Table Of ContentDeformed W algebras from elliptic sl(N) algebras
N
J. Avan
LPTHE, CNRS-URA 280, Universit´es Paris VI/VII, France
L. Frappat, M. Rossi, P. Sorba
8
9
9 Laboratoire de Physique Th´eorique ENSLAPP, CNRS-URA 1436,
1 E´cole Normale Sup´erieure de Lyon and Universit´e de Savoie, France
n
a
J
2
2
] Abstract
A
Q
We extend to the sl(N) case the results that we previously obtained on the construction
.
h of Wq,p algebras from the elliptic algebra Aq,p(sl(2)c). The elliptic algebra Aq,p(sl(N)c) at the
t critical level c = −N has an extended center containing trace-like operators t(z). Families of
a
m Poisson structures indexed by N(N−1)/2 integbers, defining q-deformations of thebW algebra,
N
1
[ are constructed. Theoperators t(z) also close an exchange algebra when(−p2)NM = q−c−N for
1 M ∈ Z. ItbecomesAbelianwheninadditionp = qNh wherehisanon-zerointeger. ThePoisson
v
structuresobtainedintheseclassicallimitscontain differentq-deformedW algebrasdepending
5 N
0 on the parity of h, characterizing the exchange structures at p 6= qNh as new Wq,p(sl(N))
1 algebras.
1
0
R´esum´e
8
9
/ Nous ´etendons au cas sl(N) les r´esultats que nous avons obtenus pr´ec´edemment concernant
h
laconstructiondesalg`ebres W `apartirdel’alg`ebre elliptiqueA (sl(2) ). L’alg`ebre elliptique
t q,p q,p c
a
A (sl(N) ) au niveau critique c = −N poss`ede un centre ´etendu contenant des op´erateurs de
m q,p c
trace t(z). On construit sur ce centre des familles de structures de Pboisson indic´ees par N(N −
:
v 1)/2 ebntiers, d´efinissant des q-d´eformations de l’alg`ebre W . Les op´erateurs t(z) engendrent
N
i
X une alg`ebre d’´echange lorsque (−p21)NM = q−c−N ou` M ∈ Z. Cette alg`ebre devient ab´elienne
ar si de plus p = qNh avec h entier non nul. Les structures de Poisson obtenues dans ces limites
classiques contiennent diff´erentes alg`ebres W q-d´eform´ees d´ependant de la parit´e de l’entier h,
N
caract´erisant les structures d’´echange `a p 6=qNh comme de nouvelles alg`ebres W (sl(N)).
q,p
ENSLAPP-AL-670/97
PAR-LPTHE 97-49
math.QA/9801105
December 1997
1 Introduction
The elliptic algebra A (sl(2) ) was introduced in [1] and further studied in [2, 3, 4] where several
q,p c
trigonometriclimitswerederivedandshowntoberelevantassymmetries fortheXXZmodel,whereas
A (sl(2) )wasproposedbasbasicsymmetry algebrafortheXYZmodel[5,6]. Morerecentlyanother,
q,p c
possibly related elliptic algebra U (sl(2) ) was introduced [7] in connection with the k-fusion RSOS
q,p c
b
model [8, 9]. A dynamical version of the algebra A (sl(2) ), making use of the dynamical “eight-
q,p c
vertex” elliptic R-matrix[10, 11], wasbalso recently proposed in [12]. The domainwas unified recently
[13, 14] where both A (sl(2) ) algebra (and its extensibon to sl(N)) and Felder’s elliptic dynamical
q,p c
algebra [11, 15] were shown to be obtained from application of universal twisting operators (see also
[16]) to the quantum algebbra U (sl(2) ). As a consequence the construction [1] was validated and
q c
extended to the sl(N) case.
The q-Virasoro algebra was intbroduced in [17] as an extension to the Ruijsenaar–Schneider model
of the Virasoro algebra arising in the collective theory formulation of the Calogero–Moser model
[18, 19, 20]. It arose simultaneously as quantization of a classical quadratic Poisson structure on
the center at c = −2 of the quantum affine algebra U (sl(2) ) [21]. It was shown [22, 23] to be the
q c
symmetry algebra of the restricted ABF model, itself connected as previously said to the elliptic
U (sl(2) ) algebra. On the other hand, the limit of q-Virasoro current coincides [22, 2] with the
q,p c
field obtained by concatenation of vertex operators of the degeneracy limit of A (sl(2) ), hinting
q,p c
indeebd to a deep connection between the two structures. This identification was recently extended
to the full q-deformed Virasoro case [24]. b
A direct connection was indeed established in two recent papers [25, 26]: we derived exchange
algebras and limit Poisson structures from the original A (sl(2) ) algebra, by establishing the ex-
q,p c
istence of a center at pm = qc+2 for any integer m. The Poisson structures all boil down to the
q-Virasoro (classical) algebra defined in [17, 21]. The exchanbge algebras are however quite distinct
from the quantization of q-Virasoro constructed in [17, 27]. Their eventual central (and linear ?)
extensions also exhibit a much richer structure than was originally derived in [27] and are currently
under investigation [28].
Itisknown thatq-Virasorostructures admitextensions tothesl(N)caseasclassicalandquantum
q-W algebras [21, 27, 29]. These have been the object of many investigations recently [30, 31]. In
N
particular, they were also shown to be in general symmetry algebras of RSOS models [32, 23]. It is a
natural question in view of our previous results to investigate whether classical and quantized q-W
N
algebras arise as structures embedded in the generalized elliptic algebra A (sl(N) ).
q,p c
The plan of the paper is as follows. We first recall in Sect. 2 useful properties of elliptic functions
b
and elliptic sl(N) R-matrix, and introduce the definition of the quantum elliptic algebra A (sl(N) )
q,p c
[14]. We then prove in Sect. 3 the existence of an extended center at c = −N and compute the
Poisson structures on this center, using the notion of sector-depending mode Poisson bracket ablready
introduced in [25]. These structures are identified as q-deformed W algebras. They are obtained
N
from analytic continuations of a classical algebra identical to the initial version of q-W given in [21].
N
1
InSect. 4we show the existence of closed (quadratic) exchange algebraswhenever (−p2)NM = q−c−N
for any integer M ∈ Z. These algebras differ from the quantum W (sl(N)) structures introduced
q,p
in [27]. They admit a classical limit (commuting algebras) at p = qNh with h ∈ Z\{0}. In Sect.
5 we compute the related Poisson structures. They include for h evebn the Poisson structures in
[21]. The exchange algebras therefore realize new quantizations of these Poisson structures. When
h is odd, by contrast, this classical limit takes a form different from the initial q-W structures.
N
This emphasizes the key role of the initial 3-parameter structure A (sl(N) ) in allowing for an
q,p c
1 b
intermediate quantum q-deformed W algebra. Finally in Sect. 6, we compute the mode expansion
N
of the quantum exchange algebra structures. As in the classical case, a “sector-type” labeling is
needed due to the singularity behaviour of the structure function viewed as an analytic continuation
andthereforeexhibiting different formalseries expansions correspondingtodifferent convergent series
expansions in distinct domains of the complex plane. We give an explicit example of this treatment
applied to the spin one field for the sake of simplicity, and we describe the essential features of the
extension of our computation to higher spin fields.
2 Notations and basic definitions
2.1 Definition of the N-elliptic R-matrix
The N-elliptic R-matrix in End(CN) ⊗End(CN), associated to the Z -vertex model, is defined as
N
follows [33, 34]:
1
ϑ 2 (ζ,τ)
1 1
R(z,q,p) = z2/N−2 (cid:20) 2 (cid:21) W (ξ,ζ,τ) I ⊗I−1 , (2.1)
κ(z2) 1 (α1,α2) (α1,α2) (α1,α2)
ϑ 21 (ξ +ζ,τ) (α1,α2X)∈ZN×ZN
e (cid:20) 2 (cid:21)
where the variables z,q,p are related to the variables ξ,ζ,τ by
z = eiπξ, q = eiπζ , p = e2iπτ . (2.2)
γ
The Jacobi theta functions with rational characteristics ϑ 1 (ξ,τ) are defined in Appendix A.
γ
2
(cid:20) (cid:21)
The normalization factor is chosen as follows:
1 (q2Nz−2;p,q2N) (q2z2;p,q2N) (pz−2;p,q2N) (pq2N−2z2;p,q2N)
∞ ∞ ∞ ∞
= . (2.3)
κ(z2) (q2Nz2;p,q2N) (q2z−2;p,q2N) (pz2;p,q2N) (pq2N−2z−2;p,q2N)
∞ ∞ ∞ ∞
The functions W are given by
(α1,α2)
1 +α /N
ϑ 2 1 (ξ +ζ/N,τ)
1 +α /N
W (ξ,ζ,τ) = (cid:20) 2 2 (cid:21) . (2.4)
(α1,α2) 1 +α /N
Nϑ 2 1 (ζ/N,τ)
1 +α /N
(cid:20) 2 2 (cid:21)
The matrices I are defined as follows:
(α1,α2)
I = gα2hα1 , (2.5)
(α1,α2)
where the N ×N matrices g and h are given by g = ωiδ and h = δ , the addition of indices
ij ij ij i+1,j
being understood modulo N.
Let us set
S(ξ,ζ,τ) = W (ξ,ζ,τ) I ⊗I−1 . (2.6)
(α1,α2) (α1,α2) (α1,α2)
(α1,α2X)∈ZN×ZN
The matrix S is Z -symmetric, that is
N
Sa+s,b+s = Sa,b (2.7)
c+s,d+s c,d
2
for any indices a,b,c,d,s ∈ Z (the addition of indices being understood modulo N) and the non-
N
vanishing elements of the matrix S are Sa,c+b. One finds explicitly:
c,a+b
Sa,c+b = W ω−bs. (2.8)
c,a+b (a−c,s)
sX∈ZN
The matrix S being Z -symmetric, it is sufficient to examine the terms Sab ≡ S0,a+b. One finds:
N a,b
N−1 k/N + 1
(b−a)/N + 1 ϑ 2 (ξ,Nτ)
ϑ 2 (ξ +ζ,Nτ) 1
1
Sab(ξ,ζ,τ) = (cid:20) 2 (cid:21) k=Y0,k6=b (cid:20) 2 (cid:21) . (2.9)
ϑ −a/N + 21 (ζ,Nτ) N−1 k/N + 1
1 ϑ 2 (0,Nτ)
1
(cid:20) 2 (cid:21) k=1 (cid:20) 2 (cid:21)
Y
It satisfies the following shift properties:
Sab(ξ,ζ +λτ,τ) = exp(−2iπλξ/N)Sa−λ,b(ξ,ζ,τ), (2.10a)
Sab(ξ +λτ,ζ,τ) = exp(−iπλ2τ −2iπλ(ξ +ζ/N + 1))Sa,b+λ(ξ,ζ,τ). (2.10b)
2
Using the following “gluing” formula:
N−1ϑ k/N + 12 (ξ,Nτ) = p214(N2−1) kN=−01(pN−k;pN)∞ ϑ 12 (ξ,τ), (2.11)
1 (pN;pN)N 1
k=0 (cid:20) 2 (cid:21) Q ∞ (cid:20) 2 (cid:21)
Y
and inserting eq. (2.9) into eqs. (2.6) and (2.1), one then finds the following expression in terms of
the Jacobi Θ functions for the Rab elements of the matrix (2.1):
Rab(z,q,p) = 1 p−aNbq2Nebz−N2(N+a−1) ΘpN(pN+b−aq2z2)ΘpN(pN)Θp(pq2)Θp(pz2) . (2.12)
κ(z2) Θ (pN+bz2)Θ (pN−aq2)Θ (pq2z2)Θ (p)
pN pN p p
e
2.2 Gauge-transformed R-matrix
In order to make the comparison with our previous results easier [25, 26], one needs to introduce the
following “gauge-transformed” matrix:
1 1 1 1
R(z,q,p) = (g2 ⊗g2)R(z,q,p)(g−2 ⊗g−2). (2.13)
e
Theorem 1 The matrix R(z,q,p) satisfies the following properties:
– Yang–Baxter equation:
R (z)R (w)R (w/z) = R (w/z)R (w)R (z), (2.14)
12 13 23 23 13 12
– Unitarity:
R (z)R (z−1) = 1, (2.15)
12 21
– Crossing-symmetry:
R (z)t2 R (z−1q−N)t2 = 1, (2.16)
12 21
3
– Antisymmetry:
R (−z) = ω(g−1 ⊗I)R (z)(g ⊗I), (2.17)
12 12
– Quasi-periodicity:
R (−zp12) = (g21hg12 ⊗I)−1R (z−1)−1(g12hg21 ⊗I), (2.18)
12 21
where b b
1
R (z) ≡ R (z,q,p) = τ (q2z−1)R (z,q,p), (2.19)
12 12 N 12
the function τ (z) being defined by
N
b b
τN(z) = zN2−2 Θq2N(qz2) . (2.20)
Θ (qz−2)
q2N
The function τ (z) is periodic with period qN: τ (qNz) = τ (z) and satisfies τ (z−1) = τ (z)−1.
N N N N N
Proof: The proof of the Yang–Baxter equation has been given in [35]. The proof of the unitarity and
the crossing-symmetry is done by a direct calculation. One has to use the following two identities
(the first one for the unitarity and the second one for the crossing-symmetry, see [36] for a proof of
eqs. (2.21) and (2.22)):
1 1
ϑ 2 (ξ +ζ,τ)ϑ 2 (−ξ +ζ,τ)
1 1
S−i−k,i−k(ξ,ζ,τ)S−j−k,k−j(−ξ,ζ,τ) = (cid:20) 2 (cid:21) (cid:20) 2 (cid:21) δ , (2.21)
1 ij
kX∈ZN ϑ 21 (ζ,τ)2
(cid:20) 2 (cid:21)
1 1
ϑ 2 (z,τ)ϑ 2 (−ξ −Nζ,τ)
1 1
Si−k,0(z,ζ,τ)Sj−k,0(−ξ −Nζ,ζ,τ) = (cid:20) 2 (cid:21) (cid:20) 2 (cid:21) δ . (2.22)
1 ij
kX∈ZN ϑ 21 (ζ,τ)2
(cid:20) 2 (cid:21)
Finally, the antisymmetry and the quasi-periodicity are explicitly checked from the expressions of
the matrix elements of R.
Remark: The crossing-symmetry and the unitarity properties of R allow exchange of inversion
12
and transposition for the matrix R as (the same property also holds for the matrix R ):
12 12
−1 t2
R (x)t2 = R (qNx)−1 . b (2.23)
12 12
(cid:16) (cid:17) (cid:16) (cid:17)
2.3 The quantum elliptic algebra A (sl(N) )
q,p c
We now define the elliptic quantum algebra A (sl(N) ) [1, 14] as an algebra of operators L (z)
q,p c ij
b
≡ L (n)zn where i,j ∈ Z , encapsulated into a N ×N matrix
n∈Z ij N
b
P
L (z) ··· L (z)
11 1N
. .
L(z) = .. .. . (2.24)
L (z) ··· L (z)
N1 NN
4
One defines A (gl(N) ) by imposing the following constraints on the L (z) (with the matrix R
q,p c ij 12
given by eq. (2.19)):
b R (z/w)L (z)L (w) = L (w)L (z)R∗ (z/w), (2.2b5)
12 1 2 2 1 12
where L (z) ≡ L(z) ⊗I, L (z) ≡ I⊗L(z) and R∗ is defined by R∗ (z,q,p) ≡ R (z,q,p∗ = pq−2c).
1 b2 12 b 12 12
This definition is the most immediate generalization to N of the definition adopted in [1] for N = 2.
The matrix R∗ obeys also the unitarity, crossbing-symmetry, anbtisymmetry abnd quasi-periodicity
12
conditionsofTheorem1(notethatthequasi-periodicitycondition(2.18)forR∗ hastobeunderstood
12
with the modbified elliptic nome p∗).
The q-determinant q-detL(z) given by b
N
q-detL(z) ≡ ε(σ) L (zqi−N−1) (2.26)
i,σ(i)
σX∈SN Yi=1
(ε(σ) being the signature of the permutation σ) is in the center of A (gl(N) ). It can be “factored
q,p c
c
out”, and set to the value q2 so as to get
b
c
A (sl(N) ) = A (gl(N) )/hq-detL−q2i. (2.27)
q,p c q,p c
It is useful to introduce the following two matrices:
b b
L+(z) ≡ L(q2cz), (2.28a)
1 1 1 1 1
L−(z) ≡ (g2hg2)L(−p2z)(g2hg2)−1. (2.28b)
They obey coupled exchange relations following from (2.25) and periodicity/unitarity properties of
the matrices R and R∗ :
12 12
R (z/w)L±(z)L±(w) = L±(w)L±(z)R∗ (z/w), (2.29a)
b b 12 1 2 2 1 12
R (q2cz/w)L+(z)L−(w) = L−(w)L+(z)R∗ (q−2cz/w). (2.29b)
12 1 2 2 1 12
b b
The parameters c,p,q in our definition are related to the corresponding parameters c′,p′,q′ of [14]
b b
by c = 2c′, p = (p′)2/N, q = (q′)1/N.
3 The center of A (sl(N) ) at the critical level c = −N
q,p c
3.1 Center of A (sl(N) )
q,p c b
Theorem 2 At the critical level c = −N, the operators generated by
b
t(z) = Tr(L(z)) = Tr L+(q2cz)L−(z)−1 (3.1)
(cid:16) (cid:17)
lie in the center of the algebra A (sl(N) ).
q,p c
Proof: Defining L±(z) ≡ (L±(z)−1b)t, one can derive from eqs. (2.29) further exchange relations
between the operators L+ and L−:
e
R12(z/w)t2 −e1 L±1(z)L±2(w) = L±2(w)L±1(z) R1∗2(z/w)t2 −1 , (3.2a)
(cid:16) (cid:17) −1 (cid:16) (cid:17) −1
Rb12(q2cz/w)t2 L+1(ze)L−2(w) =e L−2(w)L+1(z)b R1∗2(q−2cz/w)t2 . (3.2b)
(cid:16) (cid:17) (cid:16) (cid:17)
b e 5 e b
Let us now compute t(z),L+(w) . One rewrites:
h i
t(z)L+(w) = Tr L+(q2cz)L−(z)t1 L+(w) = Tr L+(q2cz)t1L−(z)L+(w) (3.3)
1 1 1 2 1 1 1 2
(cid:16) (cid:17) (cid:16) (cid:17)
since one is allowed to exchange transpoesition under a trace procedure. Ceommuting L+(w) through
2
L−(z) using eq. (3.2b), one gets:
1
e t(z)L+(w) = Tr L+(q2cz)t1(R (q2cw/z)t1)−1L+(w)L−(z)R∗ (q−2cw/z)t1 . (3.4)
2 1 1 21 2 1 21
(cid:16) (cid:17)
Using the unitarity property of R , one hbas: e b
12
L+1(q2cz)t1 (R21(q−2cwb/z)−1)t1 L+2(w) = L+2(w)(R2∗1(q−2cw/z)−1)t1 L+1(q2cz)t1 . (3.5)
Then applying eq. (2.23) to (3.5) gives:
b b
L+(q2cz)t1 (R (wq−2c−N/z)t1)−1L+(w) = L+(w)(R∗ (q−2cw/z)−1)t1 L+(q2cz)t1 , (3.6)
1 21 2 2 21 1
which leads at the critical value c = −N to
b b
L+(q2cz)t1 (R (q2cw/z)t1)−1L+(w) = L+(w)(R∗ (q−2cw/z)−1)t1L+(q2cz)t1 . (3.7)
1 21 2 2 21 1
Now, inserting (3.7) into (3.4), the first three terms in (3.4) can be rearranged and one obtains:
b b
t(z)L+(w) = L+(w)Tr (R∗ (q−2cw/z)−1)t1L+(q2cz)t1L−(z)R∗ (q−2cw/z)t1 (3.8)
2 2 1 21 1 1 21
(cid:16) (cid:17)
b e b t2
Using the fact that under a trace over the space 1 one has Tr R Q R′ = Tr Q R′ t2R t2 ,
1 21 1 21 1 1 21 21
one gets (cid:16) (cid:17) (cid:16) (cid:17)
t(z)L+(w) = L+(w)Tr L+(q2cz)t1L−(z)R∗ (q−2cw/z)t1t2(R∗ (q−2cw/z)−1)t1t2 t2 (3.9)
2 2 1 1 1 21 21
(cid:16) (cid:17)
The last two terms in the right hand side canecel eacbh other, leavingba trivial dependence in space 2
and Tr1 L+1(q2cz)t1L−1(z) ≡ t(z) in space 1. This shows the commutation of t(z) with L+(w) and
therefor(cid:16)e with L−(w) = (cid:17)(g21hg21)L+(−p12q−2cw)(g21hg12)−1, hence with the full algebra Aq,p(sl(N)c)
e
at c = −N.
b
Thisdemonstration reproduces theproofforN = 2givenin[25]; notethatonlyinthesl(N) crossing-
symmetry relation (2.16) does N appear explicitly. The form of the operator (3.1) is identical to the
form of the commuting operator derived by [37] in the case of the quantum algebra U (sl(N) ). As
q c
in [25] the center of A (sl(N) ) at c = −N may contain other generators which we have not yet
q,p c
derived. However, t(z) do close on their own a Poisson algebra as we are going to show.b
b
3.2 Exchange algebra
In order to get the Poisson structure on t(z), we need to compute the exchange algebra between the
operators t(z) and t(w) when c 6= −N. From the definition of the element t(z), one has
t(z)t(w) = L(z)i1 L(w)i2 = L+(q2cz)j1L−(z)j1L+(q2cw)j2L−(w)j2. (3.10)
i1 i2 i1 i1 i2 i2
6 e e
Suitable rewritings of the relations (2.29) lead to the following exchange relations between the oper-
ators L+ and L−:
−1
L−(z)L+(w) = R (q2cw/z)t1 L+(w)L−(z)R∗ (q−2cw/z)t1, (3.11a)
e 1 2 21 2 1 21
(cid:16) (cid:17) −1
Le+(z)L−(w) = Rb(q2cz/w)t2 L−(w)L+(z)e R∗ (bq−2cz/w)t2 , (3.11b)
1 2 12 2 1 12
−1 (cid:16) (cid:17)
L−1(z)Le−2(w) = bR12(z/w)t1t2e L−2(w)L−1(zb)R1∗2(z/w)t1t2 . (3.11c)
(cid:16) (cid:17)
The exchange relations (2.29), (3.11) and the properties of the matrix R given in Theorem 1 then
e e b e e b 12
allow to move the matrices L+(w), L−(w) to the left of the matrices L+(z), L−(z). One obtains
t(z)t(w) = Y(w/z)i1i2 L(w)j2 L(z)j1 , (3.12)
e j1j2 i2 i1 e
where Y(w/z) = T(w/z)M(w/z) with
M(w/z) = R (w/z)R (qc+Nw/z)−1R (z/w)−1 t2 R (qcz/w)t2 t2 (3.13)
21 21 12 12
(cid:18) (cid:19)
(cid:16) (cid:17)
and
1 1
τN(q2z/w)τN(q2−cw/z)
T(w/z) = . (3.14)
1 1
τN(q2w/z)τN(q2−cz/w)
3.3 Poisson structures on the center of A (sl(N) ) at c = −N
q,p c
One enounces:
b
Theorem 3 The elements t(z) form a closed algebra under the natural Poisson bracket on the center
of A (sl(N) ) given by (with x = w/z)
q,p c
d 1 d 1
b {t(z),t(w)} = −lnq x lnτ (q2x)−x−1 lnτ (q2x−1) t(z)t(w). (3.15)
dx N dx−1 N
(cid:18) (cid:19)
Proof: At the critical level c = −N, it is easy to show by direct calculation from (3.13) and (3.14)
that
T(x) = 1 and M(x) = I ⊗I (3.16)
cr cr 2 2
One recovers immediately that t(z)t(w) = t(w)t(z) at the critical level c = −N. Hence a natural
Poisson structure can be defined by
dY i1i2
t(z),t(w) = (w/z) L(w)j2L(z)j1 . (3.17)
dc i2 i1 cr
n o (cid:18) (cid:19)j1j2 (cid:12)
(cid:12)
From eq. (3.16), one has (cid:12)
dY dT dM
(x) = (x) I ⊗I + (x) . (3.18)
2 2
dc dc dc
cr cr cr
(cid:12) (cid:12) (cid:12)
Now, one has (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12)
dM d t2 d t2
= −lnq y R R (q−Nz/w)t2 −(R (z/w)−1)t2 y R
21 12 12 12
dc cr ( dy y=w/z dy y=q−Nz/w )
(cid:18) (cid:19) (cid:18) (cid:19)
(cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (3.19)
(cid:12) (cid:12) (cid:12)
7
Taking now the derivative of the relation (2.23), one obtains
d t2 d t2
R (x)t2 −1 y R R (x)t2 −1 = −y R−1 (3.20)
12 dy 12 y=x 12 dy 12 y=xqN
(cid:18) (cid:19) (cid:18) (cid:19)
(cid:12) (cid:12)
(cid:0) (cid:1) (cid:0) (cid:1)
(cid:12) (cid:12)
which can be rewritten by virtue of eq.(cid:12)(2.23) as: (cid:12)
d t2 d t2
R (xqN)−1 t2 y R R (xqN)−1 t2 = −y R−1 (3.21)
12 dy 12 y=x 12 dy 12 y=xqN
(cid:18) (cid:19) (cid:18) (cid:19)
(cid:12) (cid:12)
(cid:0) (cid:1) (cid:0) (cid:1)
(cid:12) (cid:12)
Using then eq. (3.21) for the value x =(cid:12)q−Nz/w and inserting it into eq. (3.1(cid:12)9), one finds
dM d t2 d t2
= −lnq y R + y R−1 R (q−Nz/w)t2 (3.22)
dc dy 21 dy 12 12
cr ( y=w/z y=z/w )
(cid:18) (cid:19) (cid:18) (cid:19)
(cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) dM (cid:12)dY dT
Fromthe unitarityproperty, it followsthat = 0. Hence = I ⊗I . This guarantees
2 2
dc dc dc
cr cr cr
that the Poisson bracket of t(z) closes on t(z), (cid:12)a property not obv(cid:12)ious sinc(cid:12)e we have indicated that
(cid:12) (cid:12) (cid:12)
t(z) may not exhaust the center of A (sl(N) )(cid:12). (cid:12) (cid:12)
q,p c
Finally, using the qN-periodicity property of the function τ and eq. (3.16), the derivative of T(x)
N
is given by: b
dT dlnT d 1 d 1
(x) = (x) = −lnq x lnτ (q2x)−x−1 lnτ (q2x−1) . (3.23)
dc dc dx N dx−1 N
cr cr
(cid:18) (cid:19)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
3.4 Explicit Poisson structures at c = −N
The structure function in eq. (3.15) is easily computed. One obtains:
{t(z),t(w)} = f(w/z)t(z)t(w), (3.24)
where
2x2q2Nℓ x2q2Nℓ+2 x2q2Nℓ−2
f(x) = −2lnq − −
1−x2q2Nℓ 1−x2q2Nℓ+2 1−x2q2Nℓ−2
"
ℓ≥0 (cid:18) (cid:19)
X
x2 x2q2 x2q−2
− + 1 + 1 −(x ↔ x−1) . (3.25)
1−x2 21−x2q2 21−x2q−2
(cid:21)
We now define the Poisson structure for modes of the generating function derived from (3.24, 3.25).
The modes of t(z) are defined in the sense of generating functions (or formal series expansions):
dz
t = z−nt(z) (3.26)
n
2πiz
IC
where C is a contour encircling the origin.
Mode expansions when the structure function f(x) has an infinite set of poles at x = qP(k) (where
P(k) is integer) require a specific definition using the notion of “sectors”. This was done at the
classical level in [25]. The procedure runs as follows:
8
The Poisson bracket between the modes is given by a double contour integral:
dz dw
{t ,t } = z−nw−mf(w/z)t(w)t(z). (3.27)
n m
2πiz 2πiw
IC1 IC2
The function f(x) has here simple poles at x = ±q±Nℓ and x = ±q±Nℓ±1. Hence the relative position
of the contours C and C must be specified in order to have an unambiguous result for (3.27).
1 2
In addition, the antisymmetry of the Poisson brackets is only guaranteed at the mode level by an
explicit symmetrization of (3.27) with respect to the position of the contours C and C . We shall
1 2
comment on this fact when discussing the quantum mode-exchange structure. The mode Poisson
bracket is thus defined as:
1 dz dw dz dw
{t ,t } = + z−nw−mf(w/z)t(z)t(w), (3.28)
n m
2 2πiz 2πiw 2πiz 2πiw
(cid:18)IC1 IC2 IC2 IC1 (cid:19)
where C and C are circles of radii R and R and one chooses R > R . Explicit evalutation of
1 2 1 2 1 2
(3.28) now requires to express f(w/z) as a convergent Laurent series in the appropriate domains for
|w/z|.
R
Let us define the sector (k) by 1 ∈ |q|−P(k),|q|−P(k+1) , P(k) being the ordered set of powers
R
2
of q−1 where the poles of f are locatedi(here {P(k), k ∈ hN} = {0,1;N − 1,N,N + 1;...;Nℓ −
1,Nℓ,Nℓ + 1;...} respectively). For every sector (k), eq. (3.28) defines a distinct mode Poisson
bracket.
As in the sl(2) case, one observes the difference between the analytic continuation formula (3.15),
which is unique, andtheformalseries formula (3.28), where every k-labeledconvergent Laurent series
expansions for f may be taken as the formal series expansion of f. This fact is also mentioned in
[27], considering the quantum problem.
R
Proposition 1 In the case k = 0 (i.e. 1 ∈ 1,|q|−1 ), one finds
R
2
(cid:3) (cid:2)
[(N −1)r] [r]
{t ,t } = −2lnq(q −q−1) q q t t , (3.29)
n m k=0 n−2r m+2r
[Nr]
r∈Z q
X
where the q-numbers [r] are defined as usual:
q
qr −q−r
[r] ≡ . (3.30)
q q −q−1
The proof is immediate.
When k 6= 0, one must add to (3.29) contributions arising from the poles at q−P(j) with j = 1,...,k.
Proposition 2 The convergent series expansions in any sector (k) are obtained by adding to the
coefficients of the convergent series at k = 0, coefficients obtained from the canonical formal series
expansion of the distributions δ(q−P(j)w/z) + δ(−q−P(j)w/z) −δ(qP(j)w/z) − δ(−qP(j)w/z) for j =
1,...,k, where δ(x) ≡ xn for x ∈ C.
n∈Z
P
9
