ = ZΛ0 ⊕ · · · ⊕ ZΛn ⊕ Zδ,

where Λi (hj ) = δij , Λi (d) = 0, and δ(hj ) = 0, δ(d) = 1 for j ∈ I. The dual affine weight lattice is then defined as Pˆ ∨ = Zh0 ⊕ Zh1 ⊕ · · · ⊕ Zhn ⊕ Zd. We will denote the corresponding weight (resp. root) lattice of the finite dimensional Lie algebra g by P (resp. Q). b ∗ satisfies The nondegenerate symmetric bilinear form ( | ) on h

(2.1)

(αi |αj ) = di Aij , (δ|αi ) = (δ|δ) = 0 for all i, j ∈ I,

(1)

where di = 1 except for (d0 , · · · , dn )=(1, · · · , 1, 1/2) in the case of Bn , and (1) (d0 , · · · , dn )=(1, 1/2, · · · , 1/2, 1) in the case of Cn . 1 g) is the associaLet qi = q di = q 2 (αi |αi ) , i ∈ I. The quantum affine algebra Uq (b tive algebra with 1 over C(q 1/2 ) generated by the elements ei , fi (i ∈ I) and q h

q-VERTEX OPERATORS FOR QUANTUM AFFINE ALGEBRAS

3

(h ∈ Pˆ ∨ ) with the following relations : q 0 = 1, for h, h′ ∈ Pˆ ∨ ,

′

=

q h+h ,

q h ei q −h

=

q αi (h) ei , q h fi q −h = q −αi (h) fi for h ∈ Pˆ ∨ (i ∈ I),

ei fj − fj ei

=

δij

qh qh

X

′

1 ti − t−1 hi i 2 (αi |αi )hi and i, j ∈ I, −1 , where ti = qi = q qi − qi

(m)

(−1)m ei

(n)

ej ei

= 0,

and

m+n=1−aij

X

(m)

(−1)m fi

(n)

fj fi

m+n=1−aij

= 0 for i 6= j,

qik − qi−k . For qi − qi−1 simplicity we will denote [k]i = [k] for i = 1, · · · , n − 1. The derived subalgebra generated by ei , fi , ti (i ∈ I) is denoted by Uq′ (b g). The algebra Uq (b g) has a Hopf algebra structure with comultiplication ∆, counit u∗ , and antipode S defined by (k)

where ei

(k)

= eki /[k]i !, fi

= fik /[k]i !, [m]i ! =

∆(q h ) = ∆(ei ) = ∆(fi ) = (2.2)

u∗ (q h ) = u∗ (ei ) = S(q h ) = S(ei ) =

Qm

k=1 [k]i , and [k]i =

q h ⊗ q h for h ∈ Pˆ ∨ , e i ⊗ 1 + ti ⊗ e i ,

fi ⊗ t−1 i + 1 ⊗ fi for i ∈ I, 1 for h ∈ Pˆ ∨ ,

u∗ (fi ) = 0 for i ∈ I, q −h for h ∈ Pˆ ∨ ,

−ti−1 ei , S(fi ) = −fi ti for i ∈ I.

Let V, W be two Uq (b g)-modules. The tensor product V ⊗ W is defined as the Uq (b g)-module via the coproduct ∆. The (restricted) dual Uq (b g)-module V ∗ is defined by (x · v ∗ )(u) = v ∗ (S(x) · u) for x ∈ Uq (b g), u ∈ V , and v ∗ ∈ V ∗ .

3. Drinfeld realization and level zero modules We now recall Drinfeld’s realization of the quantum affine algebra Uq (b g) (and ′ of Uq (b g))(cf. [D2, B, J3]). Let U be the associative algebra with 1 over C(q 1/2 ) ±1 ±1/2 generated by the elements x± , q ±d (i = 1, 2, · · · , n, k ∈ Z, l ∈ i (k), ai (l), Ki , γ

4


Z \ {0}) with the following defining relations : [γ ±1/2 , u] =

[ai (k), aj (l)] = [ai (k), Kj±1 ] =

[q ±d , Kj±1 ] = 0,

−d q d x± i (k)q

=

d −d q k x± = q l ai (l), i (k), q ai (l)q

−1 Ki x± j (k)Ki

=

q ±(αi |αj ) x± j (k),

[ai (k), x± j (l)] = ± x± i (k + 1)xj (l) −

= (3.1)

0 for all u ∈ U, [Aij k]i γ k − γ −k , δk+l,0 k qj − qj−1

− [x+ i (k), xj (l)] =

[Aij k]i ∓|k|/2 ± γ xj (k + l), k ± q ±(αi |αj ) x± j (l)xi (k + 1)

±

± ± ± q ±(αi |αj ) x± i (k)xj (l + 1) − xj (l + 1)xi (k), k−l l−k δij 2 ψ (k + l) − γ 2 ϕ (k + l) γ , i i qi − qi−1

where ψi (m) and ϕi (−m) (m ∈ Z≥0 ) are defined by ∞ X

m=0 ∞ X

m=0

ψi (m)z −m = Ki exp (qi − qi−1 )

k=1

ϕi (−m)z m = Ki−1 exp −(qi − qi−1 )

and the Serre relations are: m=1−Aij

Symk1 ,··· ,km

(3.2)

P∞

·

X

(−1)r

m r

r=0 ± xi (kr+1 ) · · · x± i (km )

i

ai (k)z −k ,

P∞

k=1

ai (−k)z k ,

± ± x± i (k1 ) · · · xi (kr )xj (l)

=0

if i 6= j.

±1 We denote by U′ the subalgebra of U generated by the elements x± i (k), ai (l), Ki , ±1/2 γ (i = 1, 2, · · · , n, k ∈ Z, l ∈ Z \ {0}). Based on the general result of [D2] one can compute directly the isomorphism as given in [J3]. The ǫ-sequences correspond to reduced expressions of the longest element in the Weyl group. For details see [J3].

Proposition 3.1. [J3] Let i1 , i2 , · · · , ih−1 be a sequence of indices given in the Table. Then the C(q 1/2 )-algebra isomorphism Ψ : Uq (b g) → U is given by ei (3.3)

e0 f0 t0

− 7→ x+ i (0), fi 7→ xi (0), ti 7→ Ki for i = 1, · · · , n,

−1 − − ǫ 7→ [x− ih−1 (0), · · · , xi2 (0), xi1 (1)]qǫ1 ···q h−2 γKθ , ,

+ + ǫ 7→ a(−q)−ǫ γ −1 Kθ [x+ ih−1 (0), · · · , xi2 (0), xi1 (−1)]qǫ1 ···q h−2 ,

7→ γKθ−1 , q d 7→ q d ,

P where Kθ = Ki1 · · · Kih−1 , h is the Coxeter number, ǫ = h−2 i=1 ǫi . The constant a (1) (1) (1) (1) is 1 for simply laced types An , Dn , a = [2]1 for Cn , and a = [2]1−δ1,i1 for Bn . ′ ′ ′ The restriction of Ψ to Uq (b g) defines an isomorphism of Uq (b g) and U . Table: ǫ-Sequences for simple Lie algebras


ǫ-Sequence. ǫ =

g

−1

−1

P

ǫi

α1 → · · · → αn

An

0

ǫ -n+1

−1

−1

−1

−1

−1/2

−1

−1/2

−1/2

−1

−1

−1

−1

α1 → · · · → αn−1 → αn → · · · → α2

Bn

-2n+4

0

α1 −→ · · · → αn −→ · · · −→ α2 → α1

Cn

−1

α1 → · · · → αn → αn−2 → · · · → α2

Dn

5

-n+1 -2n+4

Through the isomorphism of Proposition 3.1 the coproduct will be carried over to the Drinfeld realization. The following result is true for all type one quantum affine algebras. s s Theorem 3.1. Let k ∈ Z≥0 , l ∈ N, and let N+ (resp. N− ) be the subalgebra of U + + − generated by the elements xi1 (−m1 ) · · · xis (−ms ) (resp. x− i1 (m1 ) · · · xis (ms )) with mi ∈ Z≥0 . Then the comultiplication ∆ of the algebra U has the following form:

∆(x+ i (k)) =

+ k 2k x+ i (k) ⊗ γ + γ Ki ⊗ xi (k)

+

k−1 X

k−j 2

γ

j=0

∆(x+ i (−l))

=

x+ i (−l)

⊗ γ −l + Ki−1 ⊗ x+ i (−l)

+

l−1 X

γ

l−j 2

j=1

∆(x− i (l))

=

x− i (l)

l−1 X

γ l−j x− i (j) ⊗ γ

k−1 X

− γ j−k x− i (−j) ⊗ γ

j=1

=

x− i (k)

j=0

∆(ai (−l)) =

j−l 2

2 ψi (l − j) (mod N− ⊗ N+ ),

⊗ γ −2k Ki−1 + γ −k ⊗ x− i (−k) +

∆(ai (l)) =

2 ϕ(−l + j) ⊗ γ −l+j x+ i (−j) (mod N− ⊗ N+ ),

⊗ Ki + γ l ⊗ x− i (l) +

∆(x− i (−k))

2 ψ(k − j) ⊗ γ k−j x+ i (j) (mod N− ⊗ N+ ),

l

ai (l) ⊗ γ 2 + γ

ai (−l) ⊗ γ

− 3l 2

3l 2

k+3j 2

2 ϕi (j − k) (mod N− ⊗ N+ ),

⊗ ai (l) (mod N− ⊗ N+ ), l

+ γ − 2 ⊗ ai (−l) (mod N− ⊗ N+ ).

Moreover the same formulas are true for the derived subalgebra U′ . Let V be the finite dimensional representation described by the following representation graphs. Note that the graphs are crystal graphs, and they are also perfect (1) crystals [KMN] except in the case of Cn . Let vi be the basis elements (i ∈ I), and I is the index set. Let Eij be the matrix units. We can read the actions from the graphs. The Uq (b g)-module structure on the affinization or evaluation module of V [KMN] is given as follows. We equip the affinization Vz = V ⊗ C(q 1/2 )[z, z −1 ] with the following actions: (3.4)

ei (v ⊗ z m )

= ei v ⊗ z m+δi,0 , fi (v ⊗ z m ) = fi v ⊗ z m−δi,0 ,

q d (v ⊗ z m )

= qm v ⊗ z m,

ti (v ⊗ z m )

for i ∈ I, v ∈ V .

= ti v ⊗ z m ,

6


0 (1)

An

1

1-

2-

2

n-1-

pppppppppp

n

n-

n+1

0 (1)

Bn

n n 1 2 n-1 n-1 2 1 1 - 2 - p p p p p -n -0 -n - p p p p p -2 -1 Y 0

0 (1)

Cn

1 n 2 n-1 n-1 2 1 1 - 2 - p p p p p p p p -n -n - p p p p p p p p -2 -1

0 (1) Dn

1 Y

1 -

n-1 n n @ 2 R @ n-2 n-2 2 - p p - n−1 n−1 - p p n@ n-1 Rn @

2 1 2 -1

0

Figure 1. Representation graphs

The evaluation module Vz is a level zero Uq (b g)-module, i.e., γ acts as identity (= q 0 ). Through the isomorphism Ψ the evaluation module is also a U-module. The action of the Drinfeld generators are given by the following result. Define τ = τ (A) as follows:   n+1    2n − 1 τ=  2n + 2    2n − 2

(1)

An (1) Bn (1) Cn (1) Dn .

We define the content of each vertex in the representation graph by c(i) =

i τ −i

if i < n if i ∈ {1, · · · , n}

Here we adopt the convention that i = i.


7

Theorem 3.2. [Ko1, JKK, JM, JK] For j ∈ {1, · · · , n}, the Drinfeld generators act on the evaluation module Vz as follows. x+ j (k) = x− j (k) = aj (l) = x+ n (k) = (3.5) x− n (k) =

an (l) =

P

P(i(j),e(j))

c(j)

(qj

z)k Ei(j),e(j)

c(j)

(q z)k Ee(j),i(j) P(i(j),e(j)) [l]j c(j) l −l (q z) (qj Ei(j),i(j) − qjl Ee(j),e(j) )  (i(j),e(j)) l j (1) n k n−1 k  z) E0,n , Bn  (q z) [2]n En,0 + (q (1) (q n−1 z)k En,n , Cn  (1)  n−1 k Dn z) (En−1,n + En,n−1 ),  (q (1) n−1 k  z) [2]n En,0 + (q n z)k E0,n Bn  (q (1) Cn (q n−1 z)k En,n ,  (1)  n−1 k Dn (q z) (En,n−1 + En−1,n ),  [2l]  l n (q n−1 z)l ((En,n − q l E0,0 ) + (E0,0 − q l En,n ))    [l] n−1 l −l z) (q Enn − q l En,n ), l (q [l] n−1 l −l  (q z) (q En−1,n−1 − q l En,n    l +q −l En,n − q l En−1,n−1 )

(1)

Bn (1) Cn

(1)

Dn

j

where the summation runs through all possible pairs i(j) −→ e(j) in the represen(1) tation graphs connected by j. j = 1, 2, · · · , n − 1 (but we allow j = n for type An ), k ∈ Z and l ∈ Z \ {0}. Theorem 3.3. [Ko1, JKK, JM, JK] The Drinfeld generators act on the dual evaluation module Vz∗ = V ∗ ⊗ C(q 1/2 )[z, z −1] as follows. (3.6) x+ j (k) = x− j (k) = aj (l) = x+ n (k) =

x− n (k) =

an (l) =

P −c(j) k ∗ z) Ee(j),i(j) (−q −1 ) (i(j),e(j)) (qj P −c(j) k ∗ z) Ei(j),e(j) (−q) (i(j),e(j)) (qj P [l] −c(j) l −l ∗ ∗ z) (qj Ee(j),e(j) − qjl Ei(j),i(j) ) (q  (i(j),e(j)) l j (1) −1 −n k ∗ −n+1 k ∗  (−q )((q z) [2] E + (q z) qE ), Bn n 0,n n,0  (1) ∗ (−q −1 )(q −n+1 z)k En,n , Cn  (1)  (−q −1 )(q −n+1 z)k (E ∗ ∗ Dn n,n−1 + En−1,n ),  (1) ∗ −n+1 k ∗ + (q −n z)k En,0 ) Bn z) [2]n q −1 E0,n   (−q)((q (1) ∗ (−q)(q −n+1 z)k En,n , Cn  (1)  (−q)(q −n+1 z)k (E ∗ ∗ Dn n−1,n + En,n−1 ),  (1) ∗ ∗ ∗ ∗  [2l]n (q −n+1 z)l ((q −l E0,0 − E0,0 )), Bn − En,n ) + (q −l En,n  l   (1) [l]  (q −n+1 z)l (q −l E ∗ − q l E ∗ ), Cn nn n,n l [l] −n+1 l −l ∗ ∗  z) (q En,n − q l En−1,n−1  l (q   (1) ∗ ∗  − q l En,n ) Dn +q −l En−1,n−1 j

where the summation runs through all possible pairs i(j) −→ e(j) in the represen(1) tation graphs connected by j. j = 1, 2, · · · , n − 1 (but we allow j = n for type An ), k ∈ Z and l ∈ Z \ {0}.

8


4. Vertex representation of Uq (b g) (1)

(1)

In this section we recall the level one realizations of Uq (b g) for b g = An , Dn as (1) given in [FJ] and b g = Bn in [JM1] (cf. [Be] for V (Λ0 )). We also recall the level (1) −1/2 realization of Uq (Cn ) as given in [JKM]. The quantum affine algebra Uq (b g) (1) has level one irreducible representations: V (Λi ), i = 0, 1, · · · , n for An ; i = 0, 1, n (1) (1) for Bn and i = 0, 1, n − 1, n for Dn . The weight lattice P has two coset representatives Q and Q+λn , where λi denote the fundamental weights in P . We will express any level one fundamental weight λi as a linear combination of the simple roots αi and λn . Let ε: Q × Q −→ Z2 = {±1} be a cocycle such that ε(α, β)ε(β, α) = (−1)(α|β)+(α|α)(β|β),

(4.1)

whose existence can be constructed directly on the simple roots. Then there corresponds a central extension of the group algebra C(Q) associated with ε such that 1 −→ Z2 −→ C{Q} −→ C(Q) −→ 1. α If we still use e to denote the generators of C{Q}, then we have eα eβ = (−1)(α|β)+(α|α)(β|β)eβ eα .

(4.2)

for any α, β ∈ Q. Moreover we define the vector space C{P } = C{Q} ⊕ C{Q}eλn as a C{Q}module by formally adjoining the element eλn . For α ∈ P define the operator ∂α on C{P} by ∂α eβ = (α|β)eβ

(4.3)

ˆ be the infinite dimensional Heisenberg algebra generated by ai (k) and Let Uq (h) the central element γ (k ∈ Z\{0}, i = 1, · · · , n) subject to the relations (4.4)

[ai (k), aj (l)] = δk+l,0

[Aij k]i γ k − γ −k . k qj − qj−1

ˆ acts on the space of symmetric algebra Sym(h ˆ − ) generated by The algebra Uq (h) ai (−k) (k ∈ N, i = 1, · · · , n) with γ = q by the action: ai (−k) 7→ multiplication by ai (−k), X [Aij k]i q k − q −k d ai (k) 7→ . −1 k qj − qj d aj (−k) j

(1)

For the case of Bn we need to consider an extra fermionic field. For Z = Z + s (s = 1/2, NS-case; s = 0, R-case), let Cqs be the q-deformed Clifford algebra generated by κ(k), k ∈ Z satisfying the relations: (4.5)

{κ(k), κ(l)} = (q k + q −k )δk,−l ,

where k, l ∈ Z and {a, b} = ab + ba. Let Λ(Cq− )s be the polynomial algebra generated by κ(−k), k ∈ Z>0 , and Λ(Cq− )s0 (resp. Λ(Cq− )s1 ) be the subalgebra generated by products of even (resp. odd) number of generators κ(−k)’s. Then Λ(Cq− )s = Λ(Cq− )s0 ⊕ Λ(Cq− )s1 .


9

The algebra Cqs acts on Λ(Cq− )s canonically by (k ∈ Z>0 ) κ(−k) 7→ multiplication by κ(−k), d κ(k) 7→ − . d κ(−k) We define that V (Λi ) = V (Λ0 ) = V (Λ1 ) = V (Λn ) =

(1)

(1)

ˆ − ) ⊗ (C{Q}eλi , An , Dn Symh − ˆ ) ⊗ (C{Q0 } ⊗ Λ(C − )1/2 ⊕ C{Q0 }eλ1 ⊗ Λ(C − )1/2 ), Bn(1) Sym(h q 1 q 0 1/2

ˆ − ) ⊗ (C{Q0 }eλ1 ⊗ Λ(C − ) Sym(h q 0

1/2

(1)

⊕ C{Q0 } ⊗ Λ(Cq− )1 ), Bn

(1)

ˆ − ) ⊗ C{Q}eλn ⊗ Λ(C − )0 , Bn Sym(h q (1)

(1)

where i = 0, · · · , n for An and i = 0, 1, n − 1, n for Dn . C{Q0 } is the group (1) algebra of the sublattice of long roots in the case of Bn . Here for convenience we denote λ0 = 0. Proposition 4.1. [FJ][JM1] The space V (Λi ) of level one irreducible modules of the quantum affine Lie algebra Uq (b g) is realized under the following action: 7→ q, Kj 7→ q ∂αj , aj (k) 7→ aj (k) (1 ≤ j ≤ n), ∞ ∞ X X ai (−k) ∓k/2 k ai (k) ∓k/2 −k ± x± (z) → 7 X (z) = exp(± q z )exp(∓ q z ) i i [k] [k] γ

k=1

×e

±αi ±∂αi +1

z

,

k=1 (1) Bn , 1 ∞ X

1 ≤ i ≤ n; when gb = ∞ X an (−k) ∓k/2 k q z )exp(∓ x± 7 Xn± (z) = exp(± n (z) → [2k]n k=1

where κ(z) =

×e P

±αn ±∂αn +1/2

m∈Z

z

κ(m)z

−m

(±κ(z)),

for b g=

k=1 (1) Bn

≤i≤n−1

an (k) ∓k/2 −k q z ) [2k]n

. The highest weight vectors are |Λi i = eλi . (1)

We now recall the level −1/2 realization of Uq (Cn ) [JKM]. There are four level −1/2 weights: µ1 = − 21 Λ0 , µ2 = − 32 Λ0 + Λ1 − 21 δ, µ3 = − 21 Λn , µ4 = − 32 Λn + Λn−1 . Let ai (m) be the bosonic operators satisfying (4.4) with γ = q −1/2 . Let bi (m) be another set of bosonic operators satisfying the following defining relations. (4.6)

[bi (m), bj (l)] = [ai (m), bj (l)] =

mδij δm+l,0 , 0.

We define the Fock space Fα,β for α ∈ P + Z2 λn , β ∈ P by the defining relations ai (m)|α, βi = 0

(m > 0) ,

bi (m)|α, βi = 0

(m > 0) ,

ai (0)|α, βi = (αi |α)|α, βi , bi (0)|α, βi = (2εi |β)|α, βi , where |α, βi is the vacuum vector, and εi are the usual orthonormal vectors. We set M Fe = Fα,β . α∈P + 12 Zλn ,β∈P

Let eai = eαi and ebi (1 ≤ i ≤ n) be operators on Fe given by: eαi |α, βi = |α + εi , βi

,

ebi |α, βi = |α, β + εi i.

10


Let : : be the usual bosonic normal ordering defined by : ai (m)aj (l) := ai (m)aj (l) (m ≤ l), aj (l)ai (m) (m > l),

: ea ai (0) :=: ai (0)ea := ea ai (0) . and similar normal products for the bi (m)’s. Let ∂ = ∂q1/2 be the q-difference operator: f (q 1/2 z) − f (q −1/2 z) ∂q1/2 (f (z)) = (q 1/2 − q −1/2 )z We introduce the following operators. ∞ ∞ X X ai (k) ± k k ai (k) ± k −k ±ai ∓2ai (0) 4 z ) exp(∓ Yi± (z) = exp(± q q 4 z )e z , 1 [− 2 k] [− 21 k] k=1 k=1 Zi± (z)

= exp(±

∞ X bi (−k) k=1

We define functions.

k

z k ) exp(∓

k=1

± the operators (i = 1, · · · P xi (m) ± −m−1 Xi (z) = m∈Z x± (m)z . i

Xi+ (z) =

− ∂Zi+ (z)Zi+1 (z)Yi+ (z),

Xi− (z)

+ (z)Yi− (z), Zi− (z)∂Zi+1

=

Xn+ (z) = Xn− (z) =

1 1

1

∞ X bi (k)

k

z −k )e±bi z ±bi (0) .

, n, m ∈ Z) by the following generating

i = 1, · · · n − 1

i = 1, · · · n − 1

1 1 : Zn+ (z)∂ 2 Zn+ (z) : − : ∂Zn+ (q 2 z)∂Zn+ (q − 2 z) : Yn+ (z)

q 2 + q− 2 1 − 12 − − 12 z) : Yn− (z). 1 : Zn (q z)Zn (q 1 − q2 + q 2

Remark. Our ai (k) differs from that in [JKM], where we took ai (k)/[di ] for ai (k). Furthermore, we know that Fe contains the four irreducible highest weight mod1 ules ([JKM]). Let Fα,β be the subspace of Fα,β generated by ai (m) (i = 1, · · · , n ∈ 2,j Z, m ∈ Z) Similarly, let Fα,β (j = 1, · · · , n) be the subspace of Fα,β generated by bj (m) (m ∈ Z). We can define the following isomorphism by |α, βi ⊗ |α′ , β ′ i → |α + α′ , β + β ′ i. 2,n 2,1 1 −→ Fα,β1 +···+βn . ⊗ · · · ⊗ F0,β Fα,0 ⊗ F0,β n 1

2,j 2,j Let Q− j be the operator from Fα,β to Fα,β−εj defined by I 1 √ Q− = Zi− (z)dz. i 2π −1 We set subspaces Fi (i = 1, 2, 3, 4) of Fe as follows.

F1 =

F3 = where

M

α∈Q M

α∈Q

′ Fα,α ,

F2 =

′ , Fα− 1 2 λn ,α

F4 =

′ 1 Fα,β = Fα,0 ⊗

n O j=1

M

′ Fα+ε 1 ,α+ε1

α∈Q M

α∈Q+εn

Ker F 2,j

0,lj εj

′ Fα− 1 λn ,α , 2

Q− j ,

for β = l1 ε1 + · · · + ln εn . Then we have the following proposition.


11

Proposition 4.1. ([JKM]) Each Fi (i = 1, 2, 3, 4) is an irreducible highest weight Uq -module isomorphic to V (µi ), The highest weight vectors are given by |µ1 i = |0, 0i, |µ2 i = b1 (−1)|λ1 , λ1 i, |µ3 i = | − 12 λn , 0i, |µ4 i = | − 12 λn − εn , −εn i. 5. Realization of the level one vertex operators We first recall the notion of Frenkel-Reshetikhin vertex operators [FR]. Let V be a finite dimensional representation of the derived quantum affine Lie algebra Uq′ (b g) with the associated affinization space Vz (cf. (3). Let V (λ) and V (µ) be two integrable highest weight representations of Uq (b g). The type I (resp. type ˜ ˆ z II) vertex operator is the Uq (b g)-intertwining operator Φ(z): V (λ) −→ V (µ)⊗V ∆ −∆ µ λ ˆ (µ)), which equals to Φ(z)z (resp. V (λ) −→ Vz ⊗V where Φ(z) is the Uq′ (b g)ˆ is completed in certain sense, and intertwining operator. Here the tensor product ⊗ ˆ in the sequel. ∆µ is the conformal weight for µ. For simplicity we will write ⊗ for ⊗ we will compute the intertwining operators Φ(z) for the derived subalgebra Uq′ (b g). The existence of vertex operators is given by the fundamental fact [FR] (cf. [DJO]) (true for both types, though stated for type I): HomUq (bg) (V (λ), V (µ) ⊗ Vz ) ≃ (5.1)

{v ∈ V | wt(v) = λ − µ mod δ and hµ,hi i+1

ei

v = 0 for i = 0, · · · , n},

where the isomorphism is defined as follows. We say a pair of weights (λ, µ) is admissible if they satisfy (5.1). For each admissible pair (λ, µ) there corresponds uniquely a special vector vi in the crystal graph such that wt(vi ) = λ − µ. We then normalize the corresponding vertex operator as follows. (5.2)

ΦµV λ (z)|λi = |µi ⊗ vi + higher terms in z

Type II vertex operators assume similar normalization. For the evaluation module Vz , we define the components of the vertex operator ΦµV λ : V (λ) −→ V (µ) ⊗ Vz by X µV ΦµV (z)|ui = (5.3) Φλj (z)|ui ⊗ vj , λ j∈J

for |ui ∈ V (λ), and j runs through the index set J for the basis of V as in the Figure 1. We also consider the intertwining operators of modules of the following form: ΦµλV (z) : V (λ) ⊗ Vz −→ V (µ) ⊗ C[z, z −1 ] by means of the vertex operators with respect to the dual space Vz∗ : (5.4)

∗

ΦµλV (z)(|vi ⊗ vi ) = ΦµV λi (z)|vi

for |vi ∈ V (λ) and i belongs to J. P For a node j in the crystal graph we define the weight of j to be i (fi (j) − ei (j))Λi , where fi (j) is the number of i-arrows going out of j, and ei (j) is the number of i-arrows coming into j. By the intertwining property it is easy to see the following determination relations. For more explicit relations see [Ko1], [JM2], [JK], [JKK].

12


Proposition 5.1. The vertex operator Φ(z) of type I with respect to Vz is determined by its component Φ1 (z), where 1 is the last vertex in the representation graph. k

In particular, for each pair i −→ j in the graph, we have Φi (z) = [rj ]−1 k [Φj (z), fk ]q−(wt(j),hk ) ,

(5.5)

k

where rj counts the number of k-arrows coming into j. With respect to Vz∗ , the k

vertex operator Φ∗ (z) of type I is determined by Φ∗1 (z), and for each pair i −→ j ∗ Φ∗j (z) = [rj ]−1 k [fk , Φi (z)]q−(wt(i),hk ) ,

(5.6)

k

where rj counts the number of k-arrows coming into j. Proposition 5.2. Let Φ(z) be a type II vertex operator with respect to Vz : V (λ) → Vz ⊗ V (µ). Then Φ(z) is determined by the component Φ1 (z). More precisely, for k each with pair i −→ j we have: Φj (z) = [ri′ ]−1 k [Φi (z), ek ]q(wt(i),hk ) .

(5.7)

k

ri′

where counts the number of k-arrows going out of i. With respect to Vz∗ the vertex operator Φ∗ (z) of type II is determined by its component Φ1∗ (z), where 1 is k

the last vertex in the crystal graph. For each pair i −→ j we have ∗ Φ∗i (z) = qk2 [ri′ ]−1 k [ek , Φj (z)]q(wt(j),hk ) ,

(5.8)

k

where

ri′

counts the number of k-arrows going out of i. (1)

(1)

Proposition 5.3. Let Uq (b g) be the quantum affine algebra of type An , Bn , or (1) Dn . (a) Let Φ(z) : V (λ) −→ V (µ) ⊗ Vz be a vertex operator of type I, where (λ, µ) is an admissible pair of weights. Then we have for each j = 1, · · · , n and k ∈ N [Φ1 (z), Xj+ (w)]

=

0,

tj Φ1 (z)t−1 j

=

q δj1 Φ1 (z),

[aj (k), Φ1 (z)] = [aj (−k), Φ1 (z)] =

for type A, we let 1 = n + 1

[k] (τ +3/2)k k q z Φ1 (z), k [k] δj1 q −(τ +1/2)k z −k Φ1 (z). k δj1

(b) If Φ(z) is a vertex operator of type I associated with Vz∗ , then [Φ∗1 (z), Xj+ (w)] tj Φ∗1 (z)t−1 j

= 0, = q

δj1

for type A, we let 1 = n + 1 Φ∗1 (z),

[k] 3k k ∗ q 2 z Φ1 (z), k [k] k [aj (−k), Φ∗1 (z)] = δj1 q − 2 z −k Φ∗1 (z). k [aj (k), Φ∗1 (z)] = δj1


13

(c) If Φ(z) is a vertex operator of type II associated with Vz , then [Φ1 (z), Xj− (w)] tj Φ1 (z)t−1 j

= 0, = q −δj1 Φ1 (z),

[k] k/2 k q z Φ1 (z), k [k] [aj (−k), Φ1 (z)] = −δj1 q −3k/2 z −k Φ1 (z). k [aj (k), Φ1 (z)] = −δj1

(d) If Φ∗ (z) is a vertex operator of type II associated with Vz∗ , then [Φ∗1 (z), Xj− (w)]

=

0,

tj Φ1∗ (z)t−1 j

=

q −δj1 Φ∗1 (z), ( 2n+3 [k] q 2 k z k Φ∗1 (z), −δj1 k q −(τ −1/2)k z k Φ1∗ (z), ( 2n+5 [k] q − 2 k z −k Φ1∗ (z), −δj1 k q (τ −3/2)k z −k Φ1∗ (z),

[aj (k), Φ∗1 (z)] = [aj (−k), Φ∗1 (z)] =

(1)

type An (1) (1) type Bn or Dn (1)

type An (1) (1) type Bn or Dn .

ˆ − ) (k ∈ Z× ) such that We introduce the auxiliary elements a1 (k) ∈ Uq (h [ai (k), a1 (l)] = δi1 δk,−l , (1)

and for type An we need the elements an+1 (k) (k ∈ Z× ) such that [ai (k), an+1 (l)] = δi,n δk,−l . ¿From Propositions (5.1), (5.2) and (5.3) we only need to determine one component for each vertex operator. Theorem 5.1. The 1-components of the type I vertex operator Φ(z)µV λ with respect to Vz : V (λ) −→ V (µ) ⊗ Vz can be realized explicitly as follows: P P (τ +3/2)k −(τ +1/2)k Φ1 (z) = exp( [k] a1 (−k)z k )exp( [k] a1 (k)z −k ) k q k q ×eλ1 (q τ +1 z)∂λ1 +(λ1 |λ1 −µ) (−1)∂λn +(λn |λ1 −µ) bµλ ,

(1)

(1)

for type Bn and Dn , and where bµλ is a constant. P P (n+5/2)k −(n+3/2)k an+1 (−k)z k )exp( [k] an+1 (k)z −k ) Φn+1 (z) = exp( [k] k q k q ×eλ1 (q n+2 z)∂λ1 +(λ1 |λ1 −µ) (−1)∂λn +(λn |λ1 −µ) bµλ ,

(1)

for type An and where bµλ is a constant. ∗ The 1-component of type I vertex operator ΦµV (z) associated with Vz∗ is given λ by X [k] X [k] q 3k/2 a1 (−k)z k )exp( q −k/2 a1 (k)z −k ) Φ∗1 (z) = exp( k k ×eλ1 (qz)∂λ1 +(λ1 |λ1 −µ) (−1)∂λn +(λn |λ1 −µ) bµλ ,

where bµλ is a constant.

14


Theorem 5.2. The 1-components of the type II vertex operator ΦVλ µ (z) with respect to Vz : V (λ) −→ Vz ⊗ V (µ) can be realized explicitly as follows: X [k] −3k X [k] k Φ1 (z) = exp(− q 2 a1 (−k)z k )exp(− q 2 a1 (k)z −k ) k k ×e−λ1 (qz)−∂λ1 +(λ1 |λ1 +µ) (−1)−∂λn +(λn |λ1 +µ) bµλ , where bµλ is a constant. The 1-component of type I vertex operator Φ1∗ (z) associated with Vz∗ is given by P [k] (τ −3/2)k P [k] −(τ −1/2)k a1 (−k)z k )exp(− a1 (k)z −k ) Φ1∗ (z) = exp(− k q k q ×e−λ1 (q −2n+2 z)−∂λ1 +(λ1 |λ1 +µ) (−1)−∂λn +(λn |λ1 +µ) bµλ ,

(1)

(1)

for type Bn and Dn , and where bµλ is a constant. P [k] − 2n+5 k P [k] 2n+3 k 2 2 an+1 (−k)z k )exp(− an+1 (k)z −k ) Φ∗n+1 (z) = exp(− k q k q

×e−λ1 (q n+2 z)−∂λ1 +(λ1 |λ1 +µ) (−1)−∂λn +(λn |λ1 +µ) bµλ ,

(1)

for type An and where bµλ is a constant. (1)

For type Cn result.

level −1/2 the vertex operators are determined by the following

Theorem 5.3. [JK] For the type one vertex operators associated to the admissible pair of weights (λ, µ)= (µ1 , µ2 ), (µ2 , µ1 ), (µ3 , µ4 ), and (µ4 , µ3 ) respectively, one has ∞ X [k/2] (n+1/4−δn1 )k e µj V (z) = exp( q a1 (−k)z k ) Φ µi 1 k k=1

∞ X [k/2] −(n+3/4−δn1 )k exp( q a1 (k)z −k ) k λ1

k=1 (n+1/2)

e (q

z)−λ1 (0)+1−(λ1 |µi ) ∂Z1+ (q n+1/2 z)cij ,

where cij are constants for the four cases (µi , µj ) with c12 = 1. Moveover the type two vertex operators are given by ΦVλ µ 1 (z) = exp(−

∞ ∞ X X [k/2] −3k/4 [k/2] −k/4 q a1 (−k)z k )exp(− q a1 (k)z −k )c′ij k k k=1

e

where

c′ij

−λ1

(q

−1/2

λ1 (0)+1+(λ1 |µi )

z)

k=1 + −1/2 Z1 (q z)c′ij ,

are constants for the four cases (µi , µj ) with c′12 = 1. References

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