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Journal of Generalized Lie Theory and Applications
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Classical elliptic current algebras. I

Stanislav PAKULIAKa, Vladimir RUBTSOVb, and Alexey SILANTYEVc

aInstitute of Theoretical and Experimental Physics, 117259 Moscow, Russia Laboratory of Theoretical Physics, JINR, 141980 Dubna, Moscow region, Russia
E-mail: [email protected]

bInstitute of Theoretical and Experimental Physics, 117259 Moscow, Russia D´epartment de Math´ematiques, Universit´e d’Angers, 2 Bd. Lavoisier, 49045 Angers, France
E-mail: [email protected]

cLaboratory of Theoretical Physics, JINR, 141980 Dubna, Moscow region, Russia D´epartment de Math´ematiques, Universit´e d’Angers, 2 Bd. Lavoisier, 49045 Angers, France
E-mail: [email protected]

Received date: September 28, 2007; Revised date: December 09, 2007

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Abstract

In this paper we discuss classical elliptic current algebras and show that there are two different choices of commutative test function algebras related to a complex torus leading to two different elliptic current algebras. Quantization of these classical current algebras gives rise to two classes of quantized dynamical quasi-Hopf current algebras studied by Enriquez, Felder and Rubtsov and by Arnaudon, Buffenoir, Ragoucy, Roche, Jimbo, Konno, Odake and Shiraishi.

1 Introduction

Classical elliptic algebras are ”quasi-classical limits” of quantum algebras whose structure is defined by an elliptic R-matrix. The first elliptic R-matrix appeared as a matrix of Boltzmann weights for the eight-vertex model [3]. This matrix satisfies the Yang-Baxter equation using which one proves integrability of the model. An investigation of the eight-vertex model [4] uncovered its relation to the so-called generalized ice-type model – the Solid-On-Solid (SOS) model. This is a face type model with Boltzmann weights which form a matrix satisfying a dynamical Yang-Baxter equation.

In this paper we restrict our attention to classical current algebras (algebras which can be described by a collection of currents) related to the classical r-matrices and which are quasiclassical limits of SOS-type quantized elliptic current algebras. The latter were introduced by Felder [13, 14] and the corresponding R-matrix is called usually a Felder R-matrix. In loc. cit. the current algebras were defined by dynamical RLL-relations. At the same time Enriquez and one of authors (V.R.) developed a theory of quantum current algebras related to arbitrary genus complex curves (in particular to an elliptic curve) as a quantization of certain (twisted) Manin pairs [9] using Drinfeld’s new realization of quantized current algebras. Further, it was shown in [7] that the Felder algebra can be obtained by twisting of the Enriquez-Rubtsov elliptic algebra. This twisted algebra will be denoted by image and it is a quasi-Hopf algebra.

Originally, the dynamical Yang-Baxter equation appeared in [16, 13, 14]. The fact that elliptic algebras could be obtained as quasi-Hopf deformations of Hopf algebras was noted first in a special case in [2] and was discussed in [15]. The full potential of this idea was realized in papers [1] and [17]. It was explained in these papers how to obtain the universal dynamical Yang-Baxter equation for the twisted elliptic universal R-matrix from the Yang-Baxter equation for the universal R-matrix of the quantum affine algebra image. It was also shown that the image of the twisted R-matrix in finite-dimensional representations coincides with SOS type R-matrix.

Konno proposed in [20] an RSOS type elliptic current algebra (which will be denoted by image) generalizing some ideas of [19]. This algebra was studied in detail in [18] where it was shown that commutation relations for image expressed in terms of L-operators coincide with the commutation relations of the Enriquez-Felder-Rubtsov algebra up to a shift of the elliptic module by the central element. It was observed in [18] that this difference of central charges can be explained by different choices of contours on the elliptic curve entering in these extensions. In the case of the algebra image the elliptic module is fixed, while in the case ofimage, it turn out to be a dynamical parameter shifted by the central element. Commutation relations for these algebras coincide when the central charge is zero, but the algebras themselves are different. Furthermore, the difference between these two algebras was interpreted in [8] as a difference in definitions of half-currents (or Gauss coordinates) in L-operator representation. The roots of this difference are related to different decomposition types of so-called Green kernels introduced in [9] for quantization of Manin pairs: they are expanded into Taylor series in the case of the algebra imageand into Fourier series for image.

Here, we continue a comparative study of different elliptic current algebras. Since the Green kernel is the same in both the classical and the quantum case we restrict ourselves only to the classical case for the sake of simplicity. The classical limits of quasi-Hopf algebras image and image are quasi-Lie bialgebras denoted by image and image respectively. We will give an ”analytic” description of these algebras in terms of distributions. Then, the different expansions of Green kernels will be interpreted as the action of distributions on different test function algebras. We will call them Green distributions. The scalar products for test function algebras which define their embedding in the corresponding space of distributions are defined by integration over different contours on the surface.

Let us describe briefly the structure of the paper. Section 2 contains some basic notions and constructions which are used throughout the paper. Here, we remind some definitions from [9]. Namely, we define test function algebras on a complex curve Σ, a continuous non-degenerate scalar product, distributions on the test functions and a generalized notion of Drinfeld currents associated with these algebras and with a (possibly infinite-dimensional) Lie algebra image. Hence, our currents will be certain image-valued distributions. Then we review the case when image is a loop algebra generated by a semi-simple Lie algebra a. We also discuss a centrally and a co-centrally extended version of image and different bialgebra structures. The latter are based on the notion of Green distributions and related half-currents.

We describe in detail two different classical elliptic current algebras which correspond to two different choices of the test function algebras (in fact they correspond to two different coverings of the underlying elliptic curve).

Section 3 is devoted to the construction and comparison of classical elliptic algebras image and image. In the first two subsections we define elliptic Green distributions for both test function algebras. We pay special attention to their properties because they manifest the main differences between the corresponding elliptic algebras. Further, we describe these classical elliptic algebras in terms of the half-currents constructed using the Green distributions. We can see how the half-currents inherit the properties of Green distributions. In the last subsection we show that the half-currents describe the corresponding bialgebra structure. Namely, we recall the universal classical r-matrices for both elliptic classical algebras imageand image and make explicit their relation to the L-operators. Then, the corresponding co-brackets for half-currents are expressed in a matrix form via the L-operators.

In the next paper [21] we will describe different degenerations of the classical elliptic current algebras in terms of degenerations of Green distributions. We will discuss also the inverse problem of reconstruction of the trigonometric and elliptic classical r-matrices from the rational and trigonometric r-matrices using approach of [11].

2 Currents and half-currents

Current realization of the quantum affine algebras and Yangians was introduced by Drinfeld in [5]. In these cases the currents can be understood as elements of the space image, where A is a corresponding algebra. Here we introduce a more general notion of currents suitable even for the case when the currents are expressed by integrals instead of formal series.

Test function algebras. Let image be a function algebra on a one-dimensional complex manifold Σ ­ with a point-wise multiplication and a continuous invariant (non-degenerate) scalar product image. We shall call the pairimage a test function algebra. The non-degeneracy of the scalar product implies that the algebra image can be extended to a space image of linear continuous functionals on image. We use the notation image for the action of the distribution image on a test function image. Let image and image be dual bases of image. A typical example of the element from image is the seriesimage. This is a delta-function distribution on image because it satisfies image for any test function image.

Currents. Consider an infinite-dimensional complex Lie algebra image and an operator image. The expression image does not depend on a choice of dual bases in image and is called a current corresponding to the operator image means an action of image on image). We should interpret the current x(u) as a image-valued distribution such that image. That is the current x(u) can be regarded as a kernel of the operator image and the latter formula gives its invariant definition.

Loop algebras. Let image be a finite number of operatorsimage, where image is an infinite-dimensional space spanned by image. Consider the corresponding currents image. For these currents we impose the standard commutation relations

image (2.1)

where image are structure constants of some semi-simple Lie algebra a, dim a = n (equality (2.1) is understood in sense of distributions). These commutation relations equip image with a Lie algebra structure. The Lie algebra image defined in such a way can be viewed as a Lie algebra image with the brackets image, where image. This algebra possesses an invariant scalar product image, where (·, ·) is an invariant scalar product on a proportional to the Killing form.

Central extension. The algebra image can be extended by introducing a central element c and a co-central element d. Let us consider the space image and define an algebra structure on this space. Let the element image is given by the formula image, where imageis a derivation of s. Define the Lie bracket image requiring the scalar product defined by formulae between the elements of type

image

to be invariant. It gives the formula

image (2.2)

where [·, ·]0 is the Lie bracket in the algebra image and B(·, ·) is a standard 2-cocycle: image. The expression image depends linearly on image and, therefore, can be regarded as an action of operator image. The commutation relations for the algebraimage in terms of currents x(u) corresponding to these operators can be written in the standard form: [c, x(u)] = [c, d] = 0 and

image (2.3)

where image.

Half-currents. To describe different bialgebra structures in the current algebras we have to decompose the currents in these algebras into difference of the currents which have good analytical properties in certain domains: image. The image-valued distributions x+(u), x(u) are called half-currents. To perform such a decomposition we will use so-called Green distributions [9]. Let image be two domains separated by a hypersurfaceimage which contains the diagonal image. Assume that there exist distributions G+(u, z) and G(u, z) regular in image and image respectively such that δ(u, z) = G+(u, z) - G(u, z). To define half-currents corresponding to these Green distributions we decompose them as image and image. Then the half currents are defined as image and image. This definition does not depend on a choice of decompositions of the Green distributions. The half-currents are currents corresponding to the operators image, where image, image. One can express the half-currents through the current x(u), which we shall call a total current in contrast with the half ones:

image (2.4)

Here image.

Two elliptic classical current algebras. In this paper we will consider the case when Σ is a covering of an elliptic curve and Green distributions are regularization of certain quasi-doubly periodic meromorphic functions. We will call the corresponding centrally extended algebras of currents by elliptic classical current algebras. The main aim of this paper is to show the following facts:

• There are two essentially different choices of the test function algebras image in this case corresponding to the different covering Σ.

• The same quasi-doubly periodic meromorphic functions regularized with respect to the different test function algebras define the different quasi-Lie bialgebra structures and, therefore, the different classical elliptic current algebras.

• The internal structure of these two elliptic algebras is essentially different in spite of a similarity in the commutation relations between their half-currents.

The first choice corresponds to image = image, where image consists of complex-valued one-variable functions defined in a vicinity of origin equipped with the scalar product

image (2.5)

Here C0 is a contour encircling zero and belonging to the intersection of domains of functions s1(u), s2(u), such that the scalar product is a residue in zero. These functions can be extended up to meromorphic functions on the covering image. The regularization domains image and image for Green distributions in this case consist of the pairs (u, z) such that minimage and image, respectively, where image is an elliptic module, andimage.

The second choice corresponds to image. The algebra K consists of entire periodic functions s(u) = s(u+1) on C decaying exponentially at image equipped with an invariant scalar product

image (2.6)

These functions can be regarded as functions on cylinder Σ = Cyl. The regularization domains image, image for Green distributions consist of the pairs (u, z) such that image and image respectively and image.

Integration contour. The geometric roots of the difference between these two choices can be explained as follows. These choices of test functions on different coverings Σ of elliptic curve correspond to the homotopically different contours on the elliptic curve. Each test function can be considered as an analytical continuation of a function from this contour – a real manifold – to the corresponding covering. This covering should be chosen as a most homotopically simple covering which permits to obtain a bigger source of test functions. In the first case, this contour is a homotopically trivial and coincides with a small contour around fixed point on the torus. We can always choose a local coordinate u such that u = 0 in this point. This explains the notation image. This contour corresponds to the covering image and it enters in the pairing (2.5). In the second case, it goes along a cycle and it can not be represented as a closed contour on C. Hence the most simple covering in this case is a cylinder Σ = Cyl and the contour is that one in the pairing (2.6). This leads to essentially different properties of the current elliptic algebras based on the test function algebras image = image and image.

Restriction to the sl2 case. To make these differences more transparent we shall consider only the simplest case of Lie algebra a = sl2 defined as a three-dimensional complex Lie algebra with commutation relations [h, e] = 2e, [h, f] = −2f and [e, f] = h. We denote the constructed current algebra image for the case image = image as image and for image as image. These current algebras may be identified with classical limits of the quantized currents algebra image of [7] and image of [18] respectively. The Green distributions appear in the algebras image and image as a regularization of the same meromorphic quasi-doubly periodic functions but in different spaces: image and imagerespectively. Primes mean the extension to the space of the distributions. We call them elliptic Green distributions. We define the algebras image and image to be a priori different, because the main component of our construction, elliptic Green distributions are a priori different being understood as distributions of different types: related to algebras image and K respectively. It means, in particular, that their quantum analogs, the algebras image and image are different.

3 Half-currents and co-structures

We start with a suitable definition of theta-functions and a conventional choice of standard bases. This choice is motivated and corresponds to definitions and notations of [8].

Theta-function. Let image, image be a module of the elliptic curve image, where image is a period lattice. The odd theta function image is defined as a holomorphic function on image. with the properties

image (3.1)

3.1 Elliptic Green distributions on image

3.1.2. Dual bases. Fix a complex number image. Consider the following bases in image:

image

for image. Here image means n-times derivative. These bases are dual: image and image with respect to the scalar product (2.5), which means

image (3.2)

3.1.3. Green distributions for image and the addition theorems. Here we follow the ideas of [9] and [8]. We define the following distribution

image (3.3) image (3.4)

One can check that these series converge in sense of distributions and, therefore, define continuous functionals on image called Green distributions. Their action on a test function s(u) reads

image (3.5) image (3.6)

where integrations are taken over circles around zero which are small enough such that the corresponding inequality takes place.

One can define a ’rescaling’ of a test function s(u) as a function image, where image and therefore a ’rescaling’ of distributions by the formula image. On the contrary, we are unable to define a ’shift’ of test functions by a standard rule, because the operator image is not a continuous one 1. Nevertheless we use distributions ’shifted’ in some sense. Namely, we say that a two-variable distribution a(u, z) (a linear continuous functional image is ’shifted’ if it possesses the properties: (i) for any image the functions image and image belong to image ; (ii) imageimage. Here the subscripts u and z mean the corresponding partial action, for instance,image is a distribution acting on image by the formula

image

The condition (ii) means the equality image. The condition (i) implies that for any image the expression

image (3.7)

where image, belongs to image (as a function of z).

The Green distributions (3.3) and (3.4) are examples of the ‘shifted’ distributions. The formula (3.2) implies that

image (3.8)

The last formulae can be also obtained from (3.5), (3.6) taking into account that the function s(u) has poles only in the points u = 0. As it is seen from (3.5), the oddness of function θ(u) leads to the following connection between the image-depending Green distributions: image.

Now we define a semidirect product of two ’shifted’ distributions a(u, z) and b(v, z) as a linear continuous functional a(u, z)b(v, z) acting on image by the rule

image

Proposition 3.1. The semi-direct products of Green distributions are related by the following addition formulae

image (3.9) image (3.10) image (3.11) image (3.12)

Proof. The actions of both hand sides of (3.9), for example, can be reduced to the integration over the same contours with some kernels. One can check the equality of these kernels using the degenerated Fay’s identity [12]

image (3.13)

The other formulae can be proved in the same way.

3.1.4. Projections. Let us notice that the vectors image and image span two complementary subspaces of image . The formulae (3.3) mean that the distributionsimage and image define orthogonal projections image and image onto these subspaces. They act as imageimageand image. Similarly, the operators

image

are projections onto the Lagrangian (involutive) subspaces spanned by the vectors image and image, respectively. The fact that the corresponding spaces are complementary to each other is encoded in the formulae (3.8), which can be rewritten as image. The idempotent properties and orthogonality of these projections

image

are encoded in the formulae

image (3.14) image (3.15) image (3.16)

which immediately follow from (3.3) and also can be obtained from the relations (3.9) – (3.12) if one takes into account image.

3.2 Elliptic Green distributions on K

3.2.1. Green distributions and dual bases for K. The analogs of the Green distributions image and image are defined in this case by the following action on the space K

image (3.17) image (3.18)

where we integrate over line segments of unit length (cycles of cylinder) such that the corresponding inequality takes place. The role of dual bases in the algebra K is played by image and image, a decomposition with respect to these bases is the usual Fourier expansion. The Fourier expansions for the Green distributions are 2

image (3.19) image (3.20)

These expansions are in accordance with formulae

image (3.21) image (3.22)

where image is a delta-function on K, given by the expansion

image (3.23)  

3.2.2. Addition theorems. Now we obtain some properties of these Green distributions and compare them with the properties of their analogs image described in subsection 3.1. In particular, we shall see that some properties are essentially different. Let us start with the properties of Green distribution which are similar to the case of algebra image. They satisfy the same addition theorems that were described in the subsection 3.1.

Proposition 3.2. The semi-direct product of Green distributions for algebra K is related by the formulae (3.9)–(3.12) with the distributions image instead of image respectively.

Proof. The kernels of these distributions are the same and therefore the addition formula in this case is also based on the Fay’s identity (3.13).

3.2.3. Analogs of projections. The Green distributions define the operators on K:

image

which are similar to their analogs image and satisfy image, image (due to (3.21)), but they are not projections. This fact is reflected in the following relations:

image (3.24) image (3.25) image (3.26) image (3.27) image (3.28) image (3.29)

and image is a distribution which has the following action and expansion

image

3.2.4. Comparison of the Green distributions. Contrary to (3.14)–(3.16) the formulae (3.24)–(3.29) contain some additional terms in the right hand sides obstructing the operators image to be projections. They do not decompose the space K(Cyl) in a direct sum of subspaces as it would be in the case of projections image acting on image. Moreover, as one can see from the Fourier expansions (3.19), (3.20) of Green distributions the images of the operators coincide with whole algebra image. As we shall see this fact has a deep consequence for the half-currents of the corresponding Lie algebra image As soon as we are aware that the positive operators image as well as negative onesimage transform the algebra K to itself, we can surmise that they can be related to each other. This is actually true. ¿From formulae (3.19), (3.20) we conclude that

image (3.30)

In terms of operator’s composition these properties look as

image (3.31)

where image is a shift operator: image, and image is an integration operator: imageimage. This property is no longer true for the case of Green distributions from section 3.1.

3.3 Elliptic half-currents

3.3.5. Tensor subscripts. First introduce the following notation. Let image be a universal enveloping algebra of the considered Lie algebra image and V be a U-module. For an element

image

where image we shall use the following notation for an element of image:

 

image

where image stays in the is-th position in the tensor product and image stays in the js-th position.

3.3.6. Half-currents. The total currents h(u), e(u) and f(u) of the algebra image can be divided into half-currents using the Green distributions G(u, z), −G(z, u) for h(u); image, image for e(u); and image, image.

image (3.32) image (3.33) image (3.34)

so that image.

3.3.7. rLL-relations for image. The commutation relations between the half-currents can be written in a matrix form. Let us introduce the matrices of L-operators:

image (3.35)

as well as the r-matrices:

image (3.36)

Proposition 3.3. The commutation relations of the algebra image in terms of half-currents can be written in the form:

image (3.37) image (3.38) image (3.39)

where image and image. The L-operators satisfy an important relation

image (3.40)

Proof. Using the formulae (3.14) – (3.16) we calculate the scalar products on the half-currents: image. Differentiating these formulae by u we can obtain the values of the standard co-cycle on the half-currents: imageimage. Using the formulae (3.9)–(3.12) one can calculate the brackets [·, ·]0 on the half-currents. Representing them in the matrix form and adding the co-cycle term one can derive the relations (3.38), (3.39). Using the formulaeimageimage we obtain the relation (3.40) from (3.38), (3.39).

3.3.8. rLL-relations for image. Now consider the case of the algebra image. The halfcurrents, L-operators image and r-matrix image are defined by the same formulas as above with distributions G(u, v) and image replaced everywhere by the distributions image and image. We have

Proposition 3.4. The commutation relations of algebra image in terms of half-currents can be written in the form:

image (3.41) image (3.42)

where image. We also have in this case the relation

image (3.43)

Proof. To express the standard co-cycle on the half currents through the derivatives of the r-matrix we need the following formulae

image

Using these formulae we obtain

image

Using the formulae

image

we get the relation (3.43) from (3.41), (3.42).

3.3.9. Peculiarities of half-currents for image. To conclude this subsection we discuss the implication of the properties of Green distributions described in the end of the previous section to the Lie algebra image. The fact that the images of the operators image coincide with all the space K means that the commutation relations between the positive (or negative) halfcurrents are sufficient to describe all the Lie algebra image. This is a consequence of construction of the Lie algebra image as the central extension of image. To obtain all commutation relations given in Proposition 3.4 from relations between only positive (or negative) half-currents one can use, firstly, the connection between positive and negative ones:

image

which follows from the properties of Green distributions expressed in formulae (3.30); secondly, relations (3.43), which also follow from the relations between only positive (respectively negative) half-currents; and finally, one needs to use the equality

image

At this point we see the essential difference of the Lie algebra image with the Lie algebra image.

3.4 Coalgebra structures of image and image

We describe here the structure of quasi-Lie bialgebras for our Lie algebras image and image. We will start with an explicit expression for universal (dynamical) r-matrices for both Lie algebras.

Proposition 3.5. The universal r-matrix for the Lie algebra image defined as

image

satisfies the Classical Dynamical Yang-Baxter Equation (CDYBE)

image (3.44)

Denote by image the evaluation representation image where image and the subscript u means the argument of the functions belonging to image:

image (3.45)

and image, where image imageimage. The relations between L-operators and the universal r-matrix are given by the formulae

image (3.46)

and image. Taking into account these formulae and applying image, image to the equation (3.44) we derive the relation (3.38) with the signs ‘+’, the relation (3.38) with the signs ‘−’ and the relation (3.39) respectively. Applying imageor image to the identity image we derive the relation (3.40).

The co-bracket image and an element image are defined as image and

image

They equip the Lie algebra image with a structure of a quasi-Lie bialgebra [6]. This fact follows from the equality image where image is a tensor Casimir element of algebra image. image, image to the equation (3.44) and derive

image

We can see also that image

Proposition 3.6. The universal r-matrix for the Lie algebra image defined by formula

image

satisfies the equation

image

The relations between the universal matrix image and L-operators of the algebra image are the same as for the algebra imagewith a proper modification of the evaluation representation image defined by the same formulas (3.45) as above for image.

The bialgebra structure of image is defined in analogous way as for the algebra image and can be presented in the form

image

Acknowledgements

This paper is a part of PhD thesis of A. S. which he has prepared under co-supervision of S. P. and V. R. in the Bogoliubov Laboratory of Theoretical Physics, JINR, Dubna and in LAREMA, D´epartement de Math´ematics, Universit´e d’Angers. He is grateful to the CNRSRussia exchange program on mathematical physics and personally to J.-M. Maillet for financial and general support of this thesis project. V. R. is thankful to B. Enriquez for discussions. He had used during the project a partial financial support by ANR GIMP, Grant for support of scientific schools NSh-8065.2006.2 and a support of INFN-RFBR ”Einstein” grant (Italy-Russia). He acknowledges a warm hospitality of Erwin Schr¨odinger Institute for Mathematical Physics and the Program ”Poisson Sigma Models, Lie Algebroids, deformations and higher analogues” where this paper was finished. S.P. was supported in part by RFBR grant 06-02-17383.

Leonid Vaksman, an excellent mathematician, one of Quantum Group ”pioneers”, patient teacher and a bright person, had passed away after coward disease when this paper was finished. We dedicate it to his memory with sadness and sorrow.

1Consider, for example, the sum image. For each z there exist α such that the sum image diverges, when image.

2Fourier expansions presented in this subsection are obtained considering integration around boundary of fundamental domain.

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