DEFORMED VIRASORO and W ALGEBRAS

from ELLIPTIC ALGEBRAS

P. Sorba

Laboratoire d’Annecy-le-Vieux de Physique Théorique LAPTH, CNRS-URA 1436

LAPP, BP 110, F-74941 Annecy-le-Vieux Cedex, France

In memoriam M. SAVELIEV

Abstract

Deformed W_{n}-algebras in the classical (i.e. Poisson structures) and in the quantum cases (exchanges algebras) are constructed from the elliptic algebras A_{q,p}(ŝl(N)_{c}).

When I am thinking about Misha, and this has been happening rather frequently these recent months, it is his enthusiasm, his communicative energy which come first in my mind. Something was very important to Misha: that was friendship. We, scientists, have the priviledge to possess friends, real friends among our colleagues, and Misha was a friend. Something else was fundamental for Misha: his family,i.e. his parents, his wife, his daughter. As Professor Manin said this morning, summarising Misha’s behaviour very simply and elegantly, Misha was ”a good son, a good husband, a good father”. Misha’s parents died about three years ago, and I remember how he was sad, when meeting him in Georgia in september 96, to inform me about his father and mother death a few months before. Since then, he was concentrating all his forces to protect his wife and his daughter, the two Svetlanas, as he used to say! He was so proud of his gifted daughter: a good musician, a good painter, a brilliant student. I wish to present to them my most respectful thoughts.

Misha was a great expert in Integrable Systems. He was deeply interested in developing models, what he did so nicely in the framework of Toda theories. I learnt a lot from him on Toda theories and also on continuous Lie algebras, subjects on which we collaborated in the 90’s. Misha was highly convinced by the importance of symmetries and specially interested in the use of infinite symmetries, among them by the W algebras. The landscape of continuous symmetries and particularly of infinite dimensional symmetries is becoming richer and richer these years, and I dare to think that Misha would have been interested in the modest seminar I will present now.

The talk which follows is the result of a collaboration started about two years ago, and still in progress, with Jean Avan from Paris, Luc Frappat from Annecy and Marco Rossi from Pisa and to-day in Durham. Our results [1, 2, 3] concern the construction, from the quantum elliptic algebras A_{q,p}(ŝl(N)_{c}) of deformed Virasoro and more generally deformed W algebras. To some extent, our approach can be seen as a kind of Sugawara construction in the deformed case. One has to remark that our method differs from others [4, 5, 6] in two essential ways: first, it uses the algebras A_{q,p}(ŝl(N)_{c}); secondly, computation are directly performed from the abstract generators of the elliptic algebra, without help of a special realisation, which is the case in the above quoted papers based in particular on the q-deformed bosonisation of the quantum group U_{q}(ŝl(N)_{c}).

After introducing the algebra A_{q,p}(ŝl(N)_{c}) [7, 8], we first prove the existence of an extended center in A_{q,p}(ŝl(N)_{c}) at c = −N and compute the Poisson structures on this center. These structures are identified as q-deformed V ir_{q}(sl(N)) algebras. We then show the existence of closed (quadratic) exchange algebras whenever (−p^{\frac{1}{2}} )^{NM} = q^{−c−N} for any integer M ∈ ℤ. These algebras admit a classical limit (commuting algebras) at p = q^{Nh} with h ∈ ℤ\{0}. The related Poisson structures include for h even the Poisson structures in [4]. 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 V ir_{q}(sl(N)) structures. This emphasizes the key role of the initial 3-parameter structure A_{q,p}(ŝl(N)_{c}) in allowing for an intermediate quantum q-deformed Virasoro algebra. When computing the mode expansion of these structures (both classical and quantum), a “sector-type” labeling is needed due to the singularity behaviour of the structure function viewed as an analytic continuation and therefore exhibiting different formal series expansions corresponding to different convergent series expansions in distinct domains of the complex plane. Finally, we briefly describe the essential features of the extension of our computation to higher spin fields. We conclude by mentioning developments and projects, and pointing out the important role played by Drinfeld twists in providing a synthetic view of deformed algebras.

1 The elliptic algebra A_{q,p}(ŝl(N)_{c})

We start by defining the R-matrix of the ℤ_{N}-vertex model (ℤ_{N} ≡ ℤ/Nℤ) [7, 8]:

R̃(z, q, p) = \frac{z^{2/N−2}}{κ(z^{2})} \frac{ϑ [ \frac{1}{2} \frac{1}{2} ] (ζ, τ)}{ϑ [ \frac{1}{2} \frac{1}{2} ] (ξ + ζ, τ)} ∑_{(α_{1},α_{2})} W_{(α_{1},α_{2})}(ξ, ζ, τ) I_{(α_{1},α_{2})} ⊗ I_{(α_{1},α_{2})}^{−1} ,

where z = e^{iπξ}, q = e^{iπζ}, p = e^{2iπτ} and (α_{1}, α_{2}) ∈ ℤ_{N} × ℤ_{N}. ϑ are the Jacobi theta functions with rational characteristics (γ_{1}, γ_{2}) ∈ ℤ/N × ℤ/N:

ϑ [ \begin{matrix}γ_{1} \\ γ_{2} \end{matrix} ] (ξ, τ) = ∑_{m∈ℤ}exp (iπ(m + γ_{1})^{2}τ + 2iπ(m + γ_{1})(ξ + γ_{2})) .

The normalization factor is given by:

\frac{1}{κ(z^{2})} = \frac{(q^{2N}z^{−2}; p, q^{2N})_{∞} (q^{2}z^{2}; p, q^{2N})_{∞} (pz^{−2}; p, q^{2N})_{∞} (pq^{2N−2}z^{2}; p, q^{2N})_{∞}}{(q^{2N}z^{2}; p, q^{2N})_{∞} (q^{2}z^{−2}; p, q^{2N})_{∞} (pz^{2}; p, q^{2N})_{∞} (pq^{2N−2}z^{−2}; p, q^{2N})_{∞}} .

The functions W_{(α_{1},α_{2})} are defined as follows:

W_{(α_{1},α_{2})}(ξ, ζ, τ) = \frac{1}{N} \frac{ϑ [ \begin{matrix}\frac{1}{2} + α_{1}/N \\ \frac{1}{2} + α_{2}/N \end{matrix} ] (ξ + ζ/N, τ)}{ϑ [ \begin{matrix}\frac{1}{2} + α_{1}/N \\ \frac{1}{2} + α_{2}/N \end{matrix} ] (ζ/N, τ)} ,

and the matrices I_{(α_{1},α_{2})} by I_{(α_{1},α_{2})} = g^{α_{2}} h^{α_{1}} where g_{ij} = ω^{i}δ_{ij}, h_{ij} = δ_{i+1,j} are N ×N matrices (the addition of indices being understood modulo N) and ω = e^{2iπ/N}. The R-matrix R̃ is ℤ_{N}-symmetric:

R̃_{c+s , d+s}^{a+s , b+s} = R̃_{c , d}^{a , b} a, b, c, d, s ∈ ℤ_{N} .

We introduce the “gauge-transformed” matrix:

R(z, q, p) = (g^{\frac{1}{2}} ⊗ g^{\frac{1}{2}} )R̃(z, q, p)(g^{− \frac{1}{2}} ⊗ g^{− \frac{1}{2}} ) .

It satisfies the following properties:

– Yang–Baxter equation: R_{12}(z) R_{13}(w) R_{23}(w/z) = R_{23}(w/z) R_{13}(w) R_{12}(z),

– unitarity: R_{12}(z)R_{21}(z^{−1}) = 1,

– crossing symmetry: R_{12}(z)^{t_{2}}R_{21}(q^{−N}z^{−1})^{t_{2}} = 1,

– antisymmetry: R_{12}(−z) = ω (g^{−1} ⊗ 𝕀) R_{12}(z) (g ⊗ 𝕀),

– quasi-periodicity: R̂_{12}(−p^{\frac{1}{2}} z) = (g^{\frac{1}{2}} hg^{\frac{1}{2}} ⊗ 𝕀)^{−1} (R̂_{21}(z^{−1}))^{−1} (g^{\frac{1}{2}} hg^{\frac{1}{2}} ⊗ 𝕀),

where R̂_{12}(x) = τ_{N}(q^{1/2}x^{−1}) R_{12}(x) and τ_{N}(z) = z^{\frac{2}{N} −2} \frac{Θ_{q^{2N}} (qz^{2})}{Θ_{q^{2N}} (qz^{−2})}. The function τ_{N}(z) is periodic with period q^{N}: τ_{N}(q^{N}z) = τ_{N}(z) and satisfies τ_{N}(z^{−1}) = τ_{N}(z)^{−1}.

We now define the elliptic quantum algebra A_{q,p}(ŝl(N)_{c}) [7, 8]. It is an algebra of operators L_{ij}(z) ≡ ∑_{n∈ℤ} L_{ij}(n) z^{n} where i, j ∈ ℤ_{N}:

L(z) = ( \begin{matrix}L_{11}(z) & ⋯ & L_{1N}(z) \\ ⋮ & & ⋮ \\ L_{N1}(z) & ⋯ & L_{NN}(z) \end{matrix} ) .

The q-determinant is given by (ε(σ) being the signature of the permutation σ):

q- det L(z) ≡ ∑_{σ∈𝔖_{N}} ε(σ) ∏_{i=1}^{N} L_{i,σ(i)}(zq^{i−N−1}) .

A_{q,p}(ĝl(N)_{c}) is defined by imposing the following constraints on L(z):

R̂_{12}(z/w) L_{1}(z) L_{2}(w) = L_{2}(w) L_{1}(z) R̂_{12}^{∗}(z/w) ,

where L_{1}(z) ≡ L(z) ⊗ 𝕀, L_{2}(z) ≡ 𝕀 ⊗ L(z) and R̂_{12}^{∗}(z, q, p) ≡ R̂_{12}(z, q, p^{∗} = pq^{−2c}). The q-determinant is in the center of A_{q,p}(ĝl(N)_{c}) and one sets

A_{q,p}(ŝl(N)_{c}) = A_{q,p}(ĝl(N)_{c})/⟨q- det L − q^{\frac{c}{2}} ⟩ .

2 The center of A_{q,p}(ŝl(N)_{c})

We set L^{+}(z) ≡ L(q^{c/2}z) and L^{−}(z) ≡ g^{\frac{1}{2}} hg^{\frac{1}{2}} L(−p^{\frac{1}{2}} z) (g^{\frac{1}{2}} hg^{\frac{1}{2}} )^{−1}. Let us define the following operator in A_{q,p}(ŝl(N)_{c}):

t(z) ≡ Tr(L(z)) ≡ Tr (L^{+}(q^{c/2}z) L^{−}(z)^{−1}) .

From the defining relations of the elliptic algebra and the properties of the matrix R̂ (especially the quasi-periodicity property), one gets the following theorems: Theorem 1: At the critical level c = −N, the operators generated by t(z) lie in the center of the algebra A_{q,p}(ŝl(N)_{c}):

[t(z), L^{+}(w)] = [t(z), L^{−}(w)] = 0 .

Theorem 2: At the critical level c = −N, the operators generated by t(z) close an Abelian algebra:

[t(z), t(w)] = 0 .

This allows us to define a Poisson structure on the center of A_{q,p}(ŝl(N)_{c}) at c = −N. When c ≠ −N, one has:

t(z)t(w) = Y(w/z)_{j_{1}j_{2}}^{i_{1}i_{2}} L(w)_{i_{2}}^{j_{2}} L(z)_{i_{1}}^{j_{1}}
= t(w)t(z) + (c + N) (\frac{dY}{dc} (\frac{w}{z} ))_{j_{1}j_{2}}^{i_{1}i_{2}} L(w)_{i_{2}}^{j_{2}} L(z)_{i_{1}}^{j_{1}} |_{cr} + ⋯

The Poisson structure is then defined by:

\{t(z), t(w)\} = (\frac{dY}{dc} (\frac{w}{z} ))_{j_{1}j_{2}}^{i_{1}i_{2}} L(w)_{i_{2}}^{j_{2}} L(z)_{i_{1}}^{j_{1}} |_{cr} .

Decomposing Y(x) as Y(x) = T (x) M(x) where

T (x) = \frac{τ_{N}(q^{\frac{1}{2}} x^{−1})τ_{N}(q^{\frac{1}{2} −c}x)}{τ_{N}(q^{\frac{1}{2}} x)τ_{N}(q^{\frac{1}{2} −c}x^{−1})}

and

M(x) = ((R_{21}(x) R_{21}(q^{c+N}x)^{−1} R_{12}(x^{−1})^{−1})^{t_{2}} R_{12}(q^{c}x^{−1})^{t_{2}})^{t_{2}} ,

one gets

\frac{dY}{dc} (x)|_{cr} = \frac{dT}{dc} (x)|_{cr} 𝕀_{2} ⊗ 𝕀_{2} + \frac{dM}{dc}(x)|_{cr} ,

since T (x)_{cr} = 1 and M(x)_{cr} = 𝕀_{2} ⊗ 𝕀_{2} at the critical level c = −N. The properties of the matrix R_{12} imply the vanishing of the derivative of M with respect to c at the critical level and it follows that:

\{t(z), t(w)\} = \frac{dT}{dc} (\frac{w}{z} )|_{cr} t(z) t(w) ≡ f(w/z) t(z) t(w)

where

f(x) = −2 ln q [∑_{l≥0} (\frac{2x^{2}q^{2Nl}}{1 − x^{2}q^{2Nl}} − \frac{x^{2}q^{2Nl+2}}{1 − x^{2}q^{2Nl+2}} − \frac{x^{2}q^{2Nl−2}}{1 − x^{2}q^{2Nl−2}} )

−\frac{x^{2}}{1 − x^{2}} + \frac{1}{2} \frac{x^{2}q^{2}}{1 − x^{2}q^{2}} + \frac{1}{2} \frac{x^{2}q^{−2}}{1 − x^{2}q^{−2}} − (x ↔ x^{−1})] .

The Poisson structure on the center of A_{q,p}(ŝl(N)_{c}) at c = −N is not p depending.

We now define the Poisson structure for modes of the generating function f(x). The modes of t(z) are defined in the sense of formal series expansions: t_{m} = ∮_{C} \frac{dz}{2iπz} z^{−m} t(z), where C is a contour encircling the origin. However, the function f(x) has poles for x = ±q^{±Nl} and x = ±q^{±Nl±1} with l ≥ 0, therefore

∮_{C_{1}} \frac{dz}{2πiz} ∮_{C_{2}} \frac{dw}{2πiw} ≠ ∮_{C_{2}} \frac{dz}{2πiz} ∮_{C_{1}} \frac{dw}{2πiw} .

The relative position of the contours C_{1} and C_{2} must be specified in order to have an unambiguous result for the Poisson bracket. Moreover, the antisymmetry of the Poisson bracket is only guaranteed at the mode level by an explicit symmetrization with respect to the position of the contours C_{1} and C_{2}. The mode Poisson bracket is thus defined as:

\{t_{n}, t_{m}\} = \frac{1}{2} (∮_{C_{1}} \frac{dz}{2πiz} ∮_{C_{2}} \frac{dw}{2πiw} + ∮_{C_{2}} \frac{dz}{2πiz} ∮_{C_{1}} \frac{dw}{2πiw} )
z^{−n}w^{−m} f(w/z) t(z) t(w)

where C_{1} and C_{2} are circles of radii R_{1} and R_{2} and one chooses R_{1} > R_{2}.

The Poisson bracket depends on the relative positions of the contours C_{1} and C_{2}, circles of radii R_{1} and R_{2}. Thus we introduce the notion of sectors: the sector (k) is defined by R_{1}/R_{2} ∈ ]|q|^{−P (k)}, |q|^{−P (k+1)}[ where {P (k)} is the ordered set of powers of q^{−1} where the poles of f are located: {P (k)} = {0, 1; N − 1, N, N + 1; …; Nl − 1, Nl, Nl + 1; … }.

In the case of sl(N), for R_{1}/R_{2} ∈]1, |q|^{−1}[ (first sector), one gets the following formula:

\{t_{n}, t_{m}\}_{k=0} = −2(q − q^{−1}) ln q ∑_{r∈ℤ} \frac{[(N − 1)r]_{q}[r]_{q}}{[Nr]_{q}} t_{n−2r} t_{m+2r}

where [r]_{q} are q-numbers: [r]_{q} ≡ \frac{q^{r} − q^{−r}}{q − q^{−1}} .

In the case of sl(2), for R_{1}/R_{2} ∈]|q|^{±k}, |q|^{±(k+1)}[, it is possible to get a nice compact formula for any sector:

\{t_{n}, t_{m}\}_{k} = (−1)^{k+1}2 ln q ∮_{C_{1}} \frac{dz}{2πiz} ∮_{C_{2}} \frac{dw}{2πiw}

∑_{s∈ℤ} \frac{q^{(2k+1)s} − q^{−(2k+1)s}}{q^{s} + q^{−s}} (\frac{z}{w})^{2s} z^{−n}w^{−m} t(z) t(w) .

It follows that one obtains in this way a family of Poisson structures indexed by k ∈ ℤ, the sector k = 0 corresponding to the Poisson bracket in [4].

To realize deformed W_{N} Poisson structures, we need to introduce generating functions for the higher spin objects. Having at our disposal only one commuting generating function t(z), we are led to define shifted products, although with the same generator. We define therefore the higher spin objects for 1 ≤ i ≤ N − 1 as:

s_{i}(z) = ∏_{u=−(i−1)/2}^{(i−1)/2} t(q^{u}z)

The Poisson algebra reads as:

\{s_{i}(z), s_{j}(w)\} = ∑_{u=−(i−1)/2}^{(i−1)/2} ∑_{v=−(j−1)/2}^{(j−1)/2} f(q^{v−u} \frac{w}{z} ) s_{i}(z) s_{j}(w) .

As before, one can introduce the notion of sectors for the Poisson brackets of the mode expansions, now labelled by k(i, j) [3]. For the sector k = 0, one gets:

\{s_{i}(n), s_{j}(m)\} = − ∑_{r∈ℤ} \frac{[(N − max(i, j))r]_{q}[min(i, j)r]_{q}}{[Nr]_{q}}
×2(q − q^{−1}) ln q s_{i}(n − 2r)s_{j}(m + 2r) . (∗)

Any Poisson structure in a given sector k(i, j) can be obtained from (*) by adding to the rdependent structure coefficient contributions from the relevant singularities of the structure function, i.e. power series expansions of terms δ(±q^{±P_{ij}(s)}w/z) s_{i}(z) s_{j}(w) for 1 ≤ s ≤ k(i, j).

3 W_{q,p}(sl(N)) exchange algebras

3.1 q-Virasoro exchange algebras

While for c = −N, t(z) and L(w) commute, for (−p^{\frac{1}{2}} )^{NM} = q^{−c−N} (M ∈ ℤ), t(z) and L(w) realize an exchange algebra:

t(z) L(w) = F (M, w/z) L(w) t(z)

where

F (M, x) = \{ \begin{matrix}q^{2M(N−1)} ∏_{k=0}^{NM−1} \frac{Θ_{q^{2N}} (x^{−2}p^{−k}) Θ_{q^{2N}} (x^{2}p^{k})}{Θ_{q^{2N}} (x^{−2}q^{2}p^{−k}) Θ_{q^{2N}} (x^{2}q^{2}p^{k})} & for M > 0 , \\ q^{−2|M|(N−1)} ∏_{k=1}^{N|M|} \frac{Θ_{q^{2N}} (x^{−2}q^{2}p^{k}) Θ_{q^{2N}} (x^{2}q^{2}p^{−k})}{Θ_{q^{2N}} (x^{−2}p^{k}) Θ_{q^{2N}} (x^{2}p^{−k})} & for M < 0 . \end{matrix}

It follows that one gets the following theorem: Theorem 3: On the 2d-surface Σ_{N,M} = \{(p, q) | (−p^{\frac{1}{2}} )^{NM} = q^{−c−N}\} of the 3d-space of parameters q, p, c, the operators generated by t(z) close a quadratic algebra:

t(z) t(w) = Y_{N,p,q,M}(\frac{w}{z} ) t(w) t(z)

Y_{N,p,q,M}(x) = \{ \begin{matrix}∏_{k=1}^{NM} \frac{Θ_{q^{2N}}^{2} (x^{2}p^{−k}) Θ_{q^{2N}} (x^{2}q^{2}p^{k}) Θ_{q^{2N}} (x^{2}q^{−2}p^{k})}{Θ_{q^{2N}}^{2} (x^{2}p^{k}) Θ_{q^{2N}} (x^{2}q^{2}p^{−k}) Θ_{q^{2N}} (x^{2}q^{−2}p^{−k})} & for M > 0 , \\ ∏_{k=1}^{N|M|−1} \frac{Θ_{q^{2N}}^{2} (x^{2}p^{−k}) Θ_{q^{2N}} (x^{2}q^{2}p^{k}) Θ_{q^{2N}} (x^{2}q^{−2}p^{k})}{Θ_{q^{2N}}^{2} (x^{2}p^{k}) Θ_{q^{2N}} (x^{2}q^{2}p^{−k}) Θ_{q^{2N}} (x^{2}q^{−2}p^{−k})} & for M < 0 . \end{matrix}

For N = 2, one has

Y_{2,p,q,M}(xq^{2}) = Y_{2,p,q,M}(x) ,
Y_{2,p,q,M}(x) Y_{2,p,q,M}(xq) = 1 .

One recovers the same periodicity properties satisfied by structure function φ(x) of [5] and [6] in the case of sl(2) (when M = 1, one makes the change of variable q^{2} → p, p → q, x^{2} → x, then Y_{2,p,q,1}(x) → φ^{2}(x) of [6]). For N > 2, one has

Y_{N,p,q,M}(xq^{N}) = Y_{N,p,q,M}(x) ,
Y_{N,p,q,M}(x) Y_{N,p,q,M}(xq) … Y_{N,p,q,M}(xq^{N−1}) = 1 .

3.2 W_{q,p}(sl(N)) exchange algebras

The generalization to higher spin generators in the quantum case follows that of the classical case, although with a required notion of ordering between individual t(z)-generators:

s_{i}(z) = ↶ ; ∏ ; \frac{i−1}{2} ≥u≥− \frac{i−1}{2} t(q^{u}z) .

The exchange algebra takes now the form:

s_{i}(z)s_{j}(w) = ∏_{u=− \frac{i−1}{2}}^{\frac{i−1}{2}} ∏_{v=− \frac{j−1}{2}}^{\frac{j−1}{2}} Y_{N,p,q,M} (q^{v−u} \frac{w}{z} ) s_{j}(w) s_{i}(z) .

Furthermore one can verify that s_{N}(z) commutes with all other generators; hence we are justified in restricting i to 1, …, N − 1.

3.3 Classical limit

Theorem 4: On the surface Σ_{N,M}, when p = q^{Nh} with h ∈ ℤ\{0}:

Y_{N,p,q,M} = 1 .

Hence t(z) realizes an Abelian subalgebra in A_{q,p}(ŝl(N)_{c}) (though it is not in the center of A_{q,p}(ŝl(N)_{c}) in general).

The result of the previous theorem allows us to define Poisson structures on the corresponding Abelian algebras generated by t(z). They are obtained as limits of the exchange algebra of Theorem 3 when p = q^{Nh} with h ∈ ℤ\{0}. Setting q^{Nh} = p^{1−β}, the h-labeled Poisson structure is defined by:

\{t(z), t(w)\}^{(h)} ≡ lim_{β→0} \frac{1}{β} (t(z)t(w) − t(w)t(z)) .

For h even, {t(z), t(w)}^{(h)} coincides with the Poisson structure of the critical case c = −N, while for h odd, {t(z), t(w)}^{(h)} gives new Poisson structures (for sl(N) when N > 2). Explicitly, one has the following expression:

{t(z), t(w)}^{(h)} = f_{h}(w/z) t(z) t(w)

with

f_{h}(x) = N_{odd} [∑_{ℓ≥0} E(\frac{NM}{2} )(E(\frac{NM}{2} ) + 1) (\frac{2x^{2}q^{2Nℓ}}{1 − x^{2}q^{2Nℓ}} − \frac{x^{2}q^{2Nℓ+2}}{1 − x^{2}q^{2Nℓ+2}}

−\frac{x^{2}q^{2Nℓ−2}}{1 − x^{2}q^{2Nℓ−2}} ) + E(\frac{NM+1}{2})^{2} (\frac{2x^{2}q^{2Nℓ+N}}{1 − x^{2}q^{2Nℓ+N}} − \frac{x^{2}q^{2Nℓ+N+2}}{1 − x^{2}q^{2Nℓ+N+2}}

−\frac{x^{2}q^{2Nℓ+N−2}}{1 − x^{2}q^{2Nℓ+N−2}} ) − \frac{1}{2}E(\frac{NM}{2} )(E(\frac{NM}{2} ) + 1) (\frac{2x^{2}}{1 − x^{2}} − \frac{x^{2}q^{2}}{1 − x^{2}q^{2}}

−\frac{x^{2}q^{−2}}{1 − x^{2}q^{−2}} ) − (x ↔ x^{−1})]  for h odd

= N_{even} [∑_{ℓ≥0} (\frac{2x^{2}q^{2Nℓ}}{1 − x^{2}q^{2Nℓ}} − \frac{x^{2}q^{2Nℓ+2}}{1 − x^{2}q^{2Nℓ+2}} − \frac{x^{2}q^{2Nℓ−2}}{1 − x^{2}q^{2Nℓ−2}} )
−\frac{1}{2} (\frac{2x^{2}}{1 − x^{2}} − \frac{x^{2}q^{2}}{1 − x^{2}q^{2}} − \frac{x^{2}q^{−2}}{1 − x^{2}q^{−2}} ) − (x ↔ x^{−1})]  for h even

where N_{odd} = 2Nh ln q, N_{even} = N^{2}M(NM + 1)h ln q and the notation E(n) means the integer part of the number n.

The mode Poisson brackets in the sector k = 0 is given by:

\{t_{n}, t_{m}\}^{(h)} = −2(q − q^{−1}) ln q ∑_{r∈ℤ}(−E(\frac{NM+1}{2})^{2} \frac{[r]_{q}^{2}}{[Nr]_{q}}

+E(\frac{NM}{2} )(E(\frac{NM}{2} ) + 1)\frac{[(N − 1)r]_{q}[r]_{q}}{[Nr]_{q}} ) t_{n−2r} t_{m+2r}

for h odd
= −2(q − q^{−1}) ln q ∑_{r∈ℤ} \frac{[(N − 1)r]_{q}[r]_{q}}{[Nr]_{q}} t_{n−2r} t_{m+2r}
for h even

Since the initial non-abelian structure for t(z) is closed, the exchange algebras

t(z) t(w) = Y_{N,p,q,M}(w/z) t(w) t(z)

are inequivalent quantizations with respect to p of the Poisson bracket which defines V ir_{q}(sl(N)): t(z) generates a V ir_{q,p}(sl(N)) algebra.

The generalization for the higher spin generators s_{i}(z) presents no conceptual difficulty. The following results are obtained:

– for h even, {s_{i}(z), s_{j}(w)}^{(h)} coincides with the Poisson structure of the critical case c = −N. The Poisson bracket is again identical to the “core” contribution in [4] (i.e. without lower-spin extensions).

– for h odd, {s_{i}(z), s_{j}(w)}^{(h)} gives rise to two new types of quadratic Poisson structures (that is q-deformed classical W_{N}-algebras) whether i + j ≤ N or i + j > N.

4 Conclusion

Here above is summarized the construction, from elliptic algebras, of the deformed Virasoro and W algebras in the classical case as well as in the quantum case. We can mention that a study for consistent central extensions, providing in particular general results for the simplest cases of V ir_{q}(sl(2)) has very recently been achieved in ref. [11]. Also, a direct determination of abstract generators for q-deformed W_{N} algebras has been carried out. Such an universal construction of W_{q,p} algebras is achieved in [12].

An important application of the above developed constructions is the determination of integrable models submitted to such symmetries. It is known that the trigonometric Calogero– Moser model is related to the Virasoro algebra and the Ruijsenaars–Schneider model to the q-deformed Virasoro algebra considered in ref. [5]. One expects the existence of generalized Ruijsenaars–Schneider models the symmetries of which would be given by the just described q-Virasoro exchange algebras. For such a program, the bosonization of these algebras appears as an essential tool, which is still missing.

Let us conclude by remarking that the Yang-Baxter equation admits two classes of elliptic solutions, denoted vertex-type and face-type solutions, and to which respectively correspond the elliptic algebras A_{q,p}(ŝl(N)_{c}) and B_{q,λ}(G) , G being any simple Lie algebra. Each of these algebras can be obtained from the quantum affine algebra U_{q}(G_{c}) by performing a Drinfeld twist [8]. It might also be interesting to note that the second kind of well known algebra deformation, the Yangian Y (G),can be incorporated in this scheme, via its double, in the following way. The scaling limit q=1 of the elliptic algebra A_{q,p}(ŝl(N)_{c}) with p = q^{2}η, η fixed, leads to the deformed centrally extended double Yangian structure DY_{r}(ŝl(N))_{c},in which a treatment analogous to the one presented in the previous sections and leading to exchange subalgebras can be performed (see [13] and ref.therein). Taking the η inf. limit then provides the centrally extended double Yangian DY (ŝl(N))_{c}. We have summarised such connections between these algebras in the following tableau. We hope it will give a more synthetic view in the landscape of deformed algebras.

5 Acknowledgements

I wish to express my gratitude to Profs. C. Devchand, Yu. I. Manin and M. Olshanetsky for organizing this meeting in honour of Misha Saveliev and for giving to me the opportunity to participate.

Face Models

Vertex Models

Drinfel’d twists

B_{q,λ}(Ĝ_{c}) A_{q,p}(ŝl(N)_{c})

DY_{η}(ŝl(2)_{c})
U_{q}(Ĝ_{c})
Y (G), DY (Ĝ)_{c}
Ĝ_{c}

U_{q}(G)

G

simple finite dim. Lie algebra

[figure]

[figure]

[figure]

[figure]

[figure]

[figure]

[figure]

[figure]

[figure]

W_{q,p} algebras

W algebras

“finite” W algebras

η → ∞

p → 0

q → 1, p = q^{2η}
η fixed

A schematic view of deformed algebras in the set of current continuous symmetries

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