ITEP-TH-74/98

Analytical Properties of Elliptic Currents

J. Ding^{∗} ^{1}, S. Kharchev^{⋆} ^{2}, S. Khoroshkin^{⋆} ^{3}, S. Pakuliak^{⋆•∘} ^{4}

∗ Department of Mathematical Sciences, University of Cincinnati, PO Box 210025, Cincinnati, OH 45221-0025, USA ⋆ Institute of Theoretical & Experimental Physics, 117259 Moscow, Russia • Bogoliubov Laboratory of Theoretical Physics, JINR, 141980 Dubna, Moscow region, Russia ∘ Bogoliubov Institute for Theoretical Physics, 252143 Kiev, Ukraine

Abstract

We observe analytical properties of generating currents of elliptic affine algebras, which are defined, due to M.Jimbo et al. by precise twisting of quantum affine algebra in ”new realization”. In particular, an explicit formula for the universal R -matrix is presented.

1 Introduction

Affine Lie algebras 𝔤^{}ˆ , unlike other members of Kac-Moody algebras, admit a loop realization, that 𝔤^{}ˆ can be identified as 𝔤 ⊗ [t, t^{−1}] ⊕ ℂc, where c is the central element of 𝔤^{}ˆ . With the loop realization, we can define the so-called current operators in the form :

x(z) = ∑_{n∈ℤ} x ⊗ t^{n}z^{−n},

where x ∈ 𝔤 and z is a formal variable. The current operators have since play effective and fundamental role in representation theory of affine Lie algebras and in its applications. In particular, they provide a bridge to connect algebraic structure of affine Lie algebra to various analytic structures in mathematics and theoretical physics. For quantum affine algebras, the Drinfeld loop realization [D], as a Hopf algebra in terms of current operators, is the loop realization that is closest to the loop realization of affine Lie algebras above, which we call the quantized current algebras. The main distinctions of the quantized current algebras and the standard quantum affine algebras, which are connected through a infinite twist [KT], are in coalgebra structures and topological structures, In [DK], we study the Hopf algebra structure for the quantized current algebras and construct the universal R-matrix for admissible category of representations that are compatible with the topological structure of the quantized current algebras, where we use the Braid group action in terms of current operators and a new ordered exponent of current operators. The basic idea in [DK] comes from a transplant of an basic idea in physics, which can be ”translated” into mathematics as: the basic structure of a current algebra is determined by the analytic properties of the products of the current operators acting on the admissible category of representations, in particular, the locations of the poles of the product of current operators.

In this paper, we use the similar idea to study elliptic algebras. By elliptic algebras in general, we mean a current algebra, whose structure are given in term of elliptic functions. There has been intensive study recently on elliptic algebras, which is basically due to the motivation to solve physical models [F], [JKOS], [EF], [DI], [K]. In [JKOS], it is shown that elliptic algebras can be derived from certain twisting of the quantized current algebras. In this paper, we use a different but simpler twisting, namely the twisting coming from the Heisenberg algebra inside the quantized current algebra, Accordingly [DK1], this leads us to new elliptic algebras. The main part of the paper, along the similar line of [DK], is devoted to studying the structure of the elliptic algebras, essentially in terms of the analytic structure of the elliptic algebra.

1E-mail: Jintai.Ding@math.uc.edu

2E-mail: kharchev@vitep5.itep.ru

3E-mail: khoroshkin@heron.itep.ru

4E-mail: pakuliak@thsun1.jinr.ru

In [JKOS2], a fact of necessity to extend the elliptic algebras by certain zero mode operators was observed. There are certain algebraic reasons why any infinite dimensional algebra of affine type or their deformations should be extended by this sort of operators. This follows also from considering integrable lattice models of IRF type [LP], where elliptic algebras described in [JKOS2] appears naturally. This in turn leads to appearing of general dynamical Yang-Baxter equation. In the present paper we consider the elliptic algebras which are not extended by zero mode operators and which depend only one additional elliptic parameter associated with the central element of Cartan subalgebra and related to DYB equation where the only shifts by the central elements are presented.

The paper is arranged as follows. In Section 2, we present the basic results in [DK], where we explain the structure of quantized current algebra in terms of the analytic property of current operators. In Section 3, we describe twisting by the elements from Heisenberg subalgebra of the quantized current algebra and define elliptic current operators. they are generating functions of elliptic current algebra. In the last section, we study analytical and algebraic properties of elliptic current operators We define the corresponding braid group action on the elliptic algebra , study the analytic property of the products of the currents and finally construct the universal R-matrix in term of the current operators similar to that in [DK].

2 Braid group action in quantized current algebras

We present in this section the construction borrowed from [DK].

Let 𝔤 be a simple Lie algebra of the simply laced type^{1}. Consider the algebra generated by the central element c, gradation operator d and by the modes of the currents e_{α_{i}}(z) , f_{α_{i}}(z) and ψ_{α_{i}}^{±}(z) ,

e_{α_{i}}(z) = ∑_{k∈ℤ} e_{α_{i},k}z^{−k}, f_{α_{i}}(z) = ∑_{k∈ℤ} f_{α_{i},k}z^{−k},

ψ_{α_{i}}^{±}(z) = k_{α_{i}}^{±1} exp (±(q − q^{−1}) ∑_{n>0} a_{α_{i},±n}z^{∓n}) ,  (2.1)

where α_{i} are positive simple roots of the algebra 𝔤.

The generating elements of the quantum current algebra U_{q}(𝔤^{}ˆ ) satisfy the following defining relations ( α, β are simple roots of 𝔤):

q^{d}x(z)q^{−d} = x(qz), for x = e, f, ψ^{±} ,  (2.2)
(z − q^{(α,β)}w)e_{α}(z)e_{β}(w) = e_{β}(w)e_{α}(z)(q^{(α,β)}z − w) ,  (2.3)
(z − q^{−(α,β)}w)f_{α}(z)f_{β}(w) = f_{β}(w)f_{α}(z)(q^{−(α,β)}z − w) ,  (2.4)
\frac{(q^{c/2}z − q^{(α,β)}w)}{(q^{(α,β)+c/2}z − w)}ψ_{α}^{+}(z)e_{β}(w) = e_{β}(w)ψ_{α}^{+}(z) ,  (2.5)

ψ_{α}^{−}(z)e_{β}(w) = \frac{(q^{(α,β)−c/2}z − w)}{(q^{−c/2}z − q^{(α,β)}w)}e_{β}(w)ψ_{α}^{−}(z) ,  (2.6)

\frac{(q^{−c/2}z − q^{−(α,β)}w)}{(q^{−(α,β)−c/2}z − w)}ψ_{α}^{+}(z)f_{β}(w) = f_{β}(w)ψ_{α}^{+}(z) ,  (2.7)

1We restrict ourselves to the simply laced case only to simplify the presentation below. The general case can be treated along the same lines

ψ_{α}^{−}(z)f_{β}(w) = \frac{(q^{−(α,β)+c/2}z − w)}{(q^{c/2}z − q^{−(α,β)}w)}f_{β}(w)ψ_{α}^{−}(z) ,  (2.8)

\frac{(z − q^{(α,β)−c}w)(z − q^{−(α,β)+c}w)}{(q^{(α,β)+c}z − w)(q^{−(α,β)−c}z − w)}ψ_{α}^{+}(z)ψ_{β}^{−}(w) = ψ_{β}^{−}(w)ψ_{α}^{+}(z) ,  (2.9)
ψ_{α}^{±}(z)ψ_{β}^{±}(w) = ψ_{β}^{±}(w)ψ_{α}^{±}(z) ,  (2.10)
[e_{α}(z), f_{β}(w)] = \frac{δ_{α,β}}{q − q^{−1}} (δ(z/q^{c}w)ψ_{α}^{+}(zq^{−c/2}) − δ(zq^{c}/w)ψ_{α}^{−}(wq^{−c/2}))  (2.11)

and the Serre relations:

e_{α}(w_{1})e_{α}(w_{2})e_{β}(z) − [2]_{q}e_{α}(w_{1})e_{β}(z)e_{α}(w_{2}) + e_{β}(z)e_{α}(w_{1})e_{α}(w_{2}) + w_{1} ↔ w_{2} = 0 ,  (2.12)
f_{α}(w_{1})f_{α}(w_{2})f_{β}(z) − [2]_{q}f_{α}(w_{1})f_{β}(z)f_{α}(w_{2}) + f_{β}(z)f_{α}(w_{1})f_{α}(w_{2}) + w_{1} ↔ w_{2} = 0  (2.13)

for any α, β , (α, β) = −1 . Here δ(z) = ∑_{k∈ℤ} z^{k} .

The formulas (2.5)–(2.9) can be rewritten also using the generators a_{α,n} introduced by (2.1):

[a_{α,n}, e_{β}(z)] = \frac{[(α, β)n]}{n}q^{−c|n|/2}z^{n}e_{β}(z) ,  (2.14)

[a_{α,n}, f_{β}(z)] = −\frac{[(α, β)n]}{n}q^{c|n|/2}z^{n}f_{β}(z) ,  (2.15)
[a_{α,n}, a_{β,m}] = δ_{n+m,0} \frac{[(α, β)n][cn]}{n} .  (2.16)

2.1 The automorphisms and composed current

To define the rest of the generators of the algebra U_{q}(𝔤^{}ˆ ) which coincides by definition with universal enveloping algebra of modes of the simple currents and describe some basis of Cartan-Weyl type for U_{q}(𝔤^{}ˆ ) we need to introduce certain completion of it, see [DK1] or [M]. Here we use the completion described in [M].

We call V to be an admissible representation of the algebra U_{q}(𝔤^{}ˆ ) if for ∀v ∈ V there exists an integer number N such that xv = 0 for x = e_{α_{i},n}, f_{α_{i},n}, a_{α_{i},n} with n > N . Let us say that the elements e_{α_{i},n} , f_{α_{i},n} , a_{α_{i},n} have degree n. Let U_{±s} be subalgebras in U_{q}(𝔤^{}ˆ ) generated by the elements of degree bigger than s or smaller than −s for s ≥ 0 . Then we can introduce the system of neighborhoods

U_{kl} = ∑_{{r≥k \atop s≥l}} U_{−r} U_{q}(𝔤^{}ˆ ) U_{s}  (2.17)

Then the algebra U_{q}(𝔤^{}ˆ ) becomes a topological algebra with the fundamental system of neighborhoods and denote by Û_{q}(𝔤^{}ˆ ) its completion with respect to this topology. The completed algebra Û_{q}(𝔤^{}ˆ ) as well as original U_{q}(𝔤^{}ˆ ) is acting in the admissible representations.

The convenient basis of the generators of the algebra Û_{q}(𝔤^{}ˆ ) which corresponds to simple and composed roots of the algebra 𝔤 can be defined using following automorphisms of the algebra U_{q}(𝔤^{}ˆ ) .

Let η be a continuous anti-automorphism of U_{q}(𝔤^{}ˆ ) [M] defined by

η (e_{α_{i}}(z)) = e_{α_{i}}(z^{−1}), η (f_{α_{i}}(z)) = f_{α_{i}}(z^{−1}), η (ψ_{α_{i}}^{±}(z)) = ψ_{α_{i}}^{∓}(q^{c}z^{−1}), η (c) = c, η (d) = d. (2.18)

For any simple root α let T_{α} and T_{α}^{−1} be the following maps [DK]:

T_{α}e_{α}(z) = f_{α}(q^{−c}z)ψ_{α}^{+}(q^{−c/2}z)^{−1}, T_{α}f_{α}(z) = ψ_{α}^{−}(q^{−c/2}z)^{−1}e_{α}(q^{−c}z), T_{α}ψ_{α}^{±}(z) = ψ_{α}^{±}(z)^{−1} , (2.19)

and for the simple β , (α, β) = −1

T_{α}ψ_{β}^{±}(z) = ψ_{α}^{±}(qz)ψ_{β}^{±}(q^{2}z),  (2.20)

T_{α}e_{β}(z) = ∫ \frac{dw}{w} (e_{α}(w)e_{β}(q^{2}z) − q^{−(α,β)} \frac{1 − q^{(α,β)−2}w/z}{1 − q^{−(α,β)−2}w/z} e_{β}(q^{2}z)e_{α}(w)) ,  (2.21)

T_{α}f_{β}(z) = ∫ \frac{dw}{w} (−f_{β}(q^{2}z)f_{α}(w) + q^{(α,β)} \frac{1 − q^{−(α,β)+2}z/w}{1 − q^{(α,β)+2}z/w} f_{α}(w)f_{β}(q^{2}z)) ,  (2.22)

where the operation ∫ \frac{dw}{w} means the coefficient at w^{0} of the Laurent series in r.h.s. (2.21) and (2.22). The automorphisms T_{α}^{−1} are defined by the composition

T_{α}^{−1} = η ∘ T_{α} ∘ η  (2.23)

and the rest of Drinfeld current operators for simple roots is stable.

It is convenient to denote the action of the automorphisms T_{α}^{±1} on the currents e_{β}(z) and f_{β}(z) for simple roots α and β such that (α, β) = −1 by

T_{α}e_{β}(z) ≡ e_{α+β}(z), T_{α}f_{β}(z) ≡ f_{α+β}(z), T_{α}^{−1}e_{β}(z) ≡ ě_{α+β}(z), T_{α}^{−1}f_{β}(z) ≡ f̌_{α+β}(z)  (2.24)

and understand these currents as composed currents which correspond to the composed roots α + β . To observe that automorphism T_{α}^{−1} is an inverse to T_{α} we should use the commutation relations (2.11).

Note that the action of the automorphisms T_{α} and T_{α}^{−1} produces the normal ordered currents with respect to the rule that all the modes a_{α,n} with n > 0 are on the right, the modes a_{α,n} with n < 0 on the left and in the infinite sums appearing in (2.21) and (2.22) the modes in the products e_{α,n}e_{β,m} are ordered according to n < m but finite number of terms.

Let

w_{0} = s_{α_{i_{1}}} s_{α_{i_{2}}} ⋯ s_{α_{i_{N}}}  (2.25)

be a reduced decomposition of the longest element w_{0} of the Weyl group of Lie algebra 𝔤. Construct the set of positive roots γ_{1}, …, γ_{N} by the following prescription

γ_{N} = α_{i_{N}}
γ_{N−1} = s_{α_{i_{N}}} α_{i_{N−1}}
γ_{N−2} = s_{α_{i_{N}}} s_{α_{i_{N−1}}} α_{i_{N−2}}

and so on. For example, in case of the algebra 𝔤 = sl_{3} we have two simple roots α and β , the longest element w_{0} = s_{α}s_{β}s_{α} and abore prescription

γ_{3} = α, γ_{2} = s_{α}β = α + β, γ_{3} = s_{α}s_{β}α = s_{α}(α + β) = β

produces all positive roots of the algebra 𝔤 = sl_{3} .

The (linear independent) PBW basis in the algebra Û_{q}(𝔤^{}ˆ ) consists of the monomials of the form

∏_{i=1}^{N} f_{γ_{i},m_{1}}f_{γ_{i},m_{2}} … f_{γ_{i},m_{l}} ∏_{i=1}^{N} a_{γ_{i},p_{1}}a_{γ_{i},p_{2}} … a_{γ_{i},p_{s}} ∏_{i=1}^{N} e_{γ_{i},k_{1}}e_{γ_{i},k_{2}} … e_{γ_{i},k_{n}} ,  (2.26)

where each product is ordered with respect to linear order in the set of the roots γ_{i} , i = 1, …, N , attached to the reduced decomposition (2.25) and with respect to the indices of the generators, that is,

γ_{1} < ...γ_{N}, m_{1} ≤ m_{2} ≤ … ≤ m_{l}, p_{1} ≤ p_{2} ≤ … ≤ p_{s}, k_{1} ≤ k_{2} ≤ … ≤ k_{n}.  (2.27)

Any non-ordered monomial from the algebra Û_{q}(𝔤^{}ˆ ) can be expressed using formulas like (2.41) as a series over the monomials of the type (2.26), convergent in topology (2.17).

It is clear that composed root generators act on the admissible representations in an admissible way. For example, for two simple roots α and β consider element v in the admissible representation V such that generators e_{α,m} and e_{β,m} annihilate v for m > M . Then it is clear that composed root generators e_{α+β,m} will annihilate this element for m > 2M .

On the other hand, it is possible to express the monomials from Û_{q}(𝔤^{}ˆ ) via convergent series over degree ordered monomials

The composed root generators are introduced exactly for the reason to have root ordering in the PBW basis. It is clear, that it is impossible to construct PBW basis using only generators which corresponds to the simple roots. Among the elements of such a ‘basis’ we will have

b_{α_{1},n_{1}}b_{α_{2},n_{2}}b_{α_{1},n_{3}}

with n_{1} ≤ n_{2} ≤ n_{3} , and b = e, f, a without ordering of roots.

2.2 Universal R -matrices

The algebra U_{q}(𝔤^{}ˆ ) is a Hopf algebra with the comultiplication, antipode and counit maps:

Δ^{(1)}e_{α}(z) = e_{α}(z) ⊗ 1 + ψ_{α}^{−}(zq^{c_{1}/2}) ⊗ e_{α}(zq^{c_{1}}) ,  (2.28a)
Δ^{(1)}f_{α}(z) = 1 ⊗ f_{α}(z) + f_{α}(zq^{c_{2}}) ⊗ ψ_{α}^{+}(zq^{c_{2}/2}) ,  (2.28b)
Δ^{(1)}ψ_{α}^{±}(z) = ψ_{α}^{±}(zq^{±c_{2}/2}) ⊗ ψ_{α}^{±}(zq^{∓c_{1}/2}) ,  (2.28c)

\begin{matrix}S^{(1)} (ψ_{α}^{±}(z)) = ψ_{α}^{±}(z)^{−1}, S^{(1)} (e_{α}(z)) = −ψ_{α}^{−} (zq^{−c/2})^{−1} e_{α}(zq^{−c}), \\ S^{(1)} (f_{α}(z)) = −f_{α}(zq^{−c})ψ_{α}^{+} (zq^{−c/2})^{−1} , S^{(1)}(c) = −c, a^{(1)}(d) = −d , \end{matrix}  (2.28d)

ϵ (ψ_{α}^{±}(z)) = 1, ϵ(x) = 0, for x = c, d, e_{α}(z), f_{α}(z) .  (2.28e)

One can observe that the algebra U_{q}(𝔤^{}ˆ ) can be obtained using quantum double construction [D, DK] starting from the subalgebra generated by the modes of the currents f_{α_{i}}(z) and ψ_{α_{i}}^{+}(z) , i = 1, …, r, where r is a rank of 𝔤.

Let γ_{1}, …, γ_{N} be the system of positive roots of the algebra 𝔤 described at the end of the previous subsection. Let e_{γ_{1}}(u) , e_{γ_{2}}(u) , …, e_{γ_{N}} (u) and f_{γ_{1}}(u) , f_{γ_{2}}(u) , …, f_{γ_{N}} (u) be the composed root currents given by the prescription:

e_{γ_{N}} (u) = e_{α_{i_{N}}} (u), f_{γ_{N}} (u) = f_{α_{in}} (u) ,
e_{γ_{N−1}}(u) = T_{α_{i_{N}}} e_{α_{i_{N−1}}} (u), f_{γ_{N−1}}(u) = T_{α_{i_{N}}} f_{α_{i_{N−1}}} (u) ,
e_{γ_{N−2}}(u) = T_{α_{i_{N}}} T_{α_{i_{N−1}}} e_{α_{i_{N−2}}} (u), f_{γ_{N−2}}(u) = T_{α_{i_{N}}} T_{α_{i_{N−1}}} f_{α_{i_{N−2}}} (u)

and so on. For any simple or composed root γ_{i} we define the tensor:

\overline{R}_{γ_{i}} = P exp^{⟶}_{q^{2}} ((q^{−1} − q) ∫ f_{γ_{i}}(z) ⊗ e_{γ_{i}}(z)\frac{dz}{z} ) ,  (2.29)

where the formal operation ∫ \frac{dz}{z} means the same as in (2.21) and (2.22) and ordered exponent is defined by series

\begin{matrix}P exp^{⟶}_{q^{2}} ((q^{−1} − q) ∫ f_{γ_{i}}(z) ⊗ e_{γ_{i}}(z)\frac{dz}{z} ) = 1 + \\ + ∑_{n>0} \frac{(q^{−1} − q)^{n}}{n!} ∫ \frac{dz_{n}}{z_{n}} ∫ \frac{dz_{n−1}}{z_{n−1}} ⋯ ∫ \frac{dz_{1}}{z_{1}} f_{γ_{i}}(z_{1})f_{γ_{i}}(z_{2}) … f_{γ_{i}}(z_{n})⊗ \\ ⊗ ∏_{j<k} \frac{q^{2}z_{j} − z_{k}}{z_{j} − q^{2}z_{k}} e_{γ_{i}}(z_{n})e_{γ_{i}}(z_{n−1}) … e_{γ_{i}}(z_{1}) . \end{matrix}  (2.30)

To get n th term of the series (2.30) in terms of generators of current algebra we should expand the rational functions in the domain of the analyticity of the product e_{γ_{i}}(z_{n}) … e_{γ_{i}}(z_{1}) , namely with respect to the powers of q^{−2}z_{j}/z_{k} , j < k and pick up the coefficient in front of z_{1}^{0}z_{2}^{0} ⋯ z_{n}^{0} .

Let

K = q^{−t}q^{\frac{−c⊗d−d⊗c}{2}} ∏_{n>0} exp (−n(q − q^{−1}) ∑_{i,j=1}^{r} d_{i,j}^{(n)}a_{α_{i},n} ⊗ a_{α_{j},−n}) q^{\frac{−c⊗d−d⊗c}{2}}  (2.31)

be a tensor constructed from the Cartan part of the algebra, where t = ∑ h_{i} ⊗ h^{i} is an invariant in tensor square of Cartan subalgebra h ⊗ h and d_{i,j}^{(n)} is an inverse matrix to

b_{i,j}^{(n)} = [n(α_{i}, α_{j})]_{q} .

By the quantum double construction one can prove that the tensor

R = K · \overline{R} = K · \overrightarrow{∏}_{1≤i≤N} \overline{R}_{γ_{i}}  (2.32)

is the universal R -matrix for the algebra quantized current algebra with comultiplication Δ^{(1)} .

Note that one can define the second Drinfeld coalgebraic structures for the algebra U_{q}(𝔤^{}ˆ ) which we will distinguish from the first one by the superscript Δ^{(2)} :

Δ^{(2)}e_{α}(z) = e_{α}(z) ⊗ 1 + ψ_{α}^{+}(zq^{−c_{1}/2}) ⊗ e_{α}(zq^{−c_{1}}) ,  (2.33a)
Δ^{(2)}f_{α}(z) = 1 ⊗ f_{α}(z) + f_{α}(zq^{−c_{2}}) ⊗ ψ_{α}^{−}(zq^{−c_{2}/2}) ,  (2.33b)
Δ^{(2)}ψ_{α}^{±}(z) = ψ_{α}^{±}(zq^{±c_{2}/2}) ⊗ ψ_{α}^{±}(zq^{∓c_{1}/2}) ,  (2.33c)

\begin{matrix}S^{(2)} (ψ_{α}^{±}(z)) = ψ_{α}^{±}(z)^{−1}, S^{(2)} (e_{α}(z)) = −ψ_{α}^{−} (zq^{c/2})^{−1} e_{α}(zq^{c}), \\ S^{(2)} (f_{α}(z)) = −f_{α}(zq^{c})ψ_{α}^{+} (zq^{c/2})^{−1} , S^{(2)}(c) = −c, a^{(1)}(d) = −d \end{matrix}  (2.33d)

and counit map is the same as in (2.28e). In this case the algebra U_{q}(𝔤^{}ˆ ) can be obtained using quantum double construction starting from Hopf subalgebra generated by e_{α}(z) and ψ_{α}^{+}(z) .

The comultiplication Δ^{(2)} is related to Δ^{(1)} by the twist over tensor K

Δ^{(2)} = K^{−1}Δ_{op}^{(2)}K .  (2.34)

The corresponding universal R -matrix is

\overleftarrow{∏}_{1≤i≤N} (\overline{R}_{γ_{i}})_{21} K .

2.3 The analytical properties of the products of the currents

We are in position now to translate some of the construction of the previous section into analytical language.

The action of the current generators into highest weight modules is well defined if the appropriate ordering in the monomial is fixed. For example, the generators given by the infinite sum of the monomials e_{γ_{1},n}e_{γ_{2},m} have well defined action on highest weight modules if n ≤ m for all but finite values of the summation indices n and m. This ordering results to the fact that the product of the current e_{α}(z)e_{β}(w) is an analytical function of the variables z and w in the domain |z| >> |w|, in particular, as follows from commutation relations for |z| > |q^{(α,β)}w|.

The commutation relation (2.3) should be understood then as the rule for the analytical continuation of the product e_{α}(z)e_{β}(w) from the region of analyticity |z| > |q^{(α,β)}w| into the region |z| < |q^{(α,β)}w|, where this analytical continuation is equal to \frac{q^{(α,β)}z−w}{z−q^{(α,β)}w} e_{β}(z)e_{α}(w) . So the product e_{α}(z)e_{β}(w) can be continued

to (ℂ^{∗})^{2} as a meromorphic function with simple pole at the point z = q^{(α,β)}w. The same for the product f_{β}(w)f_{α}(z) which can be analytically continued from the domain |w| > |q^{(α,β)}z| to (ℂ^{∗})^{2} and has the pole at the same point.

Fix now the simple roots α and β such that (α, β) = −1 . Let us calculate the residue of the functions e_{α}(w)e_{β}(q^{2}z) and f_{β}(q^{2}z)f_{α}(w) at the point w = qz. We can easily verify that these residues coincide precisely with the action of the automorphisms T_{α} onto currents e_{β}(z) and f_{β}(z) . We have

e_{α+β}(z) =: e_{α}(qz)e_{β}(q^{2}z) := res_{w=qz} e_{α}(w)e_{β}(q^{2}z) = \frac{1}{2πi} ∮_{w around qz} e_{α}(w)e_{β}(q^{2}z)\frac{dw}{w} =

= \frac{1}{2πi} (∮_{C_{∞}} e_{α}(w)e_{β}(q^{2}z)\frac{dw}{w} − ∮_{C_{0}} \frac{q^{−1}w − q^{2}z}{w − qz} e_{β}(q^{2}z)e_{α}(w)\frac{dw}{w} ) ,

(2.35)

f_{α+β} =: f_{β}(q^{2}z)f_{α}(qz) := res_{w=qz} f_{β}(q^{2}z)f_{α}(w) = \frac{1}{2πi} ∮_{w around qz} f_{β}(q^{2}z)f_{α}(w)\frac{dw}{w} =

= \frac{1}{2πi} (− ∮_{C_{0}} f_{β}(q^{2}z)f_{α}(w)\frac{dw}{w} + ∮_{C_{∞}} \frac{q^{−1}w − q^{2}z}{w − qz} f_{α}(w)f_{β}(q^{2}z)\frac{dw}{w} ) .

(2.36)

In order to get two terms in the r.h.s. of (2.35) and (2.36) we split the integral over small circle around the point w = qz into the difference of two contour integrals such that the contour C_{0} is around zero and the point w = qz is outside the contour while C_{∞} is a contour close to infinity, which includes zero and the point w = qz. The rational functions in the r.h.s. of (2.35) and (2.36) appeared due to the analytical continuation of the products e_{α}(w)e_{β}(q^{2}z) and f_{β}(q^{2}z)f_{α}(w) into the corresponding domains.

One can calculate also the residues of the ‘functions’ e_{β}(q^{−2}z)e_{α}(w) and f_{α}(w)f_{β}(q^{−2}z) at the points w = q^{−1}z where these products have the simple poles. Therefore, we will have the action of the automorphisms T_{α}^{−1} onto the currents e_{β}(z) and f_{β}(z) .

ě_{α+β}(z) =: e_{β}(q^{−2}z)e_{β}(q^{−1}z) := res_{w=q^{−1}z} e_{β}(q^{−2}z)e_{α}(w) = \frac{1}{2πi} ∮_{w around q^{−1}z} e_{β}(q^{−2}z)e_{β}(w)\frac{dw}{w} =

\frac{1}{2πi} (− ∮_{C′_{0}} e_{β}(q^{−2}z)e_{α}(w)\frac{dw}{w} + ∮_{C′_{∞}} \frac{qw − q^{−2}z}{w − q^{−1}z} e_{α}(w)e_{β}(q^{−2}z)\frac{dw}{w} ) ,

(2.37)

f̌_{α+β}(z) =: f_{α}(q^{−1}z)f_{β}(q^{−2}z) := res_{w=q^{−1}z} f_{α}(w)f_{β}(q^{−2}z) = \frac{1}{2πi} ∮_{w around q^{−1}z} f_{α}(w)f_{β}(q^{−2}z)\frac{dw}{w} =

\frac{1}{2πi} (∮_{C′_{∞}} f_{α}(w)f_{β}(q^{−2}z)\frac{dw}{w} − ∮_{C′_{0}} \frac{qw − q^{−2}z}{w − q^{−1}z} f_{β}(q^{−2}z)f_{α}(w)\frac{dw}{w} ) ,

(2.38)

where C′_{0} is a contour around zero such that the point w = q^{−1}z is outside the contour while C′_{∞} is a contour close to infinity, which includes zero and the point w = q^{−1}z.

Now we would like to demonstrate that the definition of the composed currents (2.21), (2.22) or their analytical counterparts (2.35), (2.36) and (2.37), (2.38) produces for the modes of the composed currents the normal ordered expressions in the sense explained above. For example, let us consider the n th mode of the composed current e_{α+β}(z) given by (2.21). Expanding the rational function in geometrical progression with respect to the powers of q^{−1}w/z we obtain

e_{α+β,n} = q^{−2n} (e_{α,0}e_{β,n} − qe_{β,n}e_{α,0} − (q − q^{−1}) ∑_{m>0} q^{m}e_{β,n−m}e_{α,m}) .  (2.39)

The r.h.s. in (2.39) contains only finite number of non-ordered terms.

Using the commutation relations for the modes of simple currents

\begin{matrix}e_{α,k+1}e_{β,m} − q^{−1}e_{α,k}e_{β,m+1} = q^{−1}e_{β,m}e_{α,k+1} − e_{β,m+1}e_{α,k} \\ f_{α,k+1}f_{β,m} − qf_{α,k}f_{β,m+1} = qf_{β,m}f_{α,k+1} − f_{β,m+1}f_{α,k} \end{matrix}  (2.40)

we can rewrite the defining relations (2.21) and (2.22) in many equivalent ways

\begin{matrix}e_{α+β,n} = q^{−2n+p} (e_{α,p}e_{β,n−p} − qe_{β,n−p}e_{α,p} − (q − q^{−1}) ∑_{m>0} q^{m}e_{β,n−m−p}e_{α,m+p}) \\ f_{α+β,n} = q^{2n−p} (−f_{β,n−p}f_{α,p} + q^{−1}f_{α,p}e_{β,n−p} − (q − q^{−1}) ∑_{m>0} q^{−m}f_{α,p−m}e_{α,n−m+p}) \end{matrix}  (2.41)

for arbitrary integer p. This results in the analytical form of (2.21) and (2.22):

e_{α+β}(q^{−2}z)δ(qw/z) = e_{α}(w)e_{β}(z) − \frac{q − w/z}{1 − qw/z} e_{β}(z)e_{α}(w), |qw| < |z|  (2.42)

f_{α+β}(q^{−2}z)δ(qw/z) = −f_{β}(z)f_{α}(w) + \frac{q^{−1} − z/w}{1 − q^{−1}z/w} f_{α}(w)f_{β}(z), |z| < |qw|  (2.43)

and inequalities signify the direction of the expansion of the rational functions in the r.h.s. One can think about the relations (2.42) and (2.43) as follows. One starts from the analytical in the domain |w| > |q^{−1}z| function e_{α}(w)e_{β}(z) and try the analytical continuation with respect the variable w to the domain |w| < |q^{−1}z|. It is clear that at this process one will pick up the δ -function with coefficient being the residue of the function e_{α}(w)e_{β}(z) at the point w = q^{−1}z. The formulas (2.42) and (2.43) are analogues of the simple formula

\frac{1}{1 − z} = \frac{z^{−1}}{z^{−1} − 1} + δ(z)

which describe the analytical continuation of the analytical in the domain |z| < 1 function (1 − z)^{−1} into the domain |z| > 1 .

On the other hand it was proved in [DK] that so defined composed currents satisfy the commutation relations

e_{α}(z)e_{α+β}(w) = \frac{qz − w}{z − qw} e_{α+β}(w)e_{α}(z) , |z| < |qw| ,  (2.44)
e_{α+β}(z)e_{β}(w) = \frac{q^{2}z − q^{−2}w}{qz − q^{−1}w} e_{β}(w)e_{α+β}(z) , |z| < |q^{−2}w| ,  (2.45)

f_{β}(z)f_{α+β}(w) = \frac{q^{−2}z − q^{2}w}{q^{−1}z − qw} f_{α+β}(w)f_{α}(z) , |z| < |q^{2}w| ,  (2.46)

f_{α+β}(z)f_{α}(w) = \frac{q^{−1}z − w}{z − q^{−1}w} f_{α}(w)f_{α+β}(z) , |z| < |q^{−1}w| ,  (2.47)

which are equivalent to the Serre relations (2.12) and (2.13).

Using analogous analytical considerations we can rewrite the path ordered exponent entering the universal R -matrix as ordered contour integrals

P exp^{⟶}_{q^{2}} (\frac{(q^{−1} − q)}{2πi} ∮ f_{γ_{i}}(z) ⊗ e_{γ_{i}}(z)\frac{dz}{z} ) = 1+

+ ∑_{n>0} \frac{(q^{−1} − q)^{n}}{n!(2πi)^{n}} ∮_{C_{n}} \frac{dz_{n}}{z_{n}} ∮_{C_{n−1}} \frac{dz_{n−1}}{z_{n−1}} ⋯ ∮_{C_{1}} \frac{dz_{1}}{z_{1}} f_{γ_{i}}(z_{1})f_{γ_{i}}(z_{2}) … f_{γ_{i}}(z_{n})⊗

⊗e_{γ_{i}}(z_{1})e_{γ_{i}}(z_{2}) … e_{γ_{i}}(z_{n}),

where the contours C_{k} enclose an origin and are in a region where the points z_{i} = q^{2}z_{j} are outside the contours C_{i} and the points z_{i} = q^{−2}z_{j} are inside the contours C_{i} . The form of the contours is dictated by the analytical properties of the function f_{γ}(z) ⊗ e_{γ}(z) (see details in [DK]).

3 Twisting of the quantized current algebra

3.1 General principles of twisting

Here we present main facts concerning quasi-Hopf algebras [D2] relevant for the following discussion.

Definition 3.1 A quasi-triangular quasi-Hopf algebra is a set (A, Δ, ε, S, Φ, R, α, β) such that A is a unital associative algebra over field ℂ, Δ and ε are comultiplication and counit homomorphisms of the form Δ : A → A ⊗ A and ε : A → ℂ, S is antipod, that is an antiautomorphism S : A → A, Φ ∈ A^{⊗3} and R ∈ A^{⊗2} are an invertible elements called coassociator and universal R -matrix respectively, α, β ∈ A are some element such that following axioms are satisfied:

(id ⊗ Δ)Δ(x) = Φ(Δ ⊗ id)Δ(x)Φ^{−1}, ∀ x ∈ A ,  (3.1a)
(id ⊗ id ⊗ Δ)Φ · (Δ ⊗ id ⊗ id)Φ = (1 ⊗ Φ) · (id ⊗ Δ ⊗ id)Φ · (Φ ⊗ 1) ,  (3.1b)
(ε ⊗ id) ∘ Δ = (id ⊗ ε) ∘ Δ = id ,  (3.1c)
(id ⊗ ε ⊗ id)Φ = 1 ,  (3.1d)
S(x^{(1)})αx^{(2)} = ε(x)α, x^{(1)}βS(x^{(2)}) = ε(x)β, ∀ x ∈ A ,  (3.1e)
Φ^{(1)} β S(Φ^{(2)}) α Φ^{(3)} = 1, where Φ = Φ^{(1)} ⊗ Φ^{(2)} ⊗ Φ^{(3)} ,  (3.1f)
Δ^{op}(x) = R Δ(x) R^{−1}, ∀ x ∈ A ,  (3.1g)
(Δ ⊗ id)R = Φ_{312}R_{13}Φ_{132}^{−1}R_{23}Φ_{123} ,  (3.1h)
(id ⊗ Δ)R = Φ_{231}^{−1}R_{13}Φ_{213}R_{12}Φ_{123}^{−1} ,  (3.1i)

where Δ^{op} is opposite comultiplication and we used standard notations to signify the positions of the elements in the tensor products.

Definition 3.2 Let F ∈ A^{⊗2} is an invertible element such that (id ⊗ ε)F = (ε ⊗ id)F = 1 . Then to a given quasi-triangular quasi-Hopf algebra (A, Δ, ε, S, Φ, R, α, β) we can associate the twisted algebra (A, Δ^{F}, ε, S, Φ^{F}, R^{F}, α^{F}, β^{F}) where

Δ^{F}(x) = F Δ(x) F^{−1}, ∀ x ∈ A ,  (3.2a)
Φ^{F} = (F_{23}(id ⊗ Δ)F) Φ (F_{12}(Δ ⊗ id)F)^{−1} ,  (3.2b)
R^{F} = F_{21} R F_{12}^{−1} ,  (3.2c)
α^{F} = S(F^{−′}) α F^{−′′}, β^{F} = F′ β S(F′′) ,  (3.2d)

where F = F′ ⊗ F′′ and F^{−1} = F^{−′} ⊗ F^{−′′} .

It is clear that the Hopf algebras, those which have trivial coassociator Φ = 1 ⊗ 1 ⊗ 1 , are stable under the twisting if element F satisfying the cocycle condition:

F_{12}(Δ ⊗ 1)F = F_{23}(1 ⊗ Δ)F .  (3.3)

If element F(λ) depend on the element λ of abelian subalgebra 𝔥 and satisfies instead of (3.3) the shifted cocycle condition

F_{12}(λ)(Δ ⊗ 1)F(λ) = F_{23}(λ + h_{1})(1 ⊗ Δ)F(λ)  (3.4)

then the twisting of the Hopf algebra A by this element brought it to the quasi-Hopf algebra with coassociator Φ(λ) and R -matrix R(λ) which satisfy the properties

Φ(λ) = F_{23}(λ)F_{23}^{−1}(λ + h_{1}) ,  (3.5a)
(Δ(λ) ⊗ id)R(λ) = Φ_{312}(λ)R_{13}(λ)R_{23}(λ + h_{1}) ,  (3.5b)

(id ⊗ Δ(λ))R(λ) = R_{13}(λ + h_{2})R_{12}(λ)Φ_{123}^{−1}(λ) .  (3.5c)

As a corollary a dynamical Yang-Baxter equation for the R -matrix R(λ) holds

R_{12}(λ + h_{3})R_{13}(λ)R_{23}(λ + h_{1}) = R_{23}(λ)R_{13}(λ + h_{2})R_{12}(λ) .  (3.6)

In the formulas above the notation λ+h_{1} etc. means that if element λ ∈ 𝔥 has decomposition λ = ∑_{i} λ^{i}h_{i} where h_{i} is a linear independent basis in 𝔥, then λ + h_{1} means ∑_{i}(λ^{i} + h_{1}^{i})h_{i} . In the considerations below we will consider the case when subalgebra 𝔥 is one dimensional and correspond to the central element, so we have only one parameter which we call τ and which admits the shifts by the central element c.

3.2 Twisting of the quantized current algebra

Let us return to the current algebra from the first section which is a Hopf algebra with coalgebraic structure given by the formulas (2.28). In terms of the Drinfeld generators a_{α,n} this coalgebraic structure is given by

Δ a_{α,n} = a_{α,n} ⊗ q^{− \frac{c|n|}{2}} + q \frac{c|n|}{2} ⊗ a_{α,n}, ∀n ≠ 0 ,  (3.7)

where we have identified the maps Δ^{(1)} = Δ^{(2)} = Δ as follows from comparing (2.28c) and (2.33c). ¿From the general principles of twisting we can see that due to the special factorization properties of the universal R -matrix (2.32) the Cartan part of this matrix is twisted while the nonabelian part \overline{R} is dressed by the twisting element. This observation allows to guess that the solution of the shifted cocycle condition can be found using only the Cartan Drinfeld’s generators a_{α,n} , n ∈ ℤ.

Fix a complex number τ such that τ ≠ ±k^{−1} and τ + c ≠ ±k^{−1} for integer k > 0 , where by c we mean the value of the central element c on the representation of Û_{q}(𝔤^{}ˆ ) . For such values of the parameter τ we have

Lemma 3.1 The shifted cocycle condition (3.4) has two unique solution fixed by the comultiplication (3.7), the commutation relations (2.16) and by the requirement that the dressed elliptic currents satisfy the commutation relation (4.4)–(4.10):

F^{(1)}(τ) = exp (∑_{{n>0 \atop α,β}} \frac{nd_{α,β}^{(n)}q^{τn}}{[(τ + c_{1})n]} (q^{\frac{cn}{2}} a_{α,−n} ⊗ q^{\frac{cn}{2}} a_{β,n})) ,  (3.8)

F^{(2)}(τ) = exp (− ∑_{{n>0 \atop α,β}} \frac{nd_{α,β}^{(n)}q^{τn}}{[(τ + c_{1})n]} (q^{\frac{cn}{2}} a_{α,n} ⊗ q^{\frac{cn}{2}} a_{β,−n})) .  (3.9)

Definition 3.3 By the dressed elliptic currents f_{γ_{i}}(z; τ) and e_{γ_{i}}(z; τ) we understand the currents which obtained by the dressing of the tensor f_{γ_{i}}(z) ⊗ e_{γ_{i}}(z) with twisting elements F^{(1)}(τ) or F^{(2)}(τ) . For example,

f_{γ_{i}}(z; τ) ⊗ e_{γ_{1}}(z; τ + c_{1}) = F^{(i)}(τ) f_{γ_{i}}(z) ⊗ e_{γ_{1}}(z) (F^{(i)}(τ))^{−1} , i = 1, 2 .  (3.10)

It is convenient to define the elliptic current by this formula as shifted by the central element in the first tensor product c_{1} = c ⊗ 1 .

For all elements of the algebra U_{q}(𝔤^{}ˆ ) let us define the twisted comultiplications

Δ_{τ}^{(1)}(x) = F^{(1)}(τ) Δ^{(1)}(x) (F^{(1)}(τ))^{−1} , and Δ_{τ}^{(2)}(x) = F^{(2)}(τ) Δ^{(2)}(x) (F^{(2)}(τ))^{−1} .  (3.11)

We have the following Proposition 3.1 The elliptic currents defined by (3.10) with i = 1 and for simple root α can be written in normal ordered form

f_{α}(z; τ) = u_{α}^{−}(zq^{− \frac{c}{2}} ; τ) f_{α}(z), e_{α}(z; τ) = e_{α}(z) u_{α}^{+}(zq^{− \frac{c}{2}} ; τ),  (3.12)

where

u_{α}^{−}(z; τ) = exp (∑_{n>0} \frac{q^{(τ+c)n}}{[(τ + c)n]} a_{α,−n}z^{n}) ,  u_{α}^{+}(z; τ) = exp (− ∑_{n>0} \frac{q^{τn}}{[τn]} a_{α,n}z^{−n}) .  (3.13)

The comultiplication of these currents with respect to Δ_{τ}^{(1)} is given by

Δ_{τ}^{(1)}e_{α}(z; τ) = e_{α}(z; τ) ⊗ 1 + k_{α}^{−1}ψ_{α}(zq^{−τ}; τ) ⊗ e_{α}(zq^{c_{1}}; τ + c_{1}) ,  (3.14a)
Δ_{τ}^{(1)}f_{α}(z; τ) = 1 ⊗ f_{α}(z; τ + c_{1}) + f_{α}(zq^{c_{2}}; τ) ⊗ k_{α}ψ_{α}(zq^{τ+c_{2}}; τ + c_{1}) ,  (3.14b)
Δ_{τ}^{(1)}c = c ⊗ 1 + 1 ⊗ c, Δ_{τ}^{(1)}k_{α}^{±1} = k_{α}^{±1} ⊗ k_{α}^{±1}, Δ_{τ}^{(1)}ψ_{α}(z) = ψ_{α}(z; τ) ⊗ ψ_{α}(z; τ + c_{1}) ,  (3.14c)

where the elliptic Cartan current ψ_{α}(z; τ) is given by the normal ordered expression

\begin{matrix}ψ_{α}(z; τ) = k_{α}^{−1}u_{α}^{−}(zq^{−τ− \frac{3c}{2}} ; τ) ψ_{α}^{+}(zq^{−τ− \frac{c}{2}} ) u_{α}^{+}(zq^{−τ− \frac{c}{2}} ; τ) = \\ = k_{α}u_{α}^{−}(zq^{τ+ \frac{c}{2}} ; τ) ψ_{α}^{−}(zq^{τ+ \frac{c}{2}} ) u_{α}^{+}(zq^{τ− \frac{c}{2}} ; τ) = \\ = exp (∑_{n>0} \frac{q^{−cn/2}}{[(τ + c)n]} a_{α,−n}z^{n}) exp (− ∑_{n>0} \frac{q^{cn/2}}{[τn]} a_{α,n}z^{−n}) . \end{matrix}  (3.15)

Analogously, the elliptic currents defined by (3.10) for i = 2 can be written in the normal ordered form

f_{α}(z; τ) = f_{α}(z) v_{α}^{+}(zq^{\frac{c}{2}} ; τ) , e_{α}(z; τ) = v_{α}^{−}(zq^{\frac{c}{2}} ; τ) e_{α}(z) ,  (3.16)

where

v_{α}^{−}(z; τ) = exp (∑_{n>0} \frac{q^{τn}}{[τn]} a_{α,−n}z^{n}) ,  v_{α}^{+}(z; τ) = exp (− ∑_{n>0} \frac{q^{(τ+c)n}}{[(τ + c)n]} a_{α,n}z^{−n}) .  (3.17)

These currents satisfy the comultiplication rules

Δ_{τ}^{(2)}e_{α}(z; τ) = e_{α}(z; τ) ⊗ 1 + k_{α}ψ_{α}(zq^{τ}; τ) ⊗ e_{α}(zq^{−c_{1}}; τ + c_{1}) ,  (3.18a)
Δ_{τ}^{(2)}f_{α}(z; τ) = 1 ⊗ f_{α}(z; τ + c_{1}) + f_{α}(zq^{−c_{2}}; τ) ⊗ k_{α}^{−1}ψ_{α}(zq^{−τ−c_{2}}; τ + c_{1}) ,  (3.18b)
Δ_{τ}^{(2)}c = c ⊗ 1 + 1 ⊗ c, Δ_{τ}^{(2)}k_{α}^{±1} = k_{α}^{±1} ⊗ k_{α}^{±1}, Δ_{τ}^{(2)}ψ_{α}(z) = ψ_{α}(z; τ) ⊗ ψ_{α}(z; τ + c_{1}) ,  (3.18c)

with respect to the Δ_{τ}^{(2)} and the Cartan current is defined as follows

\begin{matrix}ψ_{α}(z; τ) = k_{α}^{−1}u_{α}^{−}(zq^{−τ− \frac{3c}{2}} ; τ) ψ_{α}^{+}(zq^{−τ− \frac{c}{2}} ) u_{α}^{+}(zq^{−τ− \frac{c}{2}} ; τ) = \\ = k_{α}u_{α}^{−}(zq^{τ+ \frac{c}{2}} ; τ) ψ_{α}^{−}(zq^{τ+ \frac{c}{2}} ) u_{α}^{+}(zq^{τ− \frac{c}{2}} ; τ) \\ = exp (∑_{n>0} \frac{q^{cn/2}}{[τn]} a_{α,−n}z^{n}) exp (− ∑_{n>0} \frac{q^{−cn/2}}{[(τ + c)n]} a_{α,n}z^{−n}) . \end{matrix}  (3.19)

4 Analytical properties of elliptic current algebra

The goal of this section is to describe the analytical properties of the elliptic currents and proof of the fact that so defined elliptic current algebra conserve the notion of highest weight representations of the original current algebra. In all the formulas below we understand the elliptic currents e_{α}(z; τ) , f_{α}(z; τ) , ψ_{α}(z; τ) given by formulas (3.12) and (3.15) or (3.16) and (3.19).

4.1 The commutation relations

We consider the algebra U_{q,τ} (𝔤^{}ˆ ) generated by the elements

{c, d, k_{α}^{±1}, e_{α,n}(τ), f_{α,n}(τ), ψ_{α,n}(τ)}, n ∈ ℤ, τ ∈ ℂ, |q^{τ} | ≠ 1, |q^{τ+c}| ≠ 1  (4.1)

which are decompositions of the elliptic currents into series with respect to the spectral parameter:

e^{α}(z; τ) = ∑_{n∈ℤ} e^{α,n}(τ)z^{−n}, f^{α}(z; τ) = ∑_{n∈ℤ} f^{α,n}(τ)z^{−n},  (4.2a)

ψ_{α}(z; τ) = ∑_{n∈ℤ} ψ_{α,n}(τ)z^{−n} = exp (∑_{n>0} \frac{a_{α,−n}(τ)}{[τn]}z^{n}) exp (− ∑_{n>0} \frac{a_{α,n}(τ)}{[τn]}z^{−n}) ,  (4.2b)

where we have introduced by (4.2b) the elliptic version of Drinfeld generators a_{α,n}(τ) . The value of the parameter τ are restricted to the condition formulated before Lemma 3.1.

Using formulas (2.2)–(2.13) we present the commutation relations of elliptic currents in the form:

q^{d}x(z)q^{−d} = x(qz), for x = e_{α}(z, τ), f_{α}(z, τ), ψ_{α}(z, τ) ,  (4.3)
k_{α}^{±1}e_{β}(z; τ) = q^{±(α,β)}e_{β}(z; τ)k_{α}^{±1}, k_{α}^{±1}f_{β}(z; τ) = q^{∓(α,β)}f_{β}(z; τ)k_{α}^{±1}, k_{α}^{±1}ψ_{β}(z; τ) = ψ_{β}(z; τ)k_{α}^{±1} , (4.4)
\begin{matrix}(z − q^{(α,β)}w) Υ (q^{τ} w/z; (α, β), τ) e_{α}(z; τ)e_{β}(w; τ) = \\ = (q^{(α,β)}z − w) Υ (q^{τ} z/w; (α, β), τ) e_{β}(w; τ)e_{α}(z; τ) , \end{matrix}  (4.5)

\begin{matrix}(z − q^{−(α,β)}w) Υ (q^{τ+c}w/z; −(α, β), τ + c) f_{α}(z; τ)f_{β}(w; τ) = \\ = (q^{−(α,β)}z − w) Υ (q^{τ+c}z/w; −(α, β), τ + c) f_{β}(w; τ)f_{α}(z; τ) , \end{matrix}  (4.6)
Υ (w/z; (α, β), τ) ψ_{α}(z; τ)e_{β}(w; τ) = Υ (z/w; (α, β), τ) e_{β}(w; τ)ψ_{α}(z; τ) ,  (4.7)
Υ (w/z; −(α, β), τ + c) ψ_{α}(z; τ)f_{β}(w; τ) = Υ (z/w; −(α, β), τ + c) f_{β}(w; τ)ψ_{α}(z; τ) ,  (4.8)
\frac{Υ (q^{τ} w/z; (α, β), τ)}{Υ (q^{τ+c}w/z; (α, β), τ + c)}ψ_{α}(z; τ)ψ_{β}(w; τ) = \frac{Υ (q^{τ} z/w; (α, β), τ)}{Υ (q^{τ+c}z/w; (α, β), τ + c)}ψ_{β}(w; τ)ψ_{α}(z; τ) ,  (4.9)
[e_{α}(z; τ), f_{β}(w; τ)] = \frac{δ_{α,β}}{q − q^{−1}} (δ(q^{−c}z/w)k_{α}ψ_{α}(zq^{τ}; τ) − δ(q^{c}z/w)k_{α}^{−1}ψ_{α}(zq^{−τ}; τ)) .  (4.10)

For (α, β) = −1 the currents e_{α}(z) and f_{α}(z) satisfy deformed Serre relations:

∑_{k=1}^{3} \frac{(−1)^{k}}{[k − 1]![3 − k]!} ∏_{i<j}^{3} Υ (q^{τ} w̃_{i}/w̃_{j}; (α̃_{i}, α̃_{j}), τ) ∏_{i=1}^{3} eα̃_{i}(w̃_{i}; τ) + ⋯ = 0  (4.11a)

∑_{k=1}^{3} \frac{(−1)^{k}}{[k − 1]![3 − k]!} ∏_{i<j}^{3} Υ (q^{τ+c}w̃_{j}/w̃_{i}; (α̃_{i}, α̃_{j}), τ + c)^{−1} ∏_{i=1}^{3} fα̃_{i}(w̃_{i}; τ) + ⋯ = 0  (4.11b)

where in sums over k we should understand α̃_{i} = α for i ≠ k, α̃_{k} = β , w̃_{i} = w_{i} for i < k, w̃_{k} = z, w̃_{i} = w_{i−1} for i > k and the dots in (4.11) means symmetrization with respect to the spectral parameters w_{1} and w_{2} .

In these formulas we denoted by Υ (z; a, b) the function

Υ (z; a, b) = exp (∑_{n>0} \frac{z^{n}}{n} \frac{[an]}{[bn]})  (4.12)

which always is understood as power series with respect to the parameter z and has different presentation in terms of infinite products depending whether |q^{b}| < 1 or |q^{b}| > 1 . The commutation relations of the type (4.4)–(4.11b) have been considered in [DI] for arbitrary function Υ (z; a, b) .

The commutation relations (4.7)–(4.9) can be also written in terms of the elliptic Drinfeld’s generators a_{α,n}(τ) . To make these commutation relations more symmetric it is natural to introduce another set of the Drinfeld’s generators a′_{α,n}(τ) related to a_{α,n}(τ) ones as follows:

a′_{α,n}(τ) = \frac{[(τ + c)n]}{[τc]}a_{α,n}(τ) ,  (4.13)

[a_{α,n}(τ), e_{β}(z; τ)] = \frac{[(α, β)n]}{n}z^{n}e_{β}(z) ,  (4.14)
[a′_{α,n}(τ), f_{β}(z; τ)] = −\frac{[(α, β)n]}{n}z^{n}f_{β}(z) ,  (4.15)
[a_{α,n}(τ), a_{β,m}(τ)] = δ_{n+m,0} \frac{[(α, β)n][cn]}{n} \frac{[τn]}{[(τ + c)n]} .  (4.16)

4.2 Reductions to quantized current algebra

Let us denote the values of the parameter τ such that q^{τ} = 0 and q^{−τ} = 0 , τ_{0} and τ′_{0} respectively. We can see that at the value τ = τ_{0} the elliptic algebra U_{q,τ} (𝔤^{}ˆ ) becomes current algebra U_{q}(𝔤^{}ˆ ) . The Cartan currents at τ = τ_{0} becomes:

k_{α}ψ_{α}(zq^{τ_{0}}; τ_{0}) = ψ_{α}^{+}(zq^{−c/2}),  k_{α}^{−1}ψ_{α}(zq^{−τ_{0}}; τ_{0}) = ψ_{α}^{−}(zq^{c/2})  (4.17)

or in terms of the Drinfeld generators a_{α,n}(τ)

a_{α,n}(τ_{0}) = a_{α,n}q^{c|n|/2} .  (4.18)

On the other hand the elliptic algebra U_{q,τ} (𝔤^{}ˆ ) degenerates at τ = τ′_{0} into current algebra U_{q^{−1}}(𝔤^{}ˆ ) . The Drinfeld’s generators in this case are related to the current ones as follows:

a_{α,n}(τ′_{0}) = a_{α,n}q^{−c|n|/2} .  (4.19)

4.3 The automorphisms

The elliptic current algebra defined in the previous subsection possesses the inner automorphisms^{2} η, T_{α} and T_{α}^{−1} , such that η is an anti-automorphism

η (e_{α_{i}}(z, τ)) = e_{α_{i}}(z^{−1}, τ), η (f_{α_{i}}(z, τ)) = f_{α_{i}}(z^{−1}, τ), η (ψ_{α_{i}}(z, τ)) = ψ_{α_{i}}(z^{−1},), η (k_{α}) = k_{α}^{−1}

(4.20)

and T_{α} and T_{α}^{−1} are automorphisms defined by the formulas

\begin{matrix}T_{α}k_{α} = k_{α}^{−1}, T_{α}ψ_{α}(z; τ) = :ψ_{α}^{−1}(z; τ):, \\ T_{α}e_{α}(z; τ) = :f_{α}(zq^{−c}; τ)ψ_{α}^{−1}(zq^{τ}; τ)k_{α}^{−1}: , T_{α}f_{α}(z; τ) = :k_{α}ψ_{α}^{−1}(zq^{−τ−c}; τ)e_{α}(zq^{−c}; τ): , \end{matrix}  (4.21)

and for the simple β such that (α, β) = −1 :

\begin{matrix}T_{α}k_{β} = k_{α}k_{β}, T_{α}ψ_{β}(z; τ) = :ψ_{α}(qz; τ)ψ_{β}(q^{2}z; τ): , \\ T_{α}e_{β}(z; τ) = e_{α+β}(z; τ) , T_{α}f_{β}(z; τ) = f_{α+β}(z; τ) , \end{matrix}  (4.22)

where we denoted by e_{α+β}(z; τ) and f_{α+β}(z; τ) the following series:

\begin{matrix}e_{α+β}(z) = ∫ \frac{dw}{w} ( Υ (q^{2+τ} z/w; (α, β), τ) e_{α}(w; τ)e_{β}(q^{2}z; τ) − \\ − q Υ (q^{−2−τ} w/z; (α, β), τ) e_{β}(q^{2}z; τ)e_{α}(w; τ)) , \end{matrix}  (4.23)

2We denoted them by the same letters as in the case of quantum current algebras.

\begin{matrix}f_{α+β}(z) = ∫ \frac{dw}{w} (− Υ (q^{τ+c−2}w/z; −(α, β), τ + c) f_{β}(q^{2}z; τ)f_{α}(w; τ) + \\ + q^{−1} Υ (q^{2−τ−c}z/w; −(α, β), τ + c) f_{α}(w; τ)f_{β}(q^{2}z; τ)) . \end{matrix}  (4.24)

The symbol ∫ \frac{dw}{w} means again picking up the coefficient in front of w^{0} and the function Υ (z; a, b) is understood as power series with respect to spectral parameter z.

The normal ordered in the formulas (4.21) and (4.22) signify that all the modes a_{α,n}(τ) of the Cartan current ψ_{α}(z) with n > 0 are placed on the right and with n < 0 on the left in the corresponding product. For example, the notation :ψ_{α}^{−1}(z; τ): means

:ψ_{α}^{−1}(z; τ): = exp (− ∑_{n>0} \frac{a_{α,−n}(τ)}{[τn]}z^{n}) exp (∑_{n>0} \frac{a_{α,n}(τ)}{[τn]}z^{−n}) .  (4.25)

The inverse to T_{α} automorphism is defined by the composition

T_{α}^{−1} = η ∘ T_{a} ∘ η .  (4.26)

As well as in the case of quantum current algebra the action of the inverse automorphisms T_{a}^{−1} on the currents e_{β}(z; τ) and f_{β}(z; τ) is defined by the series

\begin{matrix}ě_{α+β}(z) = T_{α}^{−1}e_{β}(z; τ) = ∫ \frac{dw}{w} (− Υ (q^{τ+2}w/z; (α, β), τ) e_{β}(q^{−2}z; τ)e_{α}(w; τ) + \\ + q Υ (q^{−τ−2}z/w; (α, β), τ) f_{α}(w; τ)f_{β}(q^{−2}z; τ)) , \end{matrix}  (4.27)

\begin{matrix}f̌_{α+β}(z) = T_{α}^{−1}f_{β}(z; τ) = ∫ \frac{dw}{w} ( Υ (q^{τ+c−2}z/w; −(α, β), τ + c) f_{α}(w; τ)f_{β}(q^{−2}z; τ) − \\ − q^{−1} Υ (q^{2−τ−c}w/z; −(α, β), τ + c) f_{β}(q^{−2}z; τ)f_{α}(w; τ)) . \end{matrix}  (4.28)

4.4 The analytical properties of quantum elliptic currents

The commutation relations above encoded the analytical structure of the product of the currents. They are given by following

Lemma 4.1 (ı) For |q^{τ} | < 1 the product e_{α}(z; τ)e_{β}(w; τ) has simple poles at the points z = q^{(α,β)}q^{2τk}w, k ≥ 0 and simple zeros at the points z = q^{−(α,β)}q^{2τk}w, k > 0 ;

(ıı) For |q^{τ} | > 1 the product e_{α}(z; τ)e_{β}(w; τ) has simple poles at the points z = q^{−(α,β)}q^{−2τk}w, k ≥ 0 and simple zeros at the points z = q^{(α,β)}q^{−2τk}w, k > 0 ;

(ııı) For |q^{τ+c}| < 1 the product f_{α}(z; τ)f_{β}(w; τ) has simple poles at the points z = q^{−(α,β)}q^{2(τ+c)k}w, k ≥ 0 and simple zeros at the points z = q^{(α,β)}q^{2(τ+c)k}w, k > 0 ;

(ıv) For |q^{τ+c}| > 1 the product f_{α}(z; τ)f_{β}(w; τ) has simple poles at the points z = q^{(α,β)}q^{−2(τ+c)k} , k ≥ 0 and simple zeros at the points z = q^{−(α,β)}q^{−2(τ+c)k}w, k > 0 .

This Lemma states that depending on whether |q^{τ} | < 1 or |q^{τ} | > 1 the product e_{α}(z; τ)e_{β}(w; τ) is analytical function in the domains |z| > |q^{(α,β)}w| and |z| < |q^{−(α,β)}w| respectively. Analogously, the product f_{α}(z; τ)f_{β}(w; τ) is analytical function in the domains |z| > |q^{−(α,β)}w| and |z| < |q^{(α,β)}w| for the conditions |q^{τ+c}| < 1 and |q^{τ+c}| > 1 respectively.

As well as in case of quantized current algebras we can calculate the residues of the products of the elliptic currents. We will demonstrate below that the residues at the points z = q^{±(α,β)}w coincide with the action of automorphisms T_{α} and T_{α}^{−1} on the currents e_{β}(z; τ) and f_{β}(z; τ) while the residues at others points are equal to the values of the corresponding product in their domains of analyticity.

For the simplicity we consider a particular case of |q^{τ} | < 1 and calculate the residue of the product e_{α}(w; τ)× e_{β}(q^{2}z; τ) for (α, β) = −1 at the point w = qz (see Fig.1, where ∘ signify the poles and • the zeros).

[figure: ·; 0; ⋯; ⋯; •; ∘; ∘; ∘; •; qz; pqz; pq^{3}z; p^{m}qz; p^{m}q^{3}z; w; Fig. 1. The form of the contours in the complex plane w, p = q^{2τ} .]

res_{w=qz} e_{α}(w)e_{β}(q^{2}z) = ∮_{w around qz} \frac{dw}{2πiw} e_{α}(w)e_{β}(q^{2}z) =

= ∮_{|w|>|qz|} \frac{dw}{2πiw} e_{α}(w)e_{β}(q^{2}z) − q ∮_{|q^{2τ+1}z|<|w|<|qz|} \frac{dw}{2πiw} \frac{Θ_{q^{2τ}} (q^{−3}w/z)}{Θ_{q^{2τ}} (q^{−1}w/z)}e_{β}(q^{2}z)e_{α}(w) =

= ∏_{k>0} \frac{(1 − q^{2kτ+2})}{(1 − q^{2kτ})}e_{α+β}(z)

(4.29)

where for |p| < 1 Θ_{p}(z) denote multiplicative θ -function given by infinite product

Θ_{p}(z) = ∏_{k=0}^{∞}(1 − zp^{k})(1 − z^{−1}p^{k+1})(1 − p^{k+1}).  (4.30)

As we can see from (4.29) the residue of the product e_{α}(w)e_{β}(q^{2}z) in the point w = qz yields the composed elliptic current given by the automorphism (4.23). The rest of the composed elliptic currents are given by the analogous residues in the point w = qz for f_{α+β} and w = q^{−1}z for ě_{α+β} and f̌_{α+β} .

Let us calculate now the residues of the product e_{α}(w; τ)e_{β}(q^{2}z; τ) in the points w = q^{2mτ+1}z for integer m > 0 . To do this we consider the operator valued function

E(w, z) = ∏_{k=1}^{m} \frac{w − q^{2τ(k−1)+1}z}{w − q^{2τk+3}z} e_{α}(w)e_{β}(q^{2}z).  (4.31)

It is obvious that this function is analytical in the domain |w| > |q^{2τm+1}z| and its residue at the point w = q^{2τm+1}z coincide with the residue which we are interesting in up to some constant. This residue can be presented as difference of two contour integrals around origin w = 0 (see Fig. 1.) such that first contour contain the point w = q^{2τm+1}z and the variable of integration in the second integral restricted to the domain |q^{2τ(m+1)+1}z| < |w| < |q^{2τm+1}z|. In this domain the function E(w, z) is replaced by its analytical continuation given by the commutation relation (4.5)

q \frac{Θ_{q^{2τ}} (q^{−3}w/z)}{Θ_{q^{2τ}} (q^{−1}w/z)} ∏_{k=1}^{m} \frac{w − q^{2τ(k−1)+1}z}{w − q^{2τk+3}z} e_{β}(q^{2}z)e_{α}(w).  (4.32)

We can now move the contour of the integration in the second integral through the pole at the point w = q^{2τm+1}z, calculate the residue of (4.32) and using once more the commutation relation (4.5) in the domain |w| > |q^{2τm+1}z| we conclude with the answer:

res_{w=q^{2τm+1}z} e^{α}(w)e^{β}(q^{2}z) = q^{2m}(q^{−1} − q) ∏_{k>0} \frac{(1 − q^{2τk+2})(1 − q^{2τk−2})}{(1 − q^{2τk})^{2}}e^{β}(q^{2}z)e^{α}(q^{2τm+1}z).  (4.33)

To make sure that our analytical considerations are correct we can calculate this residue directly using the dressing formulas for elliptic currents (3.12). To proceed with this calculation and for |q^{τ} | < 1 it is convenient to express the dressing factor u_{α}^{+}(z; τ) in terms of operators

ψ̃_{α}^{+}(z) = k_{α}^{−1}ψ_{α}^{+}(z) = exp ((q − q^{−1}) ∑_{n>0} a_{α,n}z^{−n})  (4.34)

as follows

u+ α(z; τ) = ∏_{k>0} ψ̃_{α}^{+}(zq^{−2τk})  (4.35)

Then the calculation of the residue is

qz(q^{2τm} − 1) res_{w=q^{2τm+1}z} e_{α}(w; τ)e_{β}(q^{2}z; τ) = res_{w=q^{2τm+1}z} (w − qz)e_{α}(w; τ)e_{β}(q^{2}z; τ) =

= res_{w=q^{2τm+1}z} (w − qz)e_{α}(w)e_{β}(q^{2}z) ∏_{k>0} \frac{q^{−1}w − q^{2τk+2}z}{qw − q^{2τk+2}z} ψ̃_{α}^{+} (wq^{−2τk−c/2}) ψ̃_{β}^{+} (zq^{2−2τk−c/2}) =

= res_{w=q^{2τm+1}z} (q^{−1}w − q^{2}z)e_{β}(q^{2}z)e_{α}(w) ∏_{k>0} \frac{q^{−1}w − q^{2τk+2}z}{qw − q^{2τk+2}z} ψ̃_{β}^{+} (zq^{2−2τk−c/2}) ψ̃_{α}^{+} (wq^{−2τk−c/2}) =

= res_{w=q^{2τm+1}z} (q^{−1}w − q^{2}z)e_{β}(q^{2}z; τ)e_{α}(w; τ) ∏_{k>0} \frac{w − q^{2τk+3}z}{w − q^{2τk+1}z} \frac{z − q^{2τk−3}w}{z − q^{2τk−1}w} =

= qz(q^{2τm} − 1)q^{2m}(q^{−1} − q) ∏_{k>0} \frac{(1 − q^{2τk+2})(1 − q^{2τk−2})}{(1 − q^{2τk})^{2}}e_{β}(q^{2}z)e_{α}(q^{2τm+1}z)

(4.36)

which obviously yields the same answer as (4.33).

We result of this calculations can be interpreted as follows. Although the product of elliptic currents has many poles only those at the shifted diagonal w = q^{±(α,β)}z produces new generators of the algebra which are gathered into composed currents. The definitions of these composed roots can be written in the analytical language.

\begin{matrix}e_{α+β}(q^{−1}z; τ)δ(qz/w) = Υ (q^{τ} w/z; −1, τ) e_{α}(z; τ)e_{β}(w; τ)− \\ −q Υ (q^{−τ} z/w; −1, τ) e_{β}(w; τ)e_{α}(z; τ) , \end{matrix}  (4.37a)

\begin{matrix}f_{α+β}(q^{−1}z; τ)δ(q^{−1}z/w) = Υ (q^{τ+c}w/z; 1, τ + c) f_{α}(z; τ)f_{β}(w; τ)− \\ −q^{−1} Υ (q^{−(τ+c)}z/w; 1, τ + c) f_{β}(w; τ)f_{α}(z; τ) , \end{matrix}  (4.37b)

\begin{matrix}ě_{α+β}(qz; τ)δ(q^{−1}z/w) = q Υ (q^{−τ} w/z; −1, τ) e_{α}(z; τ)e_{β}(w; τ)− \\ − Υ (q^{τ} z/w; −1, τ) e_{β}(w; τ)e_{α}(z; τ) , \end{matrix}  (4.37c)

\begin{matrix}f̌_{α+β}(qz; τ)δ(qz/w) = q^{−1} Υ (q^{−(τ+c)}w/z; 1, τ + c) f_{α}(z; τ)f_{β}(w; τ)− \\ − Υ (q^{τ+c}z/w; 1, τ + c) f_{β}(w; τ)f_{α}(z; τ) . \end{matrix}  (4.37d)

Analogously to the case of the current algebra we can prove that the relations

Υ (q^{τ} w/z; 1, τ) e_{α}(z; τ)e_{α+β}(w; τ) = q^{−1} Υ (q^{−τ} z/w; 1, τ) e_{α+β}(w; τ)e_{α}(z; τ) ,  (4.38a)
Υ (q^{τ−3}w/z; 1, τ) e_{α+β}(z; τ)e_{β}(w; τ) = q^{−1} Υ (q^{−τ+3}z/w; 1, τ) e_{β}(w; τ)e_{α+β}(z; τ) ,  (4.38b)
Υ (q^{τ+c+3}w/z; 1, τ + c) f_{β}(z; τ)f_{α+β}(w; τ) = q Υ (q^{−τ−c−3}z/w; 1, τ + c) f_{α+β}(w; τ)f_{α}(z; τ) ,  (4.38c)
Υ (q^{τ+c}w/z; 1, τ + c) f_{α+β}(z; τ)f_{α}(w; τ) = q Υ (q^{−τ−c}z/w; 1, τ + c) f_{α}(w; τ)f_{α+β}(z; τ) .  (4.38d)

are equivalent to deformed Serre relations (4.11). In formulas (4.37) and (4.38) all the functions Υ (z; a, b) should be understood as power series with respect to variable z.

Describing the commutation relations of the algebra U_{q,τ} (𝔤^{}ˆ ) and its analytical properties we did not use the explicitly elliptic functions, because the formulation which uses these functions is not universal. In particular, the explicit form of the relations depends whether q^{τ} > 1 or q^{τ} < 1 . Moreover, using the elliptic functions to define elliptic current algebras requires to add the additional information on the analytical properties of currents. For example, for q^{τ} < 1 the commutation relation (4.5) can be rewritten in the form

Θ_{q^{2τ}} (q^{(α,β)}w/z) e_{α}(z; τ)e_{β}(w; τ) = q^{(α,β)}Θ_{q^{2τ}} (q^{−(α,β)}w/z) e_{β}(w; τ)e_{α}(z; τ) ,  (4.39)

but we should say additionally to (4.39) that the product e_{α}(z; τ)e_{β}(w; τ) has zeros in the points z = q^{−(α,β)}q^{2τk}w, k > 0 . Otherwise the commutation relations (4.39) yields the algebra which is not compatible with the category of the highest weight representations. On the other hand the elliptic functions can be used to rewrite the equalities like (4.37) as equalities of the meromorphic functions. For example, (4.37a) can be written in the form

e_{α}(z; τ)e_{β}(w; τ) = q^{(α,β)} \frac{Θ_{q^{2τ}} (q^{−(α,β)}w/z)}{Θ_{q^{2τ}} (q^{(α,β)}w/z)}e_{β}(w; τ)e_{α}(z; τ) + δ(qz/w)e_{α+β}(q^{−1}z; τ) ∏_{k>0} \frac{1 − q^{2τk+2}}{1 − q^{2τk}} .

4.5 Coalgebraic structures and universal R -matrix

Using Lemma 4.1 we have

Proposition 4.1 The product of two-tensors G(z) = f_{γ}(z; τ) ⊗ e_{γ}(z; τ + c_{1}) , G(z)G(w) has the poles only in the points z = q^{±2}w, for all positive roots γ of simply laced algebra 𝔤.

Define the composed currents using the same decomposition of the longest element of the Weyl group (2.25) and the automorphisms (4.21)–(4.24). Then the universal R -matrix which correspond to the comultiplication (3.14) is given by

R(τ) = K(τ) ∏_{1≤i≤N}^{→} \overline{R}^{γ_{i}}(τ)  (4.40)

where the elliptic versions of the tensors \overline{R}_{γ}(τ) and K(τ) reads as follows:

\overline{R}_{γ_{i}}(τ) = P exp^{⟶}_{q^{2}} (\frac{(q^{−1} − q)}{2πi} ∮ f_{γ_{i}}(z; τ) ⊗ e_{γ_{i}}(z; τ + c_{1})\frac{dz}{z} ) ,  (4.41)

K(τ) = F_{21}^{(1)}(τ) K (F^{(1)}(τ))^{−1}  (4.42)

and the contours in the definition of the P -ordered exponent are the same as in (2.30) which is result of the Proposition 4.1.

The universal R -matrix which correspond to this comultiplication (3.18) is given by

∏_{1≤i≤N}^{→} (\overline{R}^{γ_{i}}(τ))_{21} · K̃(τ)  (4.43)

where \overline{R}_{γ}(τ) is given by (4.41) and K̃(τ) = F_{21}^{(2)}(τ) K (F^{(2)}(τ))^{−1} .

From the general principles of twisting one can observe that the R -matrix R(τ) satisfy the dynamical Yang-Baxter equation

R_{12}(τ + c_{3})R_{13}(τ)R_{23}(τ + c_{1}) = R_{23}(τ)R_{13}(τ + c_{2})R_{12}(τ)  (4.44)

Moreover, because the factorization property of the universal R -matrix (2.32) associated to the Drinfeld’s comultiplication in current algebra (2.28) the Cartan part of dynamical R -matrix also satisfy the same equation

K_{12}(τ + c_{3})K_{13}(τ)K_{23}(τ + c_{1}) = K_{23}(τ)K_{13}(τ + c_{2})K_{12}(τ)  (4.45)

The coassociator can be easily calculated from (3.5a) and is equal to

Φ^{(1)}(τ) = exp (∑_{{n>0 \atop α,β}} \frac{nd_{α,β}^{(n)}[c_{1}n]}{[(τ + c_{2})n][(τ + c_{1} + c_{2})n]} (1 ⊗ q^{− \frac{cn}{2}} a_{α,−n} ⊗ q^{\frac{cn}{2}} a_{β,n}))  (4.46)

Φ^{(2)}(τ) = exp (− ∑_{{n>0 \atop α,β}} \frac{nd_{α,β}^{(n)}[c_{1}n]}{[(τ + c_{2})n][(τ + c_{1} + c_{2})n]} (1 ⊗ q^{− \frac{cn}{2}} a_{α,n} ⊗ q^{\frac{cn}{2}} a_{β,−n}))  (4.47)

for comultiplication (3.14) and (3.18) respectively. It is interesting to get a closed formulas for the elements α and β (see Definition 3.1).

To get the dynamical YB equation which involves general element of Cartan subalgebra, not only the central one, we should expand the algebra by some additional operators, which where postulated in [JKOS]. We prefer to get those operators from natural algebraic construction on the level of quantum affine algebra such that the universal dynamical R -matrices which includes shifts onto general element of the Cartan subalgebra can be obtained by the same twisting procedure described in this paper. In particular this may give an universal explanation of finite-dimensional elliptic groups described in [BBB, Fe]. The meaning of this operators is very important in application of quantum elliptic algebras into integrable models.

Another problem which deserve further investigation is factorization problem of universal elliptic R - matrices (4.40), (4.43) and twisting of Drinfeld’s comultiplication to the physical one. Solving this problem allows to obtain systematically the Boltzmann weights of lattice integrable model of ABF type [ABF].

5 Acknowledgments

This investigation was started during visit one of the authors (S.P.) to Max Planck Institut für Mathematik. He would like to acknowledge a warm hospitality at this center.

The research described in this publication was made possible in part by grants RFBR-98-01-00344 (S. Khoroshkin and S. Kharchev), RFBR-97-01-01041 (S. Pakuliak), grant 96-15-96455 for support of scientific schools (S. Kharchev, S. Pakuliak) INTAS 93-10183 (S. Khoroshkin), and by Award No. RM2-150 of the U.S. Civilian Research & Development Foundation (CRDF) for the Independent States of the Former Soviet Union.

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