q-Vertex operators for quantum affine algebras

15 Pages·0.22 MB·English

by  Naihuan Jing

q-VERTEX OPERATORS FOR QUANTUM AFFINE ALGEBRAS
9
NAIHUAN JING AND KAILASH C. MISRA
9
9
1 Abstract. q-vertexoperatorsforquantumaffinealgebrashaveplayedimpor-
n tant roleinthe theory of solvablelattice models and the quantum Knizhnik-
a Zamolodchikov equation. Explicit constructions of these vertex operators for
J most level one modules are known for classical types except for type Cn(1),
6 where the level −1/2 have been constructed. In this paper we survey these
resultsforthequantum affinealgebrasoftypesA(n1),Bn(1),Cn(1) andDn(1).

]
A
Q
.
h 1. Introduction
t
a
Algebraic conformalfield theory can be simply described by the beautiful prop-
m
erties of the Virasoro algebra, vertex operators of affine Lie algebras, and the
[
Knizhnik-Zamolodchikov equation. The vertex operators or intertwining opera-

1 tors between certain representations of affine Lie algebras stand at the core of the
v foundation since they also provide solutions to the KZ equation. Most advances
6
in this part of Lie theory around 1980s circled around this interesting subject.
2
Vertex operators also helped the introduction of the vertex algebra by Borcherds
0
1 [Bo] (cf. [FLM]). In 1985 Drinfeld [D1] and Jimbo [Jb] introduced the notion of
0 quantum group, a q-deformation of enveloping algebras of Kac-Moody algebras.
9 This has inspired torrential activity to quantize existed phenomena in Lie theory.
9
Lusztig first q-deformed the integrable modules of Kac-Moody algebras [L]. Then
/
h the q-deformed level one modules of quantum affine algebras were constructed by
t vertex representations [FJ, J1]. Quantum KZ equations were defined by Frenkel-
a
m Reshetikhinusingtheq-vertexoperators[FR],andtheirmatrixcoefficientsprovide
thesolutionsfortheq-KZequations. TheKyotoschoolusedtheq-vertexoperators
:
v in solvable lattice models (see [JM]). They proposed that the half of the space of
i
X states H, on which Baxter’s cornertransfer matrix is acting, can be identified with
highest weight representations of quantum affine algebras. Then the correlation
r
a functions and the form factors can be computed in terms of the q-deformed ver-
tex 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
theexplicitbosonizationoflevelonecaseforU (A(1))firstforn=1[JMMN],then
q n
for general n by Koyama [Ko1]. Kang, Koyama, and the first author [JKK] gener-
alized the bosonization to type D(1). More recently we gave explicit bosonizations
n
1991 Mathematics Subject Classification. Primary: 17B.
Key words and phrases. q-Vertexoperators,quantum affinealgebras,
NJ:ResearchsupportedinpartbyNSAgrantMDA904-97-1-0062.
KCM:ResearchsupportedinpartbyNSAgrantMDA904-96-1-0013.
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 workwe will reviewthe techniques involvedinthe realizationofq-vertex
operators and provide a unified description for the untwisted quantum affine al-
gebras of classical types. To minimize the length of the paper we do not include
twistedquantumaffinealgebraswhichhavebeenconsideredin[JM2]. Howeverthe
basic ingredients are similar in all cases.
Currently two methods of deriving q-vertex operators are used: coproduct for-
mula (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
sufficient to determine the q-vertex operator in all the case (cf. type C(1) in [JK]).
n
We refer the reader to [JK] for more detailed information about the direct compu-
tationapproach. Due to lackof spacewe do notinclude the intertwiningoperators
forthe Drinfeld comultiplication. Theirbosonizationagreeswith ourvertexopera-
tors in the special computable component. The rest of the components in [DI] are
then derived using normal ordering and commutation relations.
This paper is arrangedas follows. In section two we review the basic notations.
Drinfeldrealization,thecoproductformulaandthelevelzeroactionoftheDrinfeld
generators are computed in section three. In section four we discuss the construc-
tions of level one modules for types A(1), B(1), D(1) and level 1/2 modules for
n n n
−
type C(1). These constructions are used to construct the q-vertex operators in the
n
last section.
2. Notation and Preliminaries
Let A=(A ),i,j I = 0,1, ,n be the generalized Cartan matrix of type
ij
∈ { ··· }
A(1),B(1),C(1), or D(1).
n n n n
Letg=g(A)be the associatedKac-Moodyalgebra,andh (i I)anddbe the
i
∈
generators[K] of the Cartan subalgebrah. We define the affine rootlattice of g to
be b b
Qˆ =Zα0 b Zαn, b
⊕···⊕
where α h∗ such that < α ,h >= A and < α ,d >= δ . The affine weight
i i j ji i i0
∈
lattice Pˆ is defined to be
b
Pˆ =ZΛ ZΛ Zδ,
0 n
⊕···⊕ ⊕
where Λ (h ) = δ ,Λ (d) = 0, and δ(h ) = 0,δ(d) = 1 for j I. The dual affine
i j ij i j
∈
weight lattice is then defined as
Pˆ∨ =Zh Zh Zh Zd.
0 1 n
⊕ ⊕···⊕ ⊕
We will denote the correspondingweight(resp. root)lattice of the finite dimen-
sional Lie algebra g by P (resp. Q).
The nondegenerate symmetric bilinear form ( ) on h∗ satisfies
|
(2.1) (α α )=d A , (δ α )=(δ δ)=0 for all i,j I,
i j i ij i
| | | b ∈
where d = 1 except for (d , ,d )=(1, ,1,1/2) in the case of B(1), and
i 0 n n
··· ···
(1)
(d , ,d )=(1,1/2, ,1/2,1)in the case of C .
0 n n
··· ···
Let qi =qdi =q21(αi|αi),i I. The quantum affine algebra Uq(g) is the associa-
∈
tive algebra with 1 over C(q1/2) generated by the elements e , f (i I) and qh
i i
∈
b
q-VERTEX OPERATORS FOR QUANTUM AFFINE ALGEBRAS 3
(h Pˆ∨) with the following relations :
∈
qhqh′ = qh+h′, q0 =1, for h,h′ Pˆ∨,
∈
qhe q−h = qαi(h)e , qhf q−h =q−αi(h)f for h Pˆ∨(i I),
i i i i
∈ ∈
t t−1
eifj −fjei = δijqi−qi−1, where ti =qihi =q12(αi|αi)hi and i,j ∈I,
i− i
( 1)me(m)e e(n) =0, and
− i j i
m+nX=1−aij
( 1)mf(m)f f(n) =0 for i=j,
− i j i 6
m+nX=1−aij
qk q−k
where e(k) = ek/[k] !, f(k) = fk/[k] !, [m] ! = m [k] , and [k] = i − i . For
i i i i i i i k=1 i i q q−1
i− i
simplicity we will denote [k] = [k] for i = 1,Q ,n 1. The derived subalgebra
i
··· −
generated by e , f , t (i I) is denoted by U′(g).
i i i ∈ q
The algebraU (g) has a Hopf algebra structure with comultiplication ∆, counit
q
u∗, and antipode S defined by b
b
∆(qh) = qh qh for h Pˆ∨,
⊗ ∈
∆(e ) = e 1+t e ,
i i i i
⊗ ⊗
∆(f ) = f t−1+1 f for i I,
i i⊗ i ⊗ i ∈
(2.2) u∗(qh) = 1 for h Pˆ∨,
∈
u∗(e ) = u∗(f )=0 for i I,
i i
∈
S(qh) = q−h for h Pˆ∨,
∈
S(e ) = t−1e , S(f )= f t for i I.
i − i i i − i i ∈
Let V,W be two U (g)-modules. The tensor product V W is defined as the
q
⊗
U (g)-module via the coproduct ∆. The (restricted) dual U (g)-module V∗ is de-
q q
fined by b
b b
(x v∗)(u)=v∗(S(x) u)
· ·
for x U (g), u V, and v∗ V∗.
q
∈ ∈ ∈
b
3. Drinfeld realization and level zero modules
We now recall Drinfeld’s realization of the quantum affine algebra U (g) (and
q
of U′(g))(cf. [D2, B, J3]). Let U be the associative algebra with 1 over C(q1/2)
q
generatedby the elements x±(k), a (l), K±1, γ±1/2, q±d (i=1,2, ,n,kb Z,l
i i i ··· ∈ ∈
b
4 NAIHUAN JING AND KAILASH C. MISRA
Z 0 ) with the following defining relations :
\{ }
[γ±1/2,u] = 0 for all u U,
∈
[A k] γk γ−k
ij i
[a (k),a (l)] = δ − ,
i j k+l,0 k q q−1
j − j
[a (k),K±1] = [q±d,K±1]=0,
i j j
qdx±(k)q−d = qkx±(k), qda (l)q−d =qla (l),
i i i i
K x±(k)K−1 = q±(αi|αj)x±(k),
i j i j
[A k]
[a (k),x±(l)] = ij iγ∓|k|/2x±(k+l),
i j ± k j
x±(k+1)x±(l) q±(αi|αj)x±(l)x±(k+1)
i j − j i
= q±(αi|αj)x±(k)x±(l+1) x±(l+1)x±(k),
i j − j i
(3.1) [x+i (k),x−j (l)] = q δijq−1 γk−2lψi(k+l)−γl−2kϕi(k+l) ,
i− i (cid:16) (cid:17)
where ψ (m) and ϕ ( m) (m Z ) are defined by
i i ≥0
− ∈
∞
ψ (m)z−m =K exp (q q−1) ∞ a (k)z−k ,
i i i− i k=1 i
m=0
X (cid:0) P (cid:1)
∞
ϕ ( m)zm =K−1exp (q q−1) ∞ a ( k)zk ,
i − i − i− i k=1 i −
m=0
X (cid:0) P (cid:1)
and the Serre relations are:
m=1−Aij m
(3.2) Sym ( 1)r x±(k ) x±(k )x±(l)
k1,···,km − r i 1 ··· i r j
r=0 (cid:20) (cid:21)i
X
x±(k ) x±(k )=0 if i=j.
· i r+1 ··· i m 6
WedenotebyU′ thesubalgebraofUgeneratedbytheelementsx±(k),a (l),K±1,
i i i
γ±1/2 (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 i ,i , ,i be a sequence of indices given in the
1 2 h−1
···
Table. Then the C(q1/2)-algebra isomorphism Ψ:U (g) U is given by
q
→
e x+(0), f x−(0), t K for i=1, ,n,
i 7→ i i 7→ i i 7→ i b ···
(3.3) e0 7→ [x−ih−1(0),··· ,x−i2(0),x−i1(1)]qǫ1···qǫh−2γKθ−1,,
f0 7→ a(−q)−ǫγ−1Kθ[x+ih−1(0),··· ,x+i2(0),x+i1(−1)]qǫ1···qǫh−2,
t γK−1, qd qd,
0 7→ θ 7→
where K = K K , h is the Coxeter number, ǫ = h−2ǫ . The constant a
θ i1··· ih−1 i=1 i
is 1 for simply laced types A(n1),Dn(1), a=[2]1 for Cn(1), anPd a=[2]1−δ1,i1 for Bn(1).
The restriction of Ψ to U′(g) defines an isomorphism of U′(g) and U′.
q q
Table: ǫ-Sequences for simple Lie algebras
b b
q-VERTEX OPERATORS FOR QUANTUM AFFINE ALGEBRAS 5
g ǫ-Sequence. ǫ= ǫ ǫ
i
−1 −1
A α α -n+1
n 1 →···→ nP
−1 −1 0 −1 −1
B α α α α -2n+4
n 1 n−1 n 2
→···→ → →···→
−1/2 −1 −1/2 −1/2 0
C α α α α -n+1
n 1 n 2 1
−→ ···→ −→ ··· −→ →
−1 −1 −1 −1 −1
D α α α α -2n+4
n 1 n n−2 2
→···→ → →···→
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.
Theorem 3.1. Let k Z , l N, and let Ns (resp. Ns) be the subalgebra of U
∈ ≥0 ∈ + −
generated by the elements x+( m ) x+( m ) (resp. x−(m ) x−(m )) with
m Z . Then the comultiip1li−catio1n·∆·· oifst−he aslgebra U hai1s th1e f·o·l·lowisingsform:
i ≥0
∈
∆(x+(k)) = x+(k) γk+γ2kK x+(k)
i i ⊗ i⊗ i
k−1
+ γk−2jψ(k−j)⊗γk−jx+i (j) (mod N−⊗N+2),
j=0
X
∆(x+( l)) = x+( l) γ−l+K−1 x+( l)
i − i − ⊗ i ⊗ i −
l−1
+ γl−2jϕ(−l+j)⊗γ−l+jx+i (−j) (mod N−⊗N+2),
j=1
X
∆(x−(l)) = x−(l) K +γl x−(l)
i i ⊗ i ⊗ i
l−1
+ γl−jx−i (j)⊗γj−2lψi(l−j) (mod N−2 ⊗N+),
j=1
X
∆(x−( k)) = x−(k) γ−2kK−1+γ−k x−( k)
i − i ⊗ i ⊗ i −
k−1
+ γj−kx−i (−j)⊗γ−k+23jϕi(j−k) (mod N−2 ⊗N+),
j=0
X
∆(ai(l)) = ai(l) γ2l +γ32l ai(l) (mod N− N+),
⊗ ⊗ ⊗
∆(ai( l)) = ai( l) γ−32l +γ−2l ai( l) (mod N− N+).
− − ⊗ ⊗ − ⊗
Moreover the same formulas are true for the derived subalgebra U′.
Let V be the finite dimensional representationdescribed by the following repre-
sentationgraphs. Notethatthegraphsarecrystalgraphs,andtheyarealsoperfect
crystals[KMN]exceptinthe caseofC(1). Letv be the basiselements (i I), and
n i
∈
I is the index set. Let E be the matrix units. We can read the actions from the
ij
graphs.
TheU (g)-modulestructureontheaffinizationorevaluationmoduleofV [KMN]
q
is given as follows. We equip the affinization V = V C(q1/2)[z,z−1] with the
z
⊗
following abctions:
e (v zm) = e v zm+δi,0, f (v zm)=f v zm−δi,0,
i i i i
⊗ ⊗ ⊗ ⊗
(3.4) t (v zm) = t v zm,
i i
⊗ ⊗
qd(v zm) = qmv zm,
⊗ ⊗
for i I, v V.
∈ ∈
6 NAIHUAN JING AND KAILASH C. MISRA
0
(cid:25)
A(1) 1 1-2 2- p p p p p p p p p p n-1-n n-n+1
n
0
(cid:25)
B(1) 1 -1 2 -2 p p p p p n--1 n -n 0 -n n n--1 p p p p p -2 2 -1 1
n Y
0
0
(cid:25)
C(1) 1 -1 2 -2 p p p p p p p p n--1 n -n n n--1 p p p p p p p p -2 2 -1 1
n
0
n
D(1) 1 -1 2(cid:25)-2 p p n--2 n 1n(cid:0)-(cid:0)(cid:18)1 @@Rn n 1n--2 p p -2 2 -1 1
n Y − n@@R (cid:0)(cid:0)(cid:18)n-1 −
n
0
Figure 1. Representation graphs
The evaluation module V is a level zero U (g)-module, i.e., γ acts as identity
z q
(= q0). Through the isomorphism Ψ the evaluation module is also a U-module.
The action of the Drinfeld generators are given bby the following result.
Define τ =τ(A) as follows:
(1)
n+1 A
n
2n 1 B(1)
τ = − n
 2n+2 Cn(1)
2n 2 D(1).
n
−

We define the content of each vertex in the representation graph by
i if i<n
c(i)=
τ i if i 1, ,n
(cid:26) − ∈{ ··· }
Here we adopt the convention that i=i.
q-VERTEX OPERATORS FOR QUANTUM AFFINE ALGEBRAS 7
Theorem 3.2. [Ko1, JKK, JM, JK] For j 1, ,n , the Drinfeld generators
∈ { ··· }
act on the evaluation module V as follows.
z
x+(k) = (qc(j)z)kE
j (i(j),e(j)) j i(j),e(j)
x−(k) = (qc(j)z)kE
j P(i(j),e(j)) j e(j),i(j)
a (l) = [l](qc(j)z)l(q−lE qlE )
j P(i(j),e(j)) l j j i(j),i(j) − j e(j),e(j)
(qnz)k[2] E +(qn−1z)kE , B(1)
P n n,0 0,n n
x+n(k) =  (qn−1z)kEn,n, Cn(1)
 (qn−1z)k(En−1,n+En,n−1), Dn(1)
(3.5) (qn−1z)k[2] E +(qnz)kE B(1)
 n n,0 0,n n
x−n(k) =  (qn−1z)kEn,n, Cn(1)
 (qn−1z)k(En,n−1+En−1,n), Dn(1)
[2l]n(qn−1z)l((E qlE )+(E qlE )) B(1)
 l n,n− 0,0 0,0− n,n n
[l](qn−1z)l(q−lE qlE ), C(1)
an(l) =  [lll](qn−1z)+l(qq−−llEEnnn−1−,n−ql1E−n,nqlEn,n) Dn(1)
n,n− n−1,n−1 n
 j
where the summation runs through all possible pairs i(j) e(j) in the represen-
−→
tation graphs connected by j. j =1,2, ,n 1(but we allow j =n for type A(1)),
n
k Z and l Z 0 . ··· −
∈ ∈ \{ }
Theorem 3.3. [Ko1,JKK,JM,JK] The Drinfeld generators act on the dual eval-
uation module V∗ =V∗ C(q1/2)[z,z−1] as follows.
z ⊗
(3.6)
x+(k) = ( q−1) (q−c(j)z)kE∗
j − (i(j),e(j)) j e(j),i(j)
x−(k) = ( q) (q−c(j)z)kE∗
j − (Pi(j),e(j)) j i(j),e(j)
a (l) = [l](q−c(j)z)l(q−lE∗ qlE∗ )
j (i(jP),e(j)) l j j e(j),e(j) − j i(j),i(j)
( q−1)((q−nz)k[2] E∗ +(q−n+1z)kqE∗ ), B(1)
P − n 0,n n,0 n
x+(k) = ( q−1)(q−n+1z)kE∗ , C(1)
n  − n,n n
 (−q−1)(q−n+1z)k(En∗,n−1+En∗−1,n), Dn(1)
( q)((q−n+1z)k[2] q−1E∗ +(q−nz)kE∗ ) B(1)
 − n 0,n n,0 n
x−(k) = ( q)(q−n+1z)kE∗ , C(1)
n  − n,n n
 (−q)(q−n+1z)k(En∗−1,n+En∗,n−1), Dn(1)
 [2ll]n(q−n+1z)l((q−lE0∗,0−En∗,n)+(q−lEn∗,n−E0∗,0)), Bn(1)
a (l) =  [ll](q−n+1z)l(q−lEn∗,n−qlEn∗n), Cn(1)
n  [ll](q−n+1z+)ql(−ql−ElE∗n∗,n−qlEqn∗l−E1∗,n−)1 D(1)
n−1,n−1− n,n n
 j
where the summation runs through all possible pairs i(j) e(j) in the represen-
−→
tation graphs connected by j. j =1,2, ,n 1(but we allow j =n for type A(1)),
n
k Z and l Z 0 . ··· −
∈ ∈ \{ }
8 NAIHUAN JING AND KAILASH C. MISRA
4. Vertex representation of U (g)
q
In this section we recall the level one realizations of U (g) for g = A(1),D(1) as
q b n n
given in [FJ] and g = B(1) in [JM1] (cf. [Be] for V(Λ )). We also recall the level
n 0
1/2 realizationof U (C(1)) as given in [JKM]. The quantubm affibne algebra U (g)
q n q
−
has level one irredbucible representations: V(Λ ), i=0,1, ,n for A(1); i=0,1,n
i n
···
for B(1) and i=0,1,n 1,n for D(1). b
n n
−
TheweightlatticeP hastwocosetrepresentativesQandQ+λ ,whereλ denote
n i
the fundamental weights in P. We will express any level one fundamental weight
λ as a linear combination of the simple roots α and λ .
i i n
Let ε: Q Q Z = 1 be a cocycle such that
2
× −→ {± }
(4.1) ε(α,β)ε(β,α)=( 1)(α|β)+(α|α)(β|β),
−
whose existence can be constructed directly on the simple roots. Then there cor-
responds a central extension of the group algebra C(Q) associated with ε such
that
1 Z C Q C(Q) 1.
2
−→ −→ { }−→ −→
If we still use eα to denote the generators of C Q , then we have
{ }
(4.2) eαeβ =( 1)(α|β)+(α|α)(β|β)eβeα.
−
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 ∂ on C P by
α
∈ { }
(4.3) ∂ eβ =(αβ)eβ
α
|
LetU (hˆ)betheinfinitedimensionalHeisenbergalgebrageneratedbya (k)and
q i
the central element γ (k Z 0 ,i=1, ,n) subject to the relations
∈ \{ } ···
[A k] γk γ−k
ij i
(4.4) [a (k),a (l)]=δ − .
i j k+l,0 k q q−1
j − j
The algebra U (hˆ) acts on the space of symmetric algebra Sym(hˆ−) generated by
q
a ( k) (k N,i=1, ,n) with γ =q by the action:
i
− ∈ ···
a ( k) multiplication by a ( k),
i i
− 7→ −
[A k] qk q−k d
ij i
a (k) − .
i 7→Xj k qj −qj−1 daj(−k)
For the case of B(1) we need to consider an extra fermionic field. For Z =Z+s
n
(s = 1/2, NS-case; s = 0, R-case), let Cs be the q-deformed Clifford algebra
q
generated by κ(k), k Z satisfying the relations:
∈
(4.5) κ(k),κ(l) =(qk+q−k)δ ,
k,−l
{ }
where k,l Z and a,b =ab+ba.
∈ { }
LetΛ(C−)sbethepolynomialalgebrageneratedbyκ( k),k Z ,andΛ(C−)s
q − ∈ >0 q 0
(resp. Λ(C−)s)bethesubalgebrageneratedbyproductsofeven(resp. odd)number
q 1
of generators κ( k)’s. Then
−
Λ(C−)s =Λ(C−)s Λ(C−)s.
q q 0⊕ q 1
q-VERTEX OPERATORS FOR QUANTUM AFFINE ALGEBRAS 9
The algebra Cs acts on Λ(C−)s canonically by (k Z )
q q ∈ >0
κ( k) multiplication by κ( k),
− 7→ −
d
κ(k) .
7→−d κ( k)
−
We define that
V(Λi) = Symhˆ−) (C Q eλi, A(n1),Dn(1)
⊗ { }
V(Λ0) = Sym(hˆ−)⊗(C{Q0}⊗Λ(Cq−)10/2⊕C{Q0}eλ1 ⊗Λ(Cq−)11/2),Bn(1)
V(Λ1) = Sym(hˆ−)⊗(C{Q0}eλ1 ⊗Λ(Cq−)10/2⊕C{Q0}⊗Λ(Cq−)11/2),Bn(1)
V(Λn) = Sym(hˆ−)⊗C{Q}eλn ⊗Λ(Cq−)0,Bn(1)
where i = 0, ,n for A(1) and i = 0,1,n 1,n for D(1). C Q is the group
n n 0
··· − { }
algebra of the sublattice of long roots in the case of B(1). Here for convenience we
n
denote λ =0.
0
Proposition 4.1. [FJ][JM1] The space V(Λ ) of level one irreducible modules of
i
the quantum affine Lie algebra U (g) is realized under the following action:
q
γ q, Kj q∂αj, aj(k) aj(k) (1 j n),
7→ 7→ b 7→ ≤ ≤
∞ ∞
a ( k) a (k)
x±(z) X±(z)=exp( i − q∓k/2zk)exp( i q∓k/2z−k)
i 7→ i ± [k] ∓ [k]
k=1 k=1
X X
e±αiz±∂αi+1, 1 i n; when g =Bn(1), 1 i n 1
× ≤ ≤ ≤ ≤ −
∞ ∞
a ( k) a (k)
x±(z) X±(z)=exp( n − q∓k/2zk)exp( n q∓k/2z−k)
n 7→ n ± [2k] b ∓ [2k]
n n
k=1 k=1
X X
e±αnz±∂αn+1/2( κ(z)), for g =Bn(1)
× ±
where κ(z)= m∈Zκ(m)z−m. The highest weight vectors are |Λii=eλi.
b
We nowrecPallthe level 1/2realizationofU (C(1))[JKM]. Therearefourlevel
q n
−
1/2weights: µ = 1Λ ,µ = 3Λ +Λ 1δ, µ = 1Λ ,µ = 3Λ +Λ .
− 1 −2 0 2 −2 0 1−2 3 −2 n 4 −2 n n−1
Let a (m) be the bosonic operators satisfying (4.4) with γ = q−1/2. Let b (m)
i i
be another set of bosonic operators satisfying the following defining relations.
[b (m),b (l)] = mδ δ ,
(4.6) i j ij m+l,0
[a (m),b (l)] = 0.
i j
Z
We define the Fock space for α P + λ , β P by the defining relations
Fα,β ∈ 2 n ∈
a (m)α,β =0 (m>0), b (m)α,β =0 (m>0),
i i
| i | i
a (0)α,β =(α α)α,β , b (0)α,β =(2ε β)α,β ,
i i i i
| i | | i | i | | i
where α,β is the vacuum vector, and ε are the usual orthonormal vectors.
i
| i
We set
= .
α,β
F F
α∈P+M12Zλn,β∈P
e
Let eai =eαi and ebi (1 i n) be operators on given by:
≤ ≤ F
eαi α,β = α+ε ,β , ebi α,β = α,β+ε .
i i
| i | i | ei | i
10 NAIHUAN JING AND KAILASH C. MISRA
Let : : be the usual bosonic normal ordering defined by
:a (m)a (l):=a (m)a (l)(m l), a (l)a (m)(m>l),
i j i j j i
≤
:eaa (0):=:a (0)ea :=eaa (0).
i i i
and similar normal products for the b (m)’s. Let ∂ = ∂ be the q-difference
i q1/2
operator:
f(q1/2z) f(q−1/2z)
∂ (f(z))= −
q1/2 (q1/2 q−1/2)z
−
We introduce the following operators.
∞ ∞
a (k) a (k)
Yi±(z) =exp(± [ i1k]q±k4zk)exp(∓ [ i1k]q±k4z−k)e±aiz∓2ai(0),
k=1 −2 k=1 −2
X X
∞ ∞
b ( k) b (k)
Z±(z) =exp( i − zk)exp( i z−k)e±biz±bi(0).
i ± k ∓ k
k=1 k=1
X X
We define the operators x±(m) (i = 1, ,n,m Z) by the following generating
i ··· ∈
functions. X±(z)= x±(m)z−m−1.
i m∈Z i
X+(z) = ∂Z+(Pz)Z− (z)Y+(z), i=1, n 1
i i i+1 i ··· −
X−(z) = Z−(z)∂Z+ (z)Y−(z), i=1, n 1
i i i+1 i ··· −
1
Xn+(z) = (cid:18)q12 +q−12 :Zn+(z)∂2Zn+(z):−:∂Zn+(q21z)∂Zn+(q−12z):(cid:19)Yn+(z)
1
Xn−(z) = q12 +q−12 :Zn−(q12z)Zn−(q−12z):Yn−(z).
Remark. Our a (k) differs from that in [JKM], where we took a (k)/[d ] for a (k).
i i i i
Furthermore,we know that contains the four irreducible highestweight mod-
F
ules([JKM]). Let 1 bethe subspaceof generatedbya (m)(i=1, ,n
Fα,β Fα,β i ··· ∈
Z, m Z) Similarly, let 2,j (je=1, ,n) be the subspace of generated by
∈ Fα,β ··· Fα,β
b (m) (m Z). We can define the following isomorphism by α,β α′,β′
j
∈ | i⊗| i →
α+α′,β+β′ .
| i
1 2,1 2,n .
Fα,0⊗F0,β1 ⊗···⊗F0,βn −→Fα,β1+···+βn
Let Q− be the operator from 2,j to 2,j defined by
j Fα,β Fα,β−εj
1
Q− = Z−(z)dz.
i 2π√ 1 i
− I
We set subspaces (i=1,2,3,4) of as follows.
i
F F
= ′ , e = ′
F1 Fα,α F2 Fα+ε1,α+ε1
α∈Q α∈Q
M M
= ′ , = ′ ,
F3 Fα−12λn,α F4 Fα−21λn,α
αM∈Q α∈MQ+εn
where
n
′ = 1 Ker Q−,
Fα,β Fα,0⊗ F02,,ljjεj j
j=1
O
for β =l ε + +l ε . Then we have the following proposition.
1 1 n n
···