
1. Introduction.
1.1.
In this paper we generalize some results concerning affine Kac-Moody algebras at the critical level to the corresponding quantized universal enveloping algebras. Here is the short description of these results for the affine algebras.
1.1.
(i) Let U ( g) cr be a completion of the universal enveloping algebra of an affine algebra g at the critical level −h ∨ (the precise definition is given in § 2). This algebra possesses a large center Z( g), which has a natural Poisson structure. B. Feigin and E. Frenkel have shown that Z( g) is isomorphic to the classical W-algebra W(g L ) associated to the simple Lie algebra g L , which is Langlands dual to g [1] .
1.1.
(ii) The W-algebra W(g L ) consists of functionals on a certain Poisson manifold C(g L ) obtained by the Drinfeld-Sokolov hamiltonian reduction [2] from a hyperplane in the dual space to the affine algebra g L . Elements of C(g L ), called g L -opers in [3] , can be considered as connections on a certain G L -bundle over the circle with some extra structure. To a g L -oper one can attach a g-module, on which the center acts according to the corresponding character. These gmodules can be considered as analogues of admissible representations of a simple group over a local non-archimedian field. They can be used in carrying out the geometric Langlands correspondence proposed by A. Beilinson and V. Drinfeld [3] . ( iii) The Wakimoto realization of g [4, 5, 1, 6 ] provides a map from U( g) cr to the tensor product of a certain Heisenberg algebra and a commutative algebra H(g). The restriction of this map to Z( g) gives us a homomorphism Z( g) → H(g), which is an analogue of the Harish-Chandra homomorphism. The corresponding map W(g L ) → H(g) is nothing but the Miura transformation, which has been defined for an arbitrary g by V. Drinfeld and V. Sokolov [2] .
1.1.
(iv) The algebra H(g) consists of functionals on a hyperplane F in the dual space to the homogeneous Heisenberg subalgebra h of g. The algebra H(g) is the classical limit of a completion of U( h), and hence it is a Heisenberg-Poisson algebra. The Miura transformation W(g L ) → H(g) is a homomorphism of Poisson algebras. For example, the center of U ( sl 2 ) cr is generated by the Sugawara operators, and C(sl 2 ) is isomorphic to a hyperplane L in the dual space to the Virasoro algebra. Thus, the center of U ( sl 2 ) cr is isomorphic to the Poisson algebra W(sl 2 ) of functionals on L. The Poisson structure on L is often called the second Gelfand-Dickey structure. We call W(sl 2 ) the classical Virasoro algebra.
1.1.
Elements of L can be considered as projective connections on the circle, i.e. differential operators of the form ∂ 2 t − q(t); these are the sl 2 -opers. On the other hand, elements of F can be considered as connections on a rank one bundle over the circle, i.e. differential operators of the form ∂ t − 1 2 χ(t). The Miura transformation sends a connection ∂ t − 1 2 χ(t) to the projective connection
1.1.
This gives us a homomorphism of Poisson algebras W(sl 2 ) → H(sl 2 ).
1.1.
In [7, 8] these results were used to give a new interpretation of the Bethe ansatz in the Gaudin models of statistical mechanics. This allowed to gain new insights on completeness of Bethe ansatz, and to relate Bethe ansatz to the geometric Langlands correspondence.
1.2.
There are many indications that these results can be generalized to the center of a completion U h ( g) cr of the quantum affine algebra U q ( g) at the critical level. An explicit construction of central elements of a quantum affine algebra at the critical level has been given by N. Reshetikhin and M. Semenov-Tian-Shansky [9] . Later, J. Ding and P. Etingof [10] showed that those elements generate all singular vectors of imaginary weight in Verma modules over U h ( g) cr . This makes us to believe that the center of U h ( g) cr is generated, in an appropriate sense, by the elements constructed in [9] .
1.2.
The center Z h ( g) of U h ( g) cr possesses a natural Poisson structure, which is a qdeformation of the Poisson structure on Z( g). A natural question is to describe Z h ( g) and its spectrum.
1.2.
In this paper we do this explicitly for U h ( sl 2 ) cr by using its Wakimoto realization. Our results for U h ( sl 2 ) cr can be summarized as follows.
1.2.
(i) The center Z h ( sl 2 ) of U h ( sl 2 ) cr contains the Fourier coefficients of a power series ℓ(z) given in [9] in terms of the Reshetikhin-Semenov-Tian-Shansky (RS) realization of U h ( sl 2 ) cr . We rewrite ℓ(z) in terms of the Drinfeld realization [11] , using the explicit isomorphism between the two realizations established by J. Ding and I. Frenkel [12] . This gives us a q-analogue of the Sugawara formula. (ii) Wakimoto realizations of U q ( sl 2 ) have been given in [13] in terms of the Drinfeld realization; we use the presentation due to H. Awata, S. Odake, and J. Shiraishi [14] . It gives us a homomorphism from U h ( g) cr to the tensor product of a certain Heisenberg algebra and a Heisenberg-Poisson algebra H h (sl 2 ). Its restriction to Z h ( sl 2 ) is a homomorphism of Poisson algebras Z h ( sl 2 ) → H h (sl 2 ), which is a q-deformation of the Miura transformation. (iii) We find the image of ℓ(z) in H h (sl 2 ):
1.2.
where Λ(z) is a generating function of elements of H h (sl 2 ). Using the Poisson structure on H h (sl 2 ) we compute the Poisson structure on Z h ( sl 2 ):
1.2.
This gives us a q-deformation of the classical Virasoro algebra. Formula (1.2) shows that if we attach to Λ(z) a first order q-difference operator
1.2.
we can attach in a natural way a second order q-difference operator of the form D q + D 
1.2.
was used by R. Baxter [15] in his study of the eight vertex model. Similar formulas appeared in [16] as the result of computation of the spectrum of the transfer-matrix of the six vertex model, an integrable model of statistical mechanics associated to U q ( sl 2 ). In this context, the function Q(z) is a product of a "vacuum value" and a polynomial, whose zeroes are solutions of Bethe ansatz equations.
1.2.
1.3. Thus, we have interpreted formulas (1.2) and (1.4) as a hamiltonian map, which can be considered as a q-analogue of the Miura transformation. In fact, the Miura transformation plays the same role as Baxter's formula (1.4) in the Gaudin models, cf. [17, 7, 8] .
1.2.
The Miura transformation (1.1) is the classical limit of the free field realization of the Virasoro algebra. Free field realizations play an important role in conformal field theory, cf. [6] . It is quite remarkable that a q-analogue of free field realization appears in the context of Bethe ansatz in statistical mechanics.
1.2.
Analogues of formula (1.2) for transfer-matrices of integrable models associated to other quantum affine algebras are known, cf. e.g. [18, 21] for U q ( sl N ), [19, 20, 22] for other U q ( g). Motivated by our computation in the case of U q ( sl 2 ) we expect that the formulas for the q-deformation of the Miura transformation of the center of U h ( g) cr coincide with the formulas for the transfer-matrices corresponding to U q ( g).
1.2.
In particular, for g = sl N we obtain the following picture.
1.2.
(i) In [9] the generating functions of central elements ℓ 1 (z), . . . , ℓ N −1 (z) of U h ( sl N ) cr corresponding to the fundamental representations have been constructed. The Fourier coefficients of
1.2.
which is closed with respect to the Poisson structure. (ii) The Wakimoto realization of U h ( sl N ) cr [14] gives rise to a homomorphism of
1.2.
Poisson algebras
1.2.
, where H h (sl N ) is a Heisenberg-Poisson algebra. This is a q-deformation of the Miura transformation. We find a formula for the image s i (z) of each generating function ℓ i (z) in H h (sl N ). These formulas match formulas for the spectra of the corresponding transfer-matrices in integrable models associated to U q ( sl N ) [18, 21] . (iii) We explicitly compute the Poisson brackets between s i (z)'s in H h (sl N ) generalizing formula (1.3). Thus, we obtain an interesting Poisson subalgebra W h (sl N ) of the Heisenberg-Poisson algebra H h (sl N ), which is a q-deformation of the classical W-algebra W(sl N ). (iv) Recall that elements of the spectrum of W(sl N ) can be considered as Nth order differential operators. We show that elements of the spectrum of W h (sl N ) can be considered as Nth order q-difference operators of the form
1.2.
We generalize (i) and (ii) to all quantum affine algebras of classical types. The computation of Poisson brackets is straightforward, and will be given in our next paper [23] along with results regarding quantum affine algebras of exceptional types.
1.2.
Using our results in the same way as in [7] we can give a new interpretation of the Bethe ansatz in integrable models associated to quantum affine algebras. This and other applications will be discussed in [23] .
1.2.
The paper is organized as follows. In § 2 we recall results concerning the center of U( sl 2 ) cr and Miura transformation. In § § 3-5 we consider the Drinfeld and the RS realizations of U h ( sl 2 ) k and the isomorphism between them. In § 6 we rewrite the RS formula for the generating function ℓ(z) of central elements in U h ( sl 2 ) cr in terms of the Drinfeld realization. In § § 7-9 we recall the Wakimoto realization of U h ( sl 2 ) k , and use it to find an explicit formula for the image of ℓ(z) in H h (sl 2 ) and to compute the Poisson bracket on Z h ( sl 2 ). In § § 10 and 11 we generalize these results to U h ( sl N ) cr and other quantum affine algebras of classical types.
1.2.
2. The center of U ( sl 2 ) at the critical level.
1.2.
2.1. The structure of the center. Let g be a finite-dimensional simple Lie algebra. The affine algebra g is the extension of g ⊗ C[t, t −1 ] by a one-dimensional center CK. For A ∈ g, n ∈ Z, denote A[n] = A ⊗ t n and put
1.2.
Introduce a completion U ( g) of U( g), the universal enveloping algebra of g:
1.2.
This is an associative algebra. It acts on g-modules M which satisfy the following property: for any
1.2.
. The center of U( g) k,loc has been described in [1] . It consists of the constants when k = −h ∨ , but becomes "large" when k = −h ∨ . Let us recall its description in the case g = sl 2 ; in this case h ∨ = 2. Let {e, h, f } be the standard basis of sl 2 . Introduce the generating function of the Sugawara operators S n by formula
1.2.
for any A ∈ g, and
1.2.
There is a natural Poisson structure on the center
1.2.
Then the formula {A, B} = C defines a Poisson bracket on Z( g), which does not depend on the liftings.
1.2.
We obtain from formula (2.1):
1.2.
Consider the hyperplane L in the dual space to the Virasoro algebra, which consists of those linear functionals on the Virasoro algebra which take value −6 on the central element (this corresponds to the factor −1/2 = −6/12 in the second term of formula (2.2) ). This hyperplane is isomorphic to the space of projective connections on the formal punctured disc ∂ 2 z − q(z), where q(z) ∈ C((z)), in the sense that the natural action of vector fields on it coincides with the coadjoint action of the Virasoro algebra on L. Let W(sl 2 ) be the Poisson algebra of local functionals on L. It is the classical limit of the local completion of the universal enveloping algebra of the Virasoro algebra. Therefore we call W(sl 2 ) the classical Virasoro algebra.
1.2.
In the case g = sl 2 the result of [1] is that the center Z( sl 2 ) of U( sl 2 ) −2,loc is isomorphic to W(sl 2 ). This isomorphism sends S n to the local functional ∂ 2 z − q(z) → q(z)z n+1 dz. According to formula (2.2), this is an isomorphism of Poisson algebras.
Wakimoto modules and Miura transformation.
Consider the Heisenberg algebra Γ with generators a n , a * n , n ∈ Z, and relations [a n ,
Wakimoto modules and Miura transformation.
[a n , a * m ] = δ n,−m , and the Heisenberg algebra H ′ (sl 2 ) with generators χ n , n ∈ Z, and 1 and relations
Wakimoto modules and Miura transformation.
Introduce the generating functions
Wakimoto modules and Miura transformation.
We define an embedding φ of sl 2 into a completion of Γ ⊗ H ′ (sl 2 ) by the formulas:
Wakimoto modules and Miura transformation.
The algebra Γ has a standard Fock representation M generated by a vector v, such that a n v = 0, n ≥ 0; a * n v = 0, n > 0. The algebra H ′ (sl 2 ) has a family of Fock representations π µ,κ , µ ∈ C, κ ∈ C, κ = 0, generated by a vector v µ,κ , such that
Wakimoto modules and Miura transformation.
It also has an infinite-dimensional family of one-dimensional representations C x(z) , where x(z) = n∈Z x n z −n−1 ∈ C((z)), on which χ n acts by multiplication by x n , and 1 acts by 0. Using the homomorphism φ we obtain representations of sl 2 of noncritical level in M ⊗ π µ,κ , and representations of critical level in M ⊗ C x(z) . These are called Wakimoto modules.
Wakimoto modules and Miura transformation.
The algebra H ′ (sl 2 ) can be considered as a deformation of the commutative algebra H(sl 2 ) = H ′ (sl 2 )/(1). It induces a Poisson structure on H(sl 2 ), which is called the Heisenberg-Poisson structure. It is determined by the following Poisson brackets of the generators:
Wakimoto modules and Miura transformation.
The homomorphism φ defines a homomorphism φ −2 from sl 2 to a completion of Γ ⊗ H(sl 2 ), which maps K to −2. One can check that under φ −2 , the Sugawara series S(z) is mapped to
Wakimoto modules and Miura transformation.
Therefore φ −2 defines an embedding of Z( sl 2 ) into a completion of H(sl 2 ), which we call the Miura transformation.
Wakimoto modules and Miura transformation.
The Poisson structure between central elements A, B ∈ Z( sl 2 ) has been defined via the commutator of their liftings to U ( sl 2 ). The result does not depend on the choice of a lifting. Moreover, we will obtain the same result if we take the commutator between the liftings of the images A ′ , B ′ of A, B in the completion of Γ ⊗ H(sl 2 ). Since the image of Z( sl 2 ) actually lies in the completion of H(sl 2 ), we can take liftings lying in the completion of H ′ (sl 2 ). But then the image of the Poisson bracket between A, B ∈ Z( sl 2 ) in H(sl 2 ) will coincide with the Poisson bracket between A ′ , B ′ ∈ H(sl 2 ). Hence the Miura transformation Z( sl 2 ) → H(sl 2 ) is a homomorphism of Poisson algebras.
Wakimoto modules and Miura transformation.
Therefore we can compute the Poisson structure on Z( sl 2 ) using formulas (2.4) and (2.3) . This gives us the Poisson bracket between the Sugawara operators, which coincides with formula (2.2).
Wakimoto modules and Miura transformation.
where A ij , i, j = 0, 1, are the entries of the Cartan matrix of sl 2 : A 00 = A 11 = −A 01 = −A 10 = 2.
Wakimoto modules and Miura transformation.
For any h, k ∈ C consider the quotient U h ( sl 2 ) k of U q ( sl 2 ) by the relations q = e h , K 0 K 1 = e hk . This is the Drinfeld-Jimbo realization of U h ( sl 2 ) k . There exist two other realizations of U h ( sl 2 ) k : Drinfeld's realization [11] , and Reshetikhin-Semenov-Tian-Shansky (RS) realization. The equivalence between the Drinfeld-Jimbo and the Drinfeld realizations has been established by V. Drinfeld [11] , cf. also [26, 27, 28] . The equivalence between the Drinfeld and the RS realizations has been established by J. Ding and I. Frenkel [12] . The equivalence between the Drinfeld-Jimbo and the RS realization follows from these two equivalences, but it can also be established directly along the lines of [12] .
Wakimoto modules and Miura transformation.
First we consider the Drinfeld realization. It is important for us because the Wakimoto realization is defined in terms of this realization.
Wakimoto modules and Miura transformation.
Introduce formal power series in x
Wakimoto modules and Miura transformation.
where
Wakimoto modules and Miura transformation.
is itself a series in q, which converges for |q| < 1 and can be analytically continued to the whole complex plane except for some roots of unity. Thus, we can extend f (x) as formal power series in x to all q except for the roots of unity. In what follows we will exclude roots of unity from consideration. Let h ∈ C\{2πiQ}, k ∈ C; put q = e h . We define an associative algebra
Wakimoto modules and Miura transformation.
The defining relations in
Wakimoto modules and Miura transformation.
These relations are understood as relations between formal powers series (cf. Remark 1).
Lemma 1. The Fourier coefficients of the power series
Consider the quotient of U h ( gl 2 ) k by the ideal generated by these elements. It has
Lemma 1. The Fourier coefficients of the power series
, n ∈ Z, as generators. There is a one-to-one correspondence between them and Drinfeld's generators which preserves relations: Drinfeld's ψ(z) (ψ + (z) of [14] ) is k
Lemma 1. The Fourier coefficients of the power series
The following Proposition then follows from [11] , cf. also [26, 27, 28] . Proposition 1. The quotient of U h ( gl 2 ) k by the ideal generated by the Fourier coefficients of the power series k
The RS realization.
Now we turn to the RS realization [9] . This realization originated from the Quantum Inverse Scattering Method, cf. [29, 30] . It is important for us because in this realization we can write explicit formulas for central elements [9] .
The RS realization.
Introduce the R-matrix
The RS realization.
where f (x) is given by formula (3.1).
The RS realization.
The matrix (4.1) is the result of computation of the universal R-matrix of U q ( sl 2 ) on the tensor product of two two-dimensional evaluation representations, cf. e.g. [31] . It satisfies the crossing-symmetry property:
The RS realization.
which follows from the existence of an isomorphism between the two-dimensional evaluation module and its double dual [31] . This is related to the fact that f (x) satisfies the q-difference equation
The RS realization.
Remark 2. Our R-matrix differs from that of [12] by the factor f (x) and by replacement of q by q −1 . It also differs from the R-matrix used in [32] by the factor which is a product of theta-functions.
The RS realization.
Let again h ∈ C\{2πiQ}, k ∈ C, q = e h . We define an associative algebra
The RS realization.
, where i, j = 1, 2, and n ∈ ∓Z + \0, and l
The RS realization.
where we put l
The RS realization.
The relations (4.2) and (4.3) are understood as relations between formal power series in z w , cf. Remark 1.
The isomorphism of two realizations.
Following [12] one can construct an explicit isomorphism between the algebras
The isomorphism of two realizations.
Consider the following decomposition
The isomorphism of two realizations.
In particular, we see that l 
Proposition 2 ([12]). The map ψ
Lemma 2. The Fourier coefficients of the power series
Proposition 2 ([12]). The map ψ
The image of the power series (5.2) under the map ψ is the series k 
Proposition 2 ([12]). The map ψ
6. The q-analogues of the Sugawara operators.
Proposition 2 ([12]). The map ψ
We define a completion U h ( g) k of U h ( g) k as follows:
Proposition 2 ([12]). The map ψ
where
Proposition 2 ([12]). The map ψ
It is easy to see that all Fourier coefficients of the power series
Proposition 2 ([12]). The map ψ
lie in U h ( sl 2 ) k . It follows from [9] that when k = −2 the coefficients of ℓ(z) are central elements of U h ( sl 2 ) cr .
Proposition 2 ([12]). The map ψ
We will now express ℓ(z) in terms of the Drinfeld realization using the isomorphism ψ. Let us put k = −2. Using formula (5.1) we obtain:
Proposition 2 ([12]). The map ψ
Applying formula (4.45) from [12] (in which the sign of the second summand in the right hand side has to be reversed) we obtain
Proposition 2 ([12]). The map ψ
and applying formula (4.25) from [12] we obtain e + (zq)k
Proposition 2 ([12]). The map ψ
Substituting (6.4) into (6.2) we obtain 6) and substituting (6.5) into (6.3) we obtain
Proposition 2 ([12]). The map ψ
Combining formula (6.1) with (6.6) and (6.7) we obtain
Proposition 2 ([12]). The map ψ
where : E(z)F (z) := e + (zq
Proposition 2 ([12]). The map ψ
Now we apply the isomorphism ψ. We see that ψ(: E(z)F (z) :) is a normally ordered product of the power series E(z) and F (z):
Proposition 2 ([12]). The map ψ
where
Proposition 2 ([12]). The map ψ
Thus, we obtain Proposition 3. The Fourier coefficients of the power series
Proposition 2 ([12]). The map ψ
These are the q-analogues of the Sugawara operators.
The Wakimoto realization of
Now we will describe a homomorphism φ h,k from U h ( sl 2 ) k to a completion of a quantum Heisenberg algebra. The map φ h,k is a q-analogue of the map φ defined in § 2. Such homomorphisms have been constructed in [13] . We will use the AwataOdake-Shiraishi construction [14] with some modifications. Introduce the quantum Heisenberg algebra A h,k (sl 2 ). The generators are λ n , b n , c n ,
The Wakimoto realization of
where q = e h and [n] q = q n − q −n q − q −1 . The generators b n and c n coincide with the corresponding generators of [14] . The generators λ n are related to the generators a n of [14] by the formula
The Wakimoto realization of
q n + q −n a n (recall that we have assumed that q is not a root of unity).
The Wakimoto realization of
We form the generating functions:
The Wakimoto realization of
The series Λ ± (z) is related to the series a ± (z) from [14] by the formula
The Wakimoto realization of
The other series are the same as in [14] . The relations between these series, in the sense of formal power series (cf. Remark 1), are the following (cf. [14] ):
The Wakimoto realization of
We define the completion A h,k (sl 2 ) of A h,k (sl 2 ) as follows:
The Wakimoto realization of
where I n is the left ideal of A h,k (sl 2 ) generated by all polynomials in λ m , b m , c m , m > 0, of degrees greater than or equal to n (we put deg
The Wakimoto realization of
The next proposition follows from [14] .
Proposition 4. There is a homomorphism
, which is defined on generators as follows:
Proposition 4. There is a homomorphism
Remark 3. Under the homomorphism φ defined in § 2.2, the affine algebra sl 2 embeds into a completion of the Heisenberg algebra Γ ⊗ H ′ (sl 2 ) generated by a n , a * n and χ n . The power series a(z) and a * (z) form the so-called βγ-system while the power series χ(z) is called a free scalar field. The βγ-system can be represented via exponentials of a pair of free scalar fields. The homomorphism φ then gives rise to a homomorphism φ ′ from sl 2 to a completion of the Heisenberg algebra generated by the Fourier coefficients of these two scalar fields and χ(z).
Proposition 4. There is a homomorphism
The power series b(z) and c(z) are q-analogues of the scalar bosonic fields representing the βγ system when q = 1. Thus, the homomorphism φ h,k is a q-deformation of φ ′ rather than φ.
Proposition 4. There is a homomorphism
When k = −2, the homomorphism φ h,k provides representations of U h ( sl 2 ) k in the Fock representation of the Heisenberg algebra A h,k , cf. [14] . These representations have one parameter -the action of λ 0 on the highest weight vector, cf. [14] . When k = −2, the generators λ n commute among themselves and generate a commutative algebra H h (sl 2 ). Therefore representations of U h ( sl 2 ) cr can be realized via φ h,k in a smaller space: the tensor product of the Fock representation of the subalgebra of A h,−2 generated by b n , c n , n ∈ Z, and a one-dimensional representation of H h (sl 2 ). For the action of U h ( sl 2 ) cr to be well-defined on this space, the action of Λ(z) should be given by a Laurent power series λ(z) (for example, this is the case if λ n , n > 0, act by 0). The corresponding representations W λ(z) are the q-analogues of the Wakimoto modules over U( sl 2 ) cr from § 2.2.
Deformation of the Miura transformation.
In this section we will apply φ cr h ≡ φ h,−2 to the generating function of central elements ℓ(z) given by (6.10). For brevity, in what follows we will use the same notation for elements of U h ( sl 2 ) cr and their images in A h,k (sl 2 ).
Deformation of the Miura transformation.
The normally ordered product : E(z)F (z) : can be written as
Deformation of the Miura transformation.
where C R and C r are circles around the origin or radii R > |w| and r < |w|, respectively.
Deformation of the Miura transformation.
Using the Wakimoto realization of U h ( sl 2 ) cr and formulas (7.2)-(7.4), we find in the region |w| > |z|, |w| > q 2 |z|, |w| > q −2 |z|:
Deformation of the Miura transformation.
We obtain the same formula for F (z)E(w) in the region |w| < |z|, |w| < q 2 |z|, |w| < q −2 |z|. We can therefore rewrite (8.1) as the integral of the this expression over the contour on the w plane surrounding the points z, zq 2 , zq −2 . Evaluating the residues, we find that
Deformation of the Miura transformation.
Using this formula we obtain
Deformation of the Miura transformation.
On the other hand, we have
Deformation of the Miura transformation.
Substituting these formulas into (6.10) we obtain a formula expressing the image of ℓ(z) in H h (sl 2 ) in terms of Λ ± (z).
Deformation of the Miura transformation.
where
Deformation of the Miura transformation.
This is a q-deformation of the Miura transformation (2.4).
Deformation of the Miura transformation.
Remark 4. There is a simpler way to obtain formula (8.6). Consider the action of s(z) = φ cr h [ℓ(z)] on the module W λ(z) introduced at the end of § 7. In the limit q → 1 this module becomes a Wakimoto module over sl 2 . Wakimoto modules are irreducible for generic values of parameters. Therefore the same is true for the modules W λ(z) . Hence any central element of U h ( sl 2 ) cr acts on W λ(z) by a constant. In particular, ℓ(z) acts by multiplication by a Laurent power series s(z). We can compute s(z) by taking the matrix element of ℓ(z) between the generating vector of W λ(z) and its dual using formulas (6.1), (5.1) and the maps ψ, φ cr h . Explicit computation shows that this matrix element is equal to λ(zq) + λ(zq −1 ) −1 . This implies that s(z) lies in H h (sl 2 ) and gives us formula (8.6) for s(z).
Deformation of the Miura transformation.
Remark 5. In [9] a generating function of central elements of U h ( g) cr has been associated to an arbitrary finite-dimensional representation of U q ( g). Thus, in the case of U h ( sl 2 ) cr we have a generating function ℓ (n) (z) of central elements associated to the representation of U q ( sl 2 ) of dimension n + 1 for each positive integer n. In particular, ℓ(z) = ℓ (1) (z). These generating functions satisfy the following relation:
Deformation of the Miura transformation.
Using this relation and formula (8.6) it is easy to find recursively:
Deformation of the Miura transformation.
(compare with [33])
Deformation of the Miura transformation.
9. Poisson bracket.
Deformation of the Miura transformation.
We can now compute the Poisson bracket between Fourier coefficients of s(z). They generate a central subalgebra of
Deformation of the Miura transformation.
Consider the algebra H h,k (sl 2 ) with generators λ n , n ∈ Z, exp(±λ 0 ) and relations (7.1). Let H h,k (sl 2 ) be its completion defined as follows:
Deformation of the Miura transformation.
where I n is the left ideal of H h,k (sl 2 ) generated by all polynomials in λ m , m > 0, of degrees greater than or equal to n.
Deformation of the Miura transformation.
The family H h,k (sl 2 ) induces a Poisson structure on H h (sl 2 ) ≡ H k,−2 (sl 2 ), such that
Deformation of the Miura transformation.
[2n] q δ n,−m (9.1) (recall that q = e h and that h ∈ 2πiQ). We define a Poisson bracket on the center of U h ( sl 2 ) cr in the same way as in § 2.1, as the leading term in the commutator of liftings of central elements. Formula (8.6) shows that the images of the Fourier coefficients of ℓ(z) under the homomorphism φ cr h lie in H h (sl 2 ). As was explained in § 2.2, we can take liftings of the images of central elements inside the deformation H h,k (sl 2 ) of H h (sl 2 ). This shows that the homomorphism from the center to H h (sl 2 ) is a homomorphism of Poisson algebras. We can use this fact to compute the Poisson brackets between Fourier coefficients of s(z).
Deformation of the Miura transformation.
From formulas (9.1) and (8.7) we obtain
Deformation of the Miura transformation.
According to formula (8.6), we have: {s(z), s(w)} = {Λ(zq), Λ(wq)} + {Λ(zq), Λ(wq
Deformation of the Miura transformation.
Substituting (9.2) into this formula, we obtain:
Deformation of the Miura transformation.
The last two terms give us
Deformation of the Miura transformation.
Finally, we obtain Theorem 2.
Deformation of the Miura transformation.
This implies that the Poisson bracket between ℓ(z) and ℓ(w) is given by formula (1.3) . Formula (9.3) gives us the following formula for the Poisson bracket between the Fourier coefficients of s(z):
Deformation of the Miura transformation.
where we put
Deformation of the Miura transformation.
The elements s n generate a Poisson algebra W h (sl 2 ), which is a q-deformation of the classical Virasoro algebra W(sl 2 ). This Poisson algebra is embedded into H h (sl 2 ) via the q-deformation of the Miura transformation (8.6). The Poisson algebra Z h ( sl 2 ) is isomorphic to W h (sl 2 ). Remark 6. In the limit h → 0 we have:
Deformation of the Miura transformation.
(cf. [10] ). On the other hand
Deformation of the Miura transformation.
Substituting these formulas into formula (8.6) and expanding in powers of h up to h 2 , we obtain:
Deformation of the Miura transformation.
which coincides with the Miura transformation (2.4). Now let us consider formula (9.4). The leading term in the expansion of the left hand side is 16h 4 {S n , S m }. Expanding the right hand side of (9.4) in powers of h up to h 4 , we obtain for the first term
Deformation of the Miura transformation.
and for the second term
Deformation of the Miura transformation.
Taking the sum, we see that the leading term in the right hand side is 16h 4 times the right hand side of formula (2.2).
Deformation of the Miura transformation.
Note that we can obtain a different Poisson algebra by placing an arbitrary overall factor in the right hand side of formula (9.3). In particular, if we put the overall factor −c/6 in the right hand side of the formula, then in the limit h → 0 we will recover the classical Virasoro algebra with central charge c.
Deformation of the Miura transformation.
We can also replace the overall factor 2h in the right hand side by (q −q −1 ) without changing the asymptotics h → 0. After that we can consider W h (sl 2 ) as a Poisson algebra over the ring of rational functions in q.
Deformation of the Miura transformation.
Remark 7. Formula (9.3) gives another asymptotics as h → 0, if we postulate that s(z) does not depend on h. If we divide the right hand side of the formula by 2h(q − q −1 ), we obtain for h = 0:
Deformation of the Miura transformation.
The corresponding limiting Poisson algebra W(sl 2 ) has a nice interpretation in terms of the algebra A k ( SL * 2 ) of functions on the Poisson-Lie group SL * 2 dual to SL 2 (this algebra is the classical limit of U h ( sl 2 ) k , cf. [9] ). Namely, the limits of the coefficients of ℓ(z) generate a central Poisson subalgebra of A 0 ( SL * 2 ) [9] . Deforming the level k we obtain a new Poisson structure on this subalgebra, which coincides with W(sl 2 ). We will discuss this in more detail in [23] .
Deformation of the Miura transformation.
10. Generalization to U q ( sl N ).
10.1.
A q-deformation of the Heisenberg-Poisson algebra. For a simple Lie algebra g of rank l, denote by B = (B ij ) i,j=1,... ,l the symmetrized Cartan matrix of g; recall that B ij = (α i , α j ). Let H h,k (g) be the Heisenberg algebra with generators a i [n], i = 1, . . . , l; n ∈ Z, and relations
10.1.
The algebra H h,k (g) appears in [14] in the construction of the Wakimoto realization
10.1.
The family H h,k (g) induces a Poisson structure on the commutative algebra H h,−h ∨ (g). The Poisson brackets between the generators of H h,−h ∨ (g) are
10.1.
Let H h (g) be the completion of H h,−h ∨ (g) defined in the same way as H h (sl 2 ).
10.1.
Consider the case g = sl N . Introduce new generators λ i [n], i = 1, . . . , N; n ∈ Z, of H h (sl N ), which are related to the generators a i [n] by the formulas
10.1.
and which satisfy the linear relation
10.1.
We find from formulas (10.2) and (10.3) the inverse change of variables:
10.1.
. . .
10.1.
From these formulas and the brackets (10.1) we find
10.1.
Introduce the generating functions
10.1.
From (10.4) and (10.5) we find: 1) ), (10.9) i = 1, . . . , N. In particular,
10.1.
etc. Formula (10.3) implies that
10.1.
These formulas coincide with formulas for spectra of transfer-matrices in integrable models associated to U q ( sl N ) [18, 21] . Note that for sl 2 we have s 1 (z) = s(zq −1 ), where s(z) is given by formula (8.6).
10.1.
The coefficients of the series s i (z), i = 1, . . . , N − 1, generate a Poisson subalgebra W h (sl N ) of H h (sl N ). The relations between them can be computed directly from formulas (10.7) and (10.8) . Introduce the functions
10.1.
The relations are:
10.1.
if i ≤ j and i + j ≤ N; and
10.1.
For example, in the case of sl 3 we have the following relations:
10.1.
The Poisson algebra W h (sl N ) is a q-deformation of the classical W-algebra W(sl N ). The asymptotic expansion of s i (z) around h = 0 has the form
10.1.
where C i is some coefficient, and W 2 (z) is the quadratic (Virasoro) generating series of the classical W-algebra W(sl N ). For each n = 2, . . . , N − 1, one can find a combination of s i (z)'s having expansion of the form M + h n W n (z) + . . . , where M is a constant and W n (z) is a multiple of an nth order generating series of W(sl N ). The Poisson structure on W(sl N ) can be recovered from that on W h (sl N ).
10.1.
Remark 8. If we replace 2h by (q − q −1 ) in the formulas above, we will obtain a Poisson algebra over the ring of rational functions in q.
10.1.
Remark 9. As in the case of sl 2 (cf. Remark 7), the Poisson algebra W h (sl N ) has another limit as h → 0, which can be interpreted in terms of the central subalgebra of the algebra of functions on the Poisson-Lie group dual to SL N [23] .
10.1.
10.3. The center of U h ( sl N ) at the critical level. Following [9] , cf. also [31, 10] , for any finite-dimensional representation W of U q ( g), one can construct matrices
10.1.
.. ,dim W consisting of generating functions of elements of U h ( sl N ) cr . It is shown in [9] that the Fourier coefficients of the power series
10.1.
be the generating function of central elements of U h ( sl N ) cr corresponding to the ith fundamental representation W ω i of U q ( sl N ). Note that for g = sl 2 we have ℓ 1 (z) = ℓ(zq −1 ), where ℓ(z) is given by (6.1). The next to leading term in the h-expansion of ℓ i (z) is a multiple of the Sugawara series of U ( sl N ) cr , cf. [10] . Higher Sugawara elements of U ( sl N ) cr can be obtained from higher order terms of the expansions of ℓ i (z)'s.
10.1.
Let Z h ( sl N ) be the completion of the central subalgebra of U h ( sl N ) cr generated by the coefficients of the series ℓ 1 (z), . . . , ℓ N −1 (z). Using the Wakimoto realization of U h ( sl N ) cr [14] , we obtain a homomorphism of Poisson algebras Z h ( sl N ) → H h (sl N ), which we call the q-deformation of the Miura transformation.
10.1.
Using the method of Remark 4 we can find the image of ℓ 1 (z) in H h (sl N ) by computing the matrix element of ℓ 1 (z) between the generating vector of a Wakimoto module over U h ( sl N ) cr and its dual. But for that we only need the diagonal part of the "Gauss decomposition" of L ± (z) [12] and a formula expressing the corresponding diagonal elements k ± i (z), i = 1, . . . , N, in terms of Λ i (z), i = 1, . . . , N − 1. This formula can be obtained from [14] and (10.6). Explicit computation shows that the image of ℓ 1 (z) in H h (sl N ) is equal to s 1 (z).
10.1.
The generating functions ℓ i (z), i = 2, . . . , N − 1, corresponding to other fundamental representations can be expressed in terms of ℓ 1 (z) by the fusion procedure, cf. [34] , [35] and references therein. Using this procedure, we can show that the image of ℓ i (z) in H h (sl N ) is equal to s i (z) given by formula (10.9) for all i = 1, . . . , N − 1. Thus, we obtain that Z h ( sl N ) is isomorphic to W h (sl N ) as a Poisson algebra. We will discuss this isomorphism in more detail in [23] .
10.1.
In conclusion of this section, recall that elements of the spectrum of W(sl N ) can be considered as Nth order differential operators, cf. [2] . The classical Miura transformation corresponds to splitting of such an operator into a product of first order operators.
10.1.
The spectrum of W h ( sl N ) and the q-deformation of the Miura transformation can be interpreted in a similar fashion. Namely, elements of the spectrum of H h (sl N ) can be considered as first order q-difference operators, and elements of the spectrum of Z h ( sl N ) can be considered as Nth order q-difference operators.
10.1.
Indeed, let Q i (z), i = 1, . . . , N − 1, be solutions of the q-difference equations
10.1.
where
10.1.
Thus, elements of the spectrum of W h (sl N ) can be considered as q-difference operators of the form
10.1.
11. Generalization to other quantum affine algebras.
10.1.
Let us first adopt notation we used in the case of sl N to the general case. We pass to another set of generators of 
10.1.
where C (m) ij is given by (10.11),
10.1.
Thus, the generators y i [n] are "dual" to the generators a i [n]. In fact, it is easy to see that
10.1.
.. ,N −1 being the Cartan matrix of sl N .
10.1.
Introduce generating functions
10.1.
We have:
10.1.
. . , N, where we put Y 0 (z) = Y N (z) = 1. Note that Y i (z) can be written as Q i (zq i+1 )/Q i (zq i−1 ) in terms of Q i (z) introduced at the end of last section.
10.1.
From formula (11.1) we find the Poisson brackets between Y i (z) and Y j (w):
10.1.
where C ij (x) is given by formula (10.10). We now define analogous generating functions Y i (z), i = 1, . . . , l, for an arbitrary simple Lie algebra g. Namely, let y i [n], i = 1, . . . , l; n ∈ Z, be the elements of H h (g) uniquely defined by the Poisson bracket (11.2). We put
10.1.
where ω i is the ith fundamental weight of g. Then we have: (11.4) where
10.1.
and the matrix (C (m) ij ) i,j=1,... ,l is defined by formula (11.3) (with N − 1 replaced by l) with respect to the symmetrized Cartan matrix B of g.
10.1.
Remark 10. It is interesting that the functions C ij (x) appear in the Thermodynamic Bethe Ansatz equations [36, 21] .
10.1.
For each dominant integral highest weight λ of g there exists an irreducible finitedimensional representation W λ of U q ( g) which satisfies the following property. Its restriction to the subalgebra U q (g) is completely reducible, the irreducible representation of U q (g) with highest weight λ has multiplicity one, and all other irreducible components of W λ have highest weights µ < λ.
10.1.
Let ℓ i (z), i = 1, . . . , l, be the generating functions of central elements of U h ( g) cr corresponding to W ω i . Let Z h ( g) be the central subalgebra of U h ( g) cr generated by the coefficients of ℓ i (z), i = 1, . . . , l. Recall from § § 1,2 that the Miura transformation is the homomorphism of Poisson algebras from the center of U ( g) to the HeisenbergPoisson algebra H(g). in H h (g) coincide with the formulas for spectra of the corresponding transfermatrices in integrable models associated to U q ( g).
10.1.
Formulas for spectra of transfer-matrices in integrable models associated to U q ( g) have been given in [19, 20, 21, 22] (although in different normalizations). We will now describe s i (z)'s for all quantum affine algebras of classical types via the series Y i (z), i = 1, . . . , l (we put Y 0 (z) = 1). The Poisson brackets between Y i (z)'s given by (11.4) where S is the set of {j 1 , . . . , j i }, such that j α < j α+1 or j α = j α+1 = n + 1, α = 1, . . . , i − 1. The formula for s n (z), which corresponds to the spinor representation of B
10.1.
n , is more complicated: where S is the set of {j 1 , . . . , j i }, such that 1 ≤ j 1 < . . . < j i ≤ 2n and if j α = l, j β = 2n + 1 − l for some l = 1, . . . , n, then l ≤ n + α − β.
The series g = D
(1) where S is the set of {j 1 , . . . , j i }, such that j α < j α+1 or j α = j α+1 + 1 = n + 1, α = 1, . . . , i − 1. The formulas for s n−1 (z) and s n (z), which correspond to the spinor representations of D (1) n , are more complicated. In these formulas the subscript ǫ means n, if ǫ = +, and n − 1, if ǫ = −. Thus, s + (z) = s n (z), s − (z) = s n−1 (z). Now let 

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