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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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.
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 . This matrix satisfies the Yang-Baxter equation using which one proves integrability of the model. An investigation of the eight-vertex model  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  using Drinfeld’s new realization of quantized current algebras. Further, it was shown in  that the Felder algebra can be obtained by twisting of the Enriquez-Rubtsov elliptic algebra. This twisted algebra will be denoted by 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  and was discussed in . The full potential of this idea was realized in papers  and . 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 . 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  an RSOS type elliptic current algebra (which will be denoted by ) generalizing some ideas of . This algebra was studied in detail in  where it was shown that commutation relations for 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  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 the elliptic module is fixed, while in the case of, 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  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  for quantization of Manin pairs: they are expanded into Taylor series in the case of the algebra and into Fourier series for .
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 and are quasi-Lie bialgebras denoted by and 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 . 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 . Hence, our currents will be certain -valued distributions. Then we review the case when is a loop algebra generated by a semi-simple Lie algebra a. We also discuss a centrally and a co-centrally extended version of 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 and . 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 and 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  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 .
Current realization of the quantum affine algebras and Yangians was introduced by Drinfeld in . In these cases the currents can be understood as elements of the space , 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 be a function algebra on a one-dimensional complex manifold Σ with a point-wise multiplication and a continuous invariant (non-degenerate) scalar product . We shall call the pair a test function algebra. The non-degeneracy of the scalar product implies that the algebra can be extended to a space of linear continuous functionals on . We use the notation for the action of the distribution on a test function . Let and be dual bases of . A typical example of the element from is the series. This is a delta-function distribution on because it satisfies for any test function .
Currents. Consider an infinite-dimensional complex Lie algebra and an operator . The expression does not depend on a choice of dual bases in and is called a current corresponding to the operator means an action of on ). We should interpret the current x(u) as a -valued distribution such that . That is the current x(u) can be regarded as a kernel of the operator and the latter formula gives its invariant definition.
Loop algebras. Let be a finite number of operators, where is an infinite-dimensional space spanned by . Consider the corresponding currents . For these currents we impose the standard commutation relations(2.1)
where 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 with a Lie algebra structure. The Lie algebra defined in such a way can be viewed as a Lie algebra with the brackets , where . This algebra possesses an invariant scalar product , where (·, ·) is an invariant scalar product on a proportional to the Killing form.
Central extension. The algebra can be extended by introducing a central element c and a co-central element d. Let us consider the space and define an algebra structure on this space. Let the element is given by the formula , where is a derivation of s. Define the Lie bracket requiring the scalar product defined by formulae between the elements of type
to be invariant. It gives the formula(2.2)
where [·, ·]0 is the Lie bracket in the algebra and B(·, ·) is a standard 2-cocycle: . The expression depends linearly on and, therefore, can be regarded as an action of operator . The commutation relations for the algebra in terms of currents x(u) corresponding to these operators can be written in the standard form: [c, x(u)] = [c, d] = 0 and(2.3)
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: . The -valued distributions x+(u), x−(u) are called half-currents. To perform such a decomposition we will use so-called Green distributions . Let be two domains separated by a hypersurface which contains the diagonal . Assume that there exist distributions G+(u, z) and G−(u, z) regular in and 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 and . Then the half currents are defined as and . This definition does not depend on a choice of decompositions of the Green distributions. The half-currents are currents corresponding to the operators , where , . One can express the half-currents through the current x(u), which we shall call a total current in contrast with the half ones:(2.4)
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 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 = , where consists of complex-valued one-variable functions defined in a vicinity of origin equipped with the scalar product(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 . The regularization domains and for Green distributions in this case consist of the pairs (u, z) such that min and , respectively, where is an elliptic module, and.
The second choice corresponds to . The algebra K consists of entire periodic functions s(u) = s(u+1) on C decaying exponentially at equipped with an invariant scalar product(2.6)
These functions can be regarded as functions on cylinder Σ = Cyl. The regularization domains , for Green distributions consist of the pairs (u, z) such that and respectively and .
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 . This contour corresponds to the covering 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 = and .
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 for the case = as and for as . These current algebras may be identified with classical limits of the quantized currents algebra of  and of  respectively. The Green distributions appear in the algebras and as a regularization of the same meromorphic quasi-doubly periodic functions but in different spaces: and respectively. Primes mean the extension to the space of the distributions. We call them elliptic Green distributions. We define the algebras and 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 and K respectively. It means, in particular, that their quantum analogs, the algebras and are different.
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 .
Theta-function. Let , be a module of the elliptic curve , where is a period lattice. The odd theta function is defined as a holomorphic function on with the properties(3.1)
3.1 Elliptic Green distributions on
3.1.2. Dual bases. Fix a complex number . Consider the following bases in :
for . Here means n-times derivative. These bases are dual: and with respect to the scalar product (2.5), which means(3.2)
One can check that these series converge in sense of distributions and, therefore, define continuous functionals on called Green distributions. Their action on a test function s(u) reads(3.5) (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 , where and therefore a ’rescaling’ of distributions by the formula . On the contrary, we are unable to define a ’shift’ of test functions by a standard rule, because the operator 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 is ’shifted’ if it possesses the properties: (i) for any the functions and belong to ; (ii) . Here the subscripts u and z mean the corresponding partial action, for instance, is a distribution acting on by the formula
The condition (ii) means the equality . The condition (i) implies that for any the expression(3.7)
where , belongs to (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(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 -depending Green distributions: .
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 by the rule
Proposition 3.1. The semi-direct products of Green distributions are related by the following addition formulae(3.9) (3.10) (3.11) (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 (3.13)
The other formulae can be proved in the same way.
3.1.4. Projections. Let us notice that the vectors and span two complementary subspaces of . The formulae (3.3) mean that the distributions and define orthogonal projections and onto these subspaces. They act as and . Similarly, the operators
are projections onto the Lagrangian (involutive) subspaces spanned by the vectors and , respectively. The fact that the corresponding spaces are complementary to each other is encoded in the formulae (3.8), which can be rewritten as . The idempotent properties and orthogonality of these projections
are encoded in the formulae(3.14) (3.15) (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 .
3.2 Elliptic Green distributions on K
3.2.1. Green distributions and dual bases for K. The analogs of the Green distributions and are defined in this case by the following action on the space K(3.17) (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 and , a decomposition with respect to these bases is the usual Fourier expansion. The Fourier expansions for the Green distributions are 2(3.19) (3.20)
These expansions are in accordance with formulae(3.21) (3.22)
where is a delta-function on K, given by the expansion(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 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 . 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 instead of 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:
which are similar to their analogs and satisfy , (due to (3.21)), but they are not projections. This fact is reflected in the following relations:(3.24) (3.25) (3.26) (3.27) (3.28) (3.29)
and is a distribution which has the following action and expansion
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 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 acting on . 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 . As we shall see this fact has a deep consequence for the half-currents of the corresponding Lie algebra As soon as we are aware that the positive operators as well as negative ones 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(3.30)
In terms of operator’s composition these properties look as(3.31)
where is a shift operator: , and is an integration operator: . 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 be a universal enveloping algebra of the considered Lie algebra and V be a U-module. For an element
where we shall use the following notation for an element of :
where stays in the is-th position in the tensor product and stays in the js-th position.
3.3.6. Half-currents. The total currents h(u), e(u) and f(u) of the algebra can be divided into half-currents using the Green distributions G(u, z), −G(z, u) for h(u); , for e(u); and , .(3.32) (3.33) (3.34)
so that .
3.3.7. rLL-relations for . The commutation relations between the half-currents can be written in a matrix form. Let us introduce the matrices of L-operators:(3.35)
as well as the r-matrices:(3.36)
Proposition 3.3. The commutation relations of the algebra in terms of half-currents can be written in the form:(3.37) (3.38) (3.39)
where and . The L-operators satisfy an important relation(3.40)
Proof. Using the formulae (3.14) – (3.16) we calculate the scalar products on the half-currents: . Differentiating these formulae by u we can obtain the values of the standard co-cycle on the half-currents: . 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 formulae we obtain the relation (3.40) from (3.38), (3.39).
3.3.8. rLL-relations for . Now consider the case of the algebra . The halfcurrents, L-operators and r-matrix are defined by the same formulas as above with distributions G(u, v) and replaced everywhere by the distributions and . We have
Proposition 3.4. The commutation relations of algebra in terms of half-currents can be written in the form:(3.41) (3.42)
where . We also have in this case the relation(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
Using these formulae we obtain
Using the formulae
we get the relation (3.43) from (3.41), (3.42).
3.3.9. Peculiarities of half-currents for . 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 . The fact that the images of the operators 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 . This is a consequence of construction of the Lie algebra as the central extension of . 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:
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
At this point we see the essential difference of the Lie algebra with the Lie algebra .
3.4 Coalgebra structures of and
We describe here the structure of quasi-Lie bialgebras for our Lie algebras and . 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 defined as
satisfies the Classical Dynamical Yang-Baxter Equation (CDYBE)(3.44)
Denote by the evaluation representation where and the subscript u means the argument of the functions belonging to :(3.45)
and , where . The relations between L-operators and the universal r-matrix are given by the formulae(3.46)
and . Taking into account these formulae and applying , 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 or to the identity we derive the relation (3.40).
The co-bracket and an element are defined as and
They equip the Lie algebra with a structure of a quasi-Lie bialgebra . This fact follows from the equality where is a tensor Casimir element of algebra . , to the equation (3.44) and derive
We can see also that
Proposition 3.6. The universal r-matrix for the Lie algebra defined by formula
satisfies the equation
The relations between the universal matrix and L-operators of the algebra are the same as for the algebra with a proper modification of the evaluation representation defined by the same formulas (3.45) as above for .
The bialgebra structure of is defined in analogous way as for the algebra and can be presented in the form
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.