FINITE DIMENSIONAL REPRESENTATIONS OF THE ELLIPTIC

FINITE DIMENSIONAL REPRESENTATIONS OF THE ELLIPTIC MODULAR DOUBLE

arXiv:1310.7570v1 [math.QA] 28 Oct 2013

S. E. DERKACHOV AND V. P. SPIRIDONOV Abstract. We describe a set of zero modes of an integral operator depending on one parameter g and defining an elliptic Fourier transformation. This operator intertwines generators of the elliptic modular double formed from a pair of Sklyanin algebras. It has zero modes for the “spin” lattices g = nη + mτ /2 and g = 1/2 + nη + mτ /2 with incommensurate 1, 2η, τ , and Im(τ ), Im(η) > 0, n, m ∈ Z≥0 , (n, m) 6= (0, 0). These modes define finite dimensional representations of the elliptic modular double of dimension d = nm for n, m ∈ Z>0 or d = n ∈ Z>0 for m = 0 and d = m ∈ Z>0 for n = 0.

1. An elliptic modular double The concept of modular double of a quantum group has been introduced by Faddeev in [9], where it was shown that the quantum algebra Uq (sl2 ) with the deformation parameter q = e2πiτ does not define uniquely its representation space. Adjoining to it via the direct product of another quantum algebra Uq˜−1 (sl2 ), where q˜ = e−2πi/τ is a modular transformed parameter, allows one to remove this non-uniqueness. Particular representations of the modular double Uq (sl2 ) ⊗ Uq˜−1 (sl2 ) were considered in [3, 4, 9, 10, 13, 14]. Quantum algebras emerged from the theory of Yang-Baxter equation (YBE) playing an important role in mathematical physics [2, 11, 12] R12 (u − v) R13 (u) R23 (v) = R23 (v) R13(u) R12 (u − v),

(1)

where operators Rjk act in the subspace Vj ⊗ Vk of the product V1 ⊗ V2 ⊗ V3 of three (in general different) spaces Vj . Variables u and v are called spectral parameters. The Sklyanin algebra [18, 19] is a one parameter deformation of Uq (sl2 ), or an elliptic deformation of the sl2 -algebra. It emerges from equation (1) when R12 (u) is given by Baxter’s 4 × 4 R-matrix [2] R12 (u) =

3 X

wa (u) σa ⊗ σa , wa (u) =

a=0

θa+1 (u + η) , θa+1 (η)

and R13 (u), R23 (v) are 2 × 2 matrices fixed as copies of the L-operator:   3 X w0 (u) S0 + w3 (u) S3 w1 (u) S1 − iw2 (u) S2 a L(u) := wa (u) σa ⊗ S = , w1 (u) S1 + iw2 (u) S2 w0 (u) S0 − w3 (u) S3

(2)

(3)

a=0

where σa are the Pauli matrices and η, τ are free parameters. Here θa (u) ≡ θa (u|τ ) are the Jacobi theta-functions 1 X 2 θ(e2πiz ; p) p− 8 2πi(n+ 21 )(z+ 21 ) πi(n+ 21 ) τ ·e = (4) , R(τ ) = e θ1 (z|τ ) = − R(τ ) i(p; p) ∞ n∈Z Q k 2πiτ with p = e , θ(t; p) = (t; p)∞ (pt−1 ; p)∞ , (t; p)∞ = ∞ k=0 (1 − tp ), and θ2 (z) = θ1 (z + 21 ),

θ3 (z) = e

πiτ 4

1

+πiz

θ2 (z + τ2 ),

θ4 (z) = θ3 (z + 12 ).

2

S. E. DERKACHOV AND V. P. SPIRIDONOV

Corresponding YBE takes the form of RLL-relation: R12 (u − v) L1(u) L2 (v) = L2 (v) L1(u) R12 (u − v) yielding the following algebraic relations for operators Sa [18]:
Sα Sβ − Sβ Sα = i S0 Sγ + Sγ S0 ,


S0 Sα − Sα S0 = i Jβγ Sβ Sγ + Sγ Sβ ,

(5)

where the triplet (α, β, γ) is an arbitrary cyclic permutation of (1, 2, 3). The structure constants J −J are Jαβ = βJγ α , γ 6= α, β, where J1 =

θ2 (2η)θ2 (0) , θ22 (η)

J2 =

θ3 (2η)θ3 (0) , θ32 (η)

J3 =

θ4 (2η)θ4 (0) . θ42 (η)

There are two Casimir operators commuting with all generators: [K0 , Sa ] = [K2 , Sa ] = 0, K0 =

3 X a=0

a

a

a

S S ;

K2 =

3 X

Jα Sα Sα .

α=1

The S -operators can be realized as finite-difference operators acting on functions of a complex variable z [19]: i (i)δa,2 θa+1 (η) h 2 2 (6) θa+1 (2z − g + η) eη∂z − θa+1 (−2z − g + η) e−η∂z e−πiz /η , Sa = eπiz /η θ1 (2z)

where e±η∂z f (z) = f (z ± η). The variable g is usually denoted as g = η(2ℓ + 1) and the parameter ℓ ∈ C is called the spin. We shall call g as the spin as well. The parameters τ , η and g characterize representations of the Sklyanin algebra, since they fix the values of the Casimir operators. Note that our operators (6) differ from the standard ones by conjugation 2 by exponentials e±πiz /η (the reason for such a choice is explained in [8]). The elliptic modular double was introduced in [23] as a direct product of two Sklyanin algebras. A particular version of this algebra degenerates in a special limit to Faddeev’s modular ˜a double [9]. The double of interest for us is generated by the operators (6) and operators S a obtained from S simply by permuting 2η and τ : h  δa,2  ˜ a = e2πiz 2 /τ (i) θa+1 (τ /2|2η) θa+1 2z − g + τ 2η e 12 τ ∂z S θ1 (2z|2η) 2  τ  − 1 τ ∂z i −2πiz 2 /τ e , (7) − θa+1 −2z − g + 2η e 2 2 where the g-spin is the same arbitrary parameter as in (6). In this note we focus on the integral operator which was introduced in [22] for defining a universal integral transform of hypergeometric type yielding an integral analogue of the Bailey chain techniques [1]. This operator acts on the functions of one complex variable Φ(z) as Z (q; q)∞ (p; p)∞ 1 Γ(±z ± x − g) [M(g)Φ](z) = Φ(x)dx, (8) 2 Γ(−2g, ±2x) 0 where Im(−g ± z) > 0. Here p = e2πiτ and q = e4πiη , Γ(a, b ± z) := Γ(a)Γ(b + z)Γ(b − z) and ∞ Y 1 − e−2πiz pn+1 q m+1 , Γ(z) := Γ(z|τ, 2η) := 1 − e2πiz pn q m n,m=0

(9)

ELLIPTIC MODULAR DOUBLE

3

is the elliptic gamma function defined for |p|, |q| < 1. This operator satisfies a very simple inversion relation resembling the key Fourier transformation property [24] M(g) M(−g) = 1l , which is true under certain constraints on the parameter g. As shown in [8] the operator (8) satisfies the following intertwining relations: ˜ a (g) = S ˜ a (−g) M(g) . M(g) Sa (g) = Sa (−g) M(g), M(g) S

(10)

(11)

Here we explicitly indicate the g-spin dependence of the Sklyanin algebra generators in order to show that g simply changes the sign under the action of M. In the conventional notation g = η(2ℓ + 1) one has the transformation ℓ → −1 − ℓ. Equalities (11) show that zero modes of the M-operator form an invariant space for the elliptic modular double, i.e. they are mapped ˜a (g) (7). A finite dimensional onto each other by the Sklyanin algebra generators Sa (g) (6) and S space of such modes was discovered in our previous work [8] and here we describe this space in more detail. The intertwining operator M(g) plays a key role in solving YBE [8] along the lines of general construction of [6,7]. The latter approach is based on a twisted representation of the generators of the permutation group. In our case the needed Coxeter relations are satisfied [8] as a consequence of the elliptic beta integral [20] and the Bailey lemma of [22]. In this note we do not apply our results to building solutions of YBE postponing this task to a separate work. 2. Contiguous relations for the intertwining operator Contiguous (or recurrence) relations connect to each other special functions with different values of parameters [1]. The first contiguous relation for elliptic hypergeometric integrals was constructed already in [20]. Such relations can be formulated for integral operators as well and we would like to do it here for the intertwining operator M(g). Recurrence relations of interest for the operator (8) have the following form [5] z2

 z2 eπiη eπi η  ¯ θk (z + g + η) eη∂z − θ¯k (z − g − η) e−η∂z e−πi η M(g) = M(g + η) θ¯k (z) , · R(τ ) θ1 (2z) where θ¯k (z) = θk (z| τ2 ), k = 3, 4, and

(12)

z2

 z2 eπiη eπi η  ¯ θ¯k (z) M(g + η) = M(g) θk (z − g) eη∂z − θ¯k (z + g) e−η∂z e−πi η . (13) R(τ ) θ1 (2z) Here R(τ ) is the constant defined in (4). On the right-hand sides of equalities (12), (13) and other expressions below we use the variable z and assume that it is an “internal” variable, i.e. it plays the role of x in the action of integral operators as in (8). The second formula (13) follows from the first one after application of the inversion relation (10). Indeed, equality (12) can be written in the form Ak (g) M(g) = M(g + η) θ¯k (z) , (14) where Ak (g) is the following difference operator  z2 eπiη πi zη2 1 ¯ θk (z + g + η) eη∂z − θ¯k (z − g − η) e−η∂z e−πi η . e Ak (g) = R(τ ) θ1 (2z) Inverting the M-operators one obtains M−1 (g + η) Ak (g) = θ¯k (z) M−1 (g) −→ M(−g − η) Ak (g) = θ¯k (z) M(−g) −→

(15)

4

S. E. DERKACHOV AND V. P. SPIRIDONOV

−→ M(g − η) Ak (−g) = θ¯k (z) M(g) −→ M(g) Ak (−g − η) = θ¯k (z) M(g + η) and the last relation coincides with equality (13). In this note we consider consequences of the relation (12) alone. In order to prove the operator identity (12) we note that it is equivalent to the following equation for the kernel of the intertwining operator  z 2 Γ(±z ± x − g) z2 1 ¯ θk (z + g + η) eη∂z − θ¯k (z − g − η) e−η∂z e−πi η = eπi η θ1 (2z) Γ(−2g, ±2x) Γ(±z ± x − g − η) ¯ = R(τ ) e−πiη · θk (x) . (16) Γ(−2g − 2η, ±2x) The derivation of the last relation is based on the key formulae Γ(z + 2η) = R(τ ) eπiz θ1 (z) Γ(z).

(17)

2 θ1 (x ± y) = 2 θ1 (x + y) θ1 (x − y) = θ¯4 (x) θ¯3 (y) − θ¯4 (y) θ¯3(x) .

(18)

and We have z2

z2

eπi η eη∂z e−πi η Γ(±z ± x − g) = R2 (τ ) e−3πiη−2πig θ1 (z − η − g ± x) Γ(±z ± x − g − η) , z2

z2

eπi η e−η∂z e−πi η Γ(±z ± x − g) = R2 (τ ) e−3πiη−2πig θ1 (z + η + g ± x) Γ(±z ± x − g − η) , and therefore z2

eπi η

 z2 1 ¯ θk (z + g + η) eη∂z − θ¯k (z − g − η) e−η∂z e−πi η Γ(±z ± x − g) = θ1 (2z) 1 = R2 (τ ) e−3πiη−2πig · Γ(±z ± x − g − η) · θ1 (2z)   · θ¯k (z + g + η) θ1 (z − η − g ± x) − θ¯k (z − g − η) θ1 (z + η + g ± x) = = R2 (τ ) e−3πiη−2πig · Γ(±z ± x − g − η) θ1 (−2g − 2η) θ¯k (x) .

In the last line we have used the equality θ¯k (z + g + η) θ1 (z − η − g ± x) − θ¯k (z − g − η) θ1 (z + η + g ± x) = θ1 (2z) θ1 (−2g − 2η) θ¯k (x) , which is derived using (18) twice. On the last step we apply the equality θ1 (−2g − 2η) R−1 (τ ) eiπ2(g+η) = Γ(−2g) Γ(−2g − 2η) and obtain identity (16). 3. Intertwiner for the two-index discrete lattices g = nη + m τ2 and g = 12 + nη + m τ2 Contiguous relations for the intertwining operator lead to particular factorized forms of M(g) at special integer points of the spin lattice g = nη + m τ2 and g = 12 + nη + m τ2 , n, m ∈ Z≥0 . One can repeat the above considerations with the change 2η ⇄ τ and obtain two types of the recurrence relations (k = 3, 4)

Ak (g) M(g) = M(g + η) θk z| τ2 ; Bk (g) M(g) = M a + τ2 θk (z|η) ,

ELLIPTIC MODULAR DOUBLE

5

where Ak (g) and Bk (g) are the following difference operators  

z2 z2 cA θk z + g + η| τ2 eη∂z − θk z − g − η| τ2 e−η∂z e−πi η , Ak (g) = eπi η θ1 (2z|τ ) h 2
τ ∂z
− τ ∂z i −2πi z2 cB 2πi zτ τ τ τ , 2 θk z + g + 2 |η e e − θk z − g − 2 |η e 2 Bk (g) = e θ1 (2z|2η) and τ eπi 2 eπiη , cB = . cA = R(τ ) R(2η) Using the initial condition M(0) = 1l, which is proved by simple residue calculus [8], it is possible to solve the recurrence relations and obtain
M(k) (nη) = Ak (nη − η) · · · Ak (η)Ak (0) · θk−n z| τ2 ,


M(k) m τ2 = Bk m τ2 − τ2 · · · Bk τ2 Bk (0) · θk−m (z|η) .

Although the form of the intertwiner should not depend on the value of k, M(k) (g) ≡ M(g), we introduced an additional upper index k in order to indicate a potential dependence
on it. Derived expressions are particular cases of the general operator M(k) nη + m τ2 . However, they are used as building blocks for constructing this general operator. First we transform these factorized operators to finite sums with explicit coefficients. Let us consider for an illustration two simple examples 
z2 z2 cA  θk (z + η| τ2 ) eη∂z − θk (z − η| τ2 ) e−η∂z θk−1 z| τ2 = e−πi η M(k) (η) eπi η = θ1 (2z) = z2

z2

e−πi η M(k) (2η) eπi η =

 cA  η∂z e − e−η∂z , θ1 (2z)

 c2A  θk (z + 2η| τ2 ) eη∂z − θk (z − 2η| τ2 ) e−η∂z · θ1 (2z)


1  η∂z e − e−η∂z θk−1 z| τ2 . θ1 (2z) Expanding the last expression we obtain a sum of four finite difference operators which can be transformed by means of (18) to the form ·

c2A · θ1 (2z − 2η)θ1 (2z)θ1 (2z + 2η)  θ1 (4η) −2η∂z − θ1 (2z) + θ1 (2z + 2η) e . θ1 (2η) z2

z2

e−πi η M(k) (2η) eπi η =  · θ1 (2z − 2η) e2η∂z

In these examples one can drop the index k in the notation M(k) (nη) because the result does not depend on k as expected. Take the general ansatz for the intertwining operator 2

−πi zη

e

2

πi zη

M(nη) e

=

n X ℓ=0

(n)

(−1)ℓ αℓ (z) e(n−2ℓ)η∂z

6

S. E. DERKACHOV AND V. P. SPIRIDONOV

and substitute it into the equality M((n + 1)η) = Ak (nη)M(nη)θk (z| τ2 )−1 . For ℓ = 1, . . . , n this yields the recurrences of the form (n)

(n+1)

αℓ

(z) =

(n)

θk (z + (n + 1)η| τ2 )αℓ (z + η) + θk (z − (n + 1)η| τ2 )αℓ−1 (z − η) cA · θ1 (2z|τ ) θk (z + (n + 1 − 2ℓ)η| τ2 )

and for ℓ = 0 and ℓ = n + 1 one has cA (n+1) (n) α0 (z) = α0 (z + η), θ1 (2z|τ )

(n+1)

αn+1 (z) =

(19)

cA αn(n) (z − η). θ1 (2z|τ )

(0)

Since α0 (z) = 1, we obtain (n)

cnA , θ (2z + 2ηk|τ ) 1 k=0

α0 (z) = Qn−1

cnA . θ (2z − 2ηk|τ ) 1 k=0

αn(n) (z) = Qn−1

These boundary conditions fix uniquely the solution of recurrence relation (19) which can be found by induction to have the following general form   θ1 (2z + 2η(n − 2ℓ) |τ ) n (n) n , (20) αℓ (z) = cA · · Qn ℓ τ,2η j=0 θ1 (2z − 2η(ℓ − j) |τ ) Qn   n j=1 θ1 (2ηj |τ ) = Qℓ . Qn−ℓ ℓ τ,2η j=1 θ1 (2ηj |τ ) · j=1 θ1 (2ηj |τ )   n (n) In the expression for αℓ (z) we extracted the elliptic binomial coefficient which does ℓ τ,2η not depend on z and explicitly wrote the rest z-depended part. The discrete intertwining operator M(nη) was obtained first in [26] in this form. Note that indeed the general result has (n) no k-dependence, which is present in the recurrence relation for αℓ (z).
A similar expression for the operator M m τ2 is obtained by a simple permutation cA ⇄ cB , n ⇄ m and τ ⇄ 2η m X 2
2πi z2 τ (m) −2πi zτ τ τ e M m2 e = (−1)ℓ βℓ (z) e(m−2ℓ) 2 ∂z , ℓ=0

where

  θ1 (2z + τ (m − 2ℓ) |2η) m , = · · Qm ℓ 2η,τ j=0 θ1 (2z − τ (ℓ − j) |2η) Qm   m j=1 θ1 (τ j |2η) = Qℓ . Qm−ℓ ℓ 2η,τ j=1 θ1 (τ j |2η) · j=1 θ1 (τ j |2η)

(m) βℓ (z)

cm B

(21)

Now we are going to describe the general case. It is easy to derive the following representation
M(k) nη + m τ2 = Ak (nη − η + m τ2 ) · · · Ak (η + m τ2 )Ak (m τ2 ) ·


(22) · Bk m τ2 − τ2 · · · Bk τ2 Bk (0) · θk−m (z|η) θk−n z| τ2 .
Of course there are many equivalent ways to represent M(k) nη + m τ2 as a product of Ak - and Bk -operators and we have described only one of them. Analogously one can consider the lattice g = nη + m τ2 + 12 . For this one uses the fact that 1 M( 12 ) = P , where P = e 2 ∂z is the shift operator of z by 1/2 [8]. Since we work in the space of functions which are meromorphic in w = e2πiz , this is the half period shift working as the

ELLIPTIC MODULAR DOUBLE

7

parity operator, P w = −w. Repeating the previous procedure once more in this setting, we obtain
M(k) nη + m τ2 + 12 = Ak (nη − η + m τ2 + 21 ) · · · Ak (η + m τ2 + 12 )Ak (m τ2 + 21 ) ·



· Bk m τ2 − τ2 + 21 · · · Bk τ2 + 21 Bk ( 12 ) · θk−m z + 12 |η θk−n z + 21 | τ2 P . (23)

Because θ3 (z ± 21 ) = θ4 (z), we see that A3,4 (g + 12 ) = A4,3 (g) and B3,4 (g + 21 ) = B4,3 (g). Therefore the intertwining operator for the second lattice is obtained from the first one simply by exchanging θ3 and θ4 and multiplication by P from the right

(24) M(3,4) nη + m τ2 + 21 = M(4,3) nη + m τ2 P. 4. A finite dimensional invariant space of the elliptic modular double

The irreducible representation of the Sklyanin algebra at (half)-integer spin ℓ = n−1 , n ∈ Z>0 , 2 + is n-dimensional and it can be realized in the space Θ2n−2 consisting of even theta functions of order 2n − 2 [19]. Denote 2 2 W(g) = e−πiz /η M(g) eπiz /η . By the factorized representation of W(ηn) we see that its action annihilates this irreducible representation space Θ+ 2n−2 . Let us demonstrate this fact by means of the recurrence relations. The function θk2 (z|τ ) for any k = 1, . . . , 4 is an even theta-function of the second order. Let us fix any two different indices k = k1 and k = k2 and denote e1 := θk21 (z|τ ) and e2 = θk22 (z|τ ). All monomials of these building blocks ek1 e2N −k for k = 0, . . . , N are even theta-functions of the order 2N which are linearly independent due to the fact that location of their zeroes is different. The number of such monomials yields the dimension of the space of even theta-functions of the order 2N and therefore these monomials form a basis of Θ+ 2N . Now using formulae 2 θ1 (x + y) θ1 (x − y) = θ¯4 (x) θ¯3 (y) − θ¯4 (y) θ¯3(x), 2 θ2 (x + y) θ2 (x − y) = θ¯3 (x) θ¯3 (y) − θ¯4 (y) θ¯4(x), 2 θ3 (x + y) θ3 (x − y) = θ¯3 (x) θ¯3 (y) + θ¯4 (y) θ¯4 (x), 2 θ4 (x + y) θ4 (x − y) = θ¯4 (x) θ¯3 (y) + θ¯4 (y) θ¯3 (x), it is possible to express any square θk2 (z|τ ) as a linear combination of θ¯3 (z) and θ¯4 (z). Therefore the set of n functions 
j 
n−1−j θ3 z| τ2 θ4 z| τ2 ; j = 0, 1, · · · , n − 1,

forms a basis in the space Θ+ 2n−2 . Then applying n − 1 times the recurrence relation and taking into account that  cA  η∂z W(η) · 1 = e − e−η∂z · 1 = 0 θ1 (2z)

we obtain


j 
n−1−j  θ4 z| τ2 = 0. W(ηn) · θ3 z| τ2 In the same way the set of m functions [θ3 (z|η)]ℓ [θ4 (z|η)]m−1−ℓ ; ℓ = 0, 1, · · · , m − 1,
is annihilated by W τ2 m
W τ2 m · [θ3 (z|η)]ℓ [θ4 (z|η)]m−1−ℓ = 0 .

8

S. E. DERKACHOV AND V. P. SPIRIDONOV

The generalization is evident. The set of n m functions 
j 
n−1−j θ3 z| τ2 θ4 z| τ2 · [θ3 (z|η)]ℓ [θ4 (z|η)]m−1−ℓ
is annihilated by W η n + τ2 m

j 
n−1−j θ4 z| τ2 · [θ3 (z|η)]ℓ [θ4 (z|η)]m−1−ℓ = 0 . W η n + τ2 m θ3 z| τ2

Similar properties hold for the operator W (nη + m τ2 + 12 ), since the shift of z by 1/2 just permutes θ3 (z) and θ4 (z), i.e. we have an involution of the basis. Described formulae show that we have a finite dimensional kernel of the intertwiner. Because the space of zero modes of the intertwining operator forms an invariant subspace under the action of Sklyanin algebra generators one obtains finite dimensional representations of this algebra. This space was partially characterized in [8]. The observation that the Sklyanin algebra has finite dimensional representations not only for half-integer values of the spin but for the whole lattices η n + τ2 m and nη + m τ2 + 21 with n, m ∈ Z≥0 , (n, m) 6= (0, 0), is a more or less evident consequence of the modular doubling of Sklyanin algebras introduced in [23], since there exists an involution permuting two Sklyanin subalgebras. This fact was noticed also in [15]. 5. A non-factorized form of the intertwiner

Now we are going to transform the general expression (22) to a “normal ordered” form. We use the formulae for the transformation of theta-functions θk (z + m τ2 | τ2 ) = (−1)mδk,4 e−

iπτ 2

m2 −2πi mz

θk (z | τ2 ) ,

k = 3, 4, m ∈ Z.

(25)

Using this formula it is easy to check that Ak (g + m τ2 ) = (−1)mδk,4 e−

iπτ 2 m −2πim(g+η)+iπmη 2

·e

iπm 2 z η

Ak (g) e−

iπm 2 z η

,

and therefore Ak (nη − η + m τ2 ) · · · Ak (η + m τ2 )Ak (m τ2 ) = iπτ

2

2

iπm 2

iπm 2

= (−1)mnδk,4 e− 2 m n−iπη n m · e η z Ak (nη − η) · · · Ak (η)Ak (0) e− η z .

Now it is possible to single out in M nη + m τ2 the operators M (nη) and M m τ2 , iπτ 2
2 M nη + m τ2 = (−1)mnδk,4 e− 2 m n−iπη n m ·


iπm 2 iπm 2 · e η z M(nη) e− η z · θkn z| τ2 M m τ2 θk−n z| τ2 .

There are evident sources of the k-dependence and, as an additional check, we have to see how this dependence disappears. Next we consider the effect of the similarity transformations for both operators. The needed formulae are e

iπ(m+1) 2 z η

e(n−2k)η∂z e−

iπ(m+1) 2 z η

2

= e−2πiz (m+1)(n−2k) e−πiη (m+1)(n−2k) e(n−2k)η∂z

and e

2πi 2 z τ



2πi 2 τ θkn z| τ2 e(m−2ℓ) 2 ∂z θk−n z| τ2 e− τ z = =e

2πi 2 [z −(z+(m−2ℓ) τ2 )2 ] τ

θkn


θkn z | τ2 τ
e(m−2ℓ) 2 ∂z = τ τ z + (m − 2ℓ) 2 | 2

= (−1)mnδk,4 e2πiz (n−1)(m−2ℓ) e

iπτ 2

(n−1)(m−2ℓ)2

τ

e(m−2ℓ) 2 ∂z .

ELLIPTIC MODULAR DOUBLE

9

We see that k-dependence disappears, as it should. Now we can write iπτ 2

2 ′ M nη + m τ2 = e− 2 m n−iπη n m · M(nη) · M m τ2 .

where transformed operators have the form n X 2 (n) M(nη) = (−1)j αj (z) e−2πiz (m+1)(n−2j) e−πiη (m+1)(n−2j) e(n−2j)η∂z , j=0

M

′

m τ2




=

m X

(m)

(−1)ℓ βℓ (z) e2πiz (n−1)(m−2ℓ) e

iπτ 2

(n−1)(m−2ℓ)2

τ

e(m−2ℓ) 2 ∂z .

ℓ=0

Let us move all shift operators to the right M nη +

m τ2




−

=e

iπτ 2 m n−iπη n2 m 2

n X (n) · (−1)k αk (z) e−2πiz (m+1)(n−2k) · k=0

2

· e−πiη (m+1)(n−2k) ·

m X

(m)

(−1)ℓ βℓ

(z + η(n − 2k)) ·

ℓ=0

· e2πi(z+η(n−2k)) (n−1)(m−2ℓ) · e

iπτ 2

(n−1)(m−2ℓ)2

τ

e(n−2k)η∂z e(m−2ℓ) 2 ∂z .

Using formulae for the transformation of theta-function 2 −4πi mz

θ1 (2z + 2ηm |2η) = (−1)m e−2πiη m

θ1 (z |2η) ,

(26)

and explicit expression for the β (m) -coefficients we obtain (m)

βℓ

2

(m)

(z + η(n − 2k)) = (−1)nm e4πiz m(n−2k) e2πiη m(n−2k) eiπτ (m−1)(m−2ℓ)(n−2k) βℓ

(z) .

Collecting everything together we obtain our operator in a normal ordered form: n X iπτ 2
2 (n) (−1)k αk (z) · M nη + m τ2 = (−1)nm e− 2 m n−iπη n m · k=0

·

m X

(m)

(−1)ℓ βℓ

(z) e(n−1)(m−2ℓ)[

πiτ 2

(m−2ℓ)+2πi(z+(n−2k)η) ]

·

ℓ=0

τ τ · e(m−1)(n−2k)[πiη (n−2k)+2πi(z+(m−2ℓ) 2 ) ] · e[(n−2k)η+(m−2ℓ) 2 ]∂z ,

(27)

where we have arranged the phase factors in the form resembling transformation laws of theta functions of definite level with respect to the shift by a period. For the lattice g = nη + m τ2 + 12 one has

M nη + m τ2 + 21 = M nη + m τ2 P, since in relation (24) the operators M(k) do not depend on k.

6. A complete factorization of the intertwiner on theta functions We are going to apply derived intertwined operators to the functions with special transformation properties under the shifts on periods τ and 2η. Consider the product FN (z) GM (z) where functions FN (z) and GM (z) are transformed in the following way (a, b ∈ Z) FN (z + τ a) = e−2N a [πiτ a+2πiz ] FN (z) , GM (z + 2ηb) = e−4M b [πiη b+πiz ] GM (z) .

10

S. E. DERKACHOV AND V. P. SPIRIDONOV

We write all phase factors in the form close to phases in (27). For holomorphic functions, FN (z) are theta functions of modulus τ and order 2N, whereas GM (z) are theta functions of modulus 2η and order 2M. Let us introduce parameters α = 0, ±1 and β = 0, ±1 such that n − α and m − β are always even integers. This means that α = 0 when n is even and α = ±1 when n is odd. Analogously, β = 0 when m is even and β = ±1 when m is odd. Now we can single out the full period shifts and find that

FN z + β τ2 + (n − 2k)η + (m − β − 2ℓ) τ2 GM z + αη + (n − α − 2k)η + (m − 2ℓ) τ2 = τ τ πiτ = e−N (m−β−2ℓ)[ 2 (m−2ℓ)+2πi(z+β 2 +(n−2k)η) ] · e−M (n−α−2k)[πiη (n−α−2k)+2πi(z+αη+(m−2ℓ) 2 ) ] ·

· FN z + β τ2 + (n − 2k)η GM z + αη + (m − 2ℓ) τ2 . Choosing N = n − 1 and M = m − 1 we see an almost complete cancellation of the phase factors: iπτ 2
2 M nη + m τ2 Fn−1 (z) Gm−1 (z) = (−1)nm e− 2 m n−iπη n m · " n X πiτ 2 (n) (−1)k αk (z) · e−4πiη β (n−1)k · e 2 β (n−1) e2πiη β (n−1)n · e2πiβ (n−1) z ·

(28)

k=0

τ 2

· Fn−1 z + β πiηα2

·e

(m−1)

πiτ α (m−1)m

e




+ (n − 2k)η

#

·

2πiα (m−1) z

·e

· Gm−1 z + αη + (m − 2ℓ) τ2




#

·

"

m X

(m)

(−1)ℓ βℓ

(z) · e−2πiτ α (m−1)ℓ

ℓ=0

.

The case α = β = 0 corresponds to even n and m, in which case all phase factors are absent. Thus we see a complete factorization of the intertwiner into two operators acting in different spaces. Note that we have described the action of M for arbitrary theta functions and, as + shown before, zero modes of M(g) single out the spaces Θ+ 2n−2 (z|τ ) for Fn−1 and Θ2m−2 (z|2η) for Gm−1 . Let us pass to the multiplicative notation which is more compact and convenient
for analytical reasons [8]. Remind first that p = e2πiτ , q = e4πiη and θ1 (z|τ ) = e−πiz θ e2πiz ; p /R(τ ). Inserting the latter relation into the definition of the elliptic binomial coefficients we see that all R(τ )coefficients cancel. Simplifying the resulting expression with the help of relation θ(z; p) = −z θ(z −1 ; p), we find
  k Y θ q b−n−1 ; p n k 21 k(n+1) = (−1) q . k τ,2η θ (q b ; p) b=1 (n)

In the rest part of the coefficients αk (z) the R(τ )-factors cancel too and we find:
θ e4πiz q n−2k ; p n 1 θ1 (2z + 2η(n − 2k) |τ ) (n−2k)(n−1) 2πi nz n Qn = q 4 q4 . e cA · Qn 4πiz q j−k ; p) θ (2z − 2η(k − j) |τ ) 1 j=0 j=0 θ (e

ELLIPTIC MODULAR DOUBLE

11

Transforming the theta-functions to an appropriate form and using the previous expression for the elliptic binomial coefficients, we obtain
n2 k
Y θ e−4πiz q b−n−1 , q b−n−1 ; p (−1)k+1 q 4 +n(k+1) e2πi (n+2)z (n) −4πiz 2k−n Qn αk (z) = . ·θ e q ;p 4πiz q j ; p) θ (e−4πiz q b , q b ; p) j=0 θ (e b=1

Finally we give explicitly the form of the intertwiner in multiplicative notation
M nη + m τ2 Fn−1 (z) Gm−1 (z) = (−1)nm e2πi z [n+m+4+α (m−1)+β (n−1)] · ·

q

(n2 −α2 )(1−m) n(n−1) +β +n 4 2

Qn

j=0

·

"

n X

q

θ (e4πiz

k=0

·

m X

; p)

·

p

Qm

4πiz j=0 θ (e

pj ; q)

−4πiz

(29)

·

k
Y θ e−4πiz q b−n−1 , q b−n−1 ; p θ e q ;p θ (e−4πiz q b , q b ; p) b=1 #
z + β τ2 + (n − 2k)η ·

kn (1−β)+βk

· Fn−1 "

qj

m(m−1) (m2 −β 2 )(1−n) +α +m 4 2

2k−n




ℓ
Y θ e−4πiz pb−m−1 , pb−m−1 ; q ℓm (1−α)+αℓ −4πiz 2ℓ−m p θ e p ;q θ (e−4πiz pb , pb ; q) ℓ=0 b=1 #
· Gm−1 z + αη + (m − 2ℓ) τ2 .




Choosing α = β = 1 in this formula we come to the intertwiner derived in [8] with the help of the residue calculus (after the change of notation w = e−2πiz , n = 2ℓq + 1 and m = 2ℓp + 1): (n2 −1)(1−m)

(n+1)n

(m2 −1)(1−n)

m(m+1)

+ 2 + 4 4 2 p (−1)nm e4πi z [n+m+1] q Q Qn M nη + · Fn−1 (z) Gm−1 (z) = m 4πiz pj ; q) 4πiz q j ; p) j=0 θ (e j=0 θ (e # " n
k −4πiz b−n−1 b−n−1 X Y

θ e q , q ; p Fn−1 z + τ2 + (n − 2k)η ·(30) · q k θ e−4πiz q 2k−n ; p −4πiz b b θ (e q , q ; p) k=0 b=1 " m #
ℓ −4πiz b−m−1 b−m−1 X Y

θ e p , p ; q · pℓ θ e−4πiz p2ℓ−m ; q Gm−1 z + η + (m − 2ℓ) τ2 . −4πiz pb , pb ; q) θ (e ℓ=0 b=1

As established above, M nη + m τ2 + 12 = M nη + m τ2 P. Therefore for this case the action of the intertwiner has the same form as given above with the shift of z in the arguments of Fn−1 and Gm−1 functions by 1/2.

m τ2




7. Conclusion We have described a finite dimensional space of zero modes of the integral operator M(g) emerging for two discrete spin lattices g. In particular, we have derived the explicit expression for M(g) acting in this space as a finite-difference operator. Since this operator is an intertwiner for the elliptic modular double, its zero modes define finite-dimensional representation of this algebra. Next, it is necessary to find some closed form for the action of generators of the Sklyanin algebras in some basis of this space. Here we stress that the choice of the basis is a

12

S. E. DERKACHOV AND V. P. SPIRIDONOV

free option and in some special cases it may drastically simplify the situation. For instance, the products of theta functions (N ) hk (w; p, q)

:=

k−1 Y j=0

j

±1

θ(q aw ; p)

NY −k−1

θ(q j bw ±1 ; p),

k, N ∈ Z≥0 ,

(31)

j=0

were used as basis vectors of Θ+ 2N -space in [16,17] for a simplified analysis of elliptic 6j-symbols (these functions emerged first as some intertwining vectors in [25]). For the elliptic modular (N,M ) (N ) (M ) double one should consider the two-index basis vectors hkj (w) := hk (w; p, q)hj (w; q, p) and generalize considerations of [16, 17] to such a case. We expect that after an appropriate choice of the measure one should be able to come to the two-index biorthogonal functions of [21]. The identity M(g)R12 = R′12 M(g), where R12 is a solution of the YBE derived in [8] and R′12 is another similar operator, shows that the kernel space of M(g)-operator is mapped onto itself by the R-matrix R12 . Therefore zero modes of M(g) form an invariant space for the action of operator R12 . The explicit form of corresponding finite dimensional R-matrices will be considered in a subsequent publication. The work of V. S. is supported by RFBR grant no. 11-01-00980 and NRU HSE scientific fund grant no. 13-09-0133. The work of S. D. is supported by RFBR grants 12-02-91052, 13-01-12405 and 14-01-00341. References [1] G. E. Andrews, R. Askey, and R. Roy, Special Functions, Encyclopedia of Math. Appl. 71, Cambridge Univ. Press, Cambridge, 1999. [2] R. J. Baxter, Exactly Solved Models in Statistical Mechanics, Academic Press, London, 1982. [3] F. J. van de Bult, Ruijsenaars’ hypergeometric function and the modular double of Uq (sl(2, C)), Adv. Math. 204 (2006), 539–571. [4] A. G. Bytsko and J. Teschner, R operator, coproduct and Haar measure for the modular double of Uq (sl(2, R)), Commun. Math. Phys. 240 (2003), 171–196. [5] D. Chicherin, S. Derkachov, D. Karakhanyan, and R. Kirschner, Baxter operators with deformed symmetry, Nucl. Phys. B868 (2013), 652–683. [6] S. E. Derkachov, Factorization of the R-matrix. I. Zap. Nauchn. Sem. POMI 335 (2006), 134–163 (J. Math. Sciences 143 (1) (2007), 2773-2790), math.QA/0503396. [7] S. Derkachov, D. Karakhanyan, and R. Kirschner, Yang-Baxter R-operators and parameter permutations, Nucl. Phys. B785 (2007), 263–285. [8] S. E. Derkachov and V. P. Spiridonov, Yang-Baxter equation, parameter permutations, and the elliptic beta integral, Russian Math. Surveys, in press, arXiv:1205.3520 [math-ph]. [9] L. D. Faddeev, Modular double of a quantum group, Conf. Mosh´e Flato 1999, vol. I, Math. Phys. Stud. 21, Kluwer, Dordrecht, 2000, pp. 149–156. [10] L. D. Faddeev, Discrete series of representations for the modular double of Uq (sl(2, R)), Funct. An. and its Appl. 42 (2008), no. 4, 98–104. [11] L. D. Faddeev and L. A. Takhtadzhan, The quantum method of inverse problem and the Heisenberg XYZ model, Uspekhi Mat. Nauk 34 (5), 13–63 (Russian Math. Surveys 34 (5) (1979), 11–68). [12] M. Jimbo (ed), Yang-Baxter equation in integrable systems, Adv. Ser. Math. Phys., 10, World Scientific (Singapore), 1990. [13] S. Kharchev, D. Lebedev, and M. Semenov-Tian-Shansky, Unitary representations of Uq (sl(2, R)), the modular double, and the multiparticle q-deformed Toda chains, Commun. Math. Phys. 225 (2002), 573– 609. [14] B. Ponsot and J. Teschner, Clebsch–Gordan and Racah–Wigner coefficients for a continuous series of representations of Uq (sl(2, R)), Commun. Math. Phys. 224 (2001), 613–655.

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13

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