Local action of the symmetric group

and the twisted Yang-Baxter relation

Alexander Odesskii

Introduction

The Yang-Baxter relation plays a central role in the Quantum Field Theory. The usual R-matrix describes the scattering of two ”particles” (Fig.1). Here u and v are some characteristics of the particles which are called the spectral parameters. R-matrix (R_{ij}^{kl}(u, v)) is a meromorphic function on the space of spectral parameters. Now suppose that the spectral parameters change after scattering (Fig.2). New spectral parameters φ(u, v), ψ(u, v) are some meromorphic functions in the variables u and v. The factorization conditions impose a strong requirement on these functions (Fig.3). One can see from Fig.3 that

φ(u, φ(v, w)) = φ(φ(u, v), φ(ψ(u, v), w))

φ(ψ(u, φ(v, w)), ψ(v, w)) = ψ(φ(u, v), φ(ψ(u, v), w))

ψ(ψ(u, v), w) = ψ(ψ(u, φ(v, w)), ψ(v, w))

If we have the solution of these equations, then we can define the analogous Yang-Baxter relation, L-operators, Zamolodchikov algebra and so on. In §1 we develop the mathematical formalism for φ and ψ and give nontrivial examples. In §2 we introduce the twisted Yang-Baxter relation and corresponding algebraic structures. The most interesting examples of φ and ψ appear in the theory of factorization of matrix polynomials (§3) and and matrix θ-functions (§4). In §5 we discus the generalized star-triangle relation and corresponding algebraic structures. In ”physical” language it is the case when our particles are not elementary but are multiplets of m elementary particles.

\xymatrix{v & u & φ(u, v) & ψ(u, v) \\ u \ar[ur] & v \ar[ul] & u \ar[ur] & v \ar[ul]}

Figure 1 Figure 2

The examples of twisted R-matrices as well as the solutions of generalized startriangle relation were found in [4] as intertwiners of cyclic representations and its tensor products of the algebra of monodromy matrices of the six-vertex model at root of unity [3]. These solutions are natural generalizations of the one from chiral Potts model [1,2,3]. One can obtain other solutions of the twisted Yang-Baxter and star-triangle relations by calculating the intertwiners of the representations of the algebras of monodromy matrices at root of unity for other trigonometric and elliptic R-matrices.

\xymatrix{ & φ(φ(u, v), φ(ψ(u, v), w)) & ψ(φ(u, v), φ(ψ(u, v), w)) & ψ(ψ(u, v), w) \\ u \ar[urrr]_{ψ(u,v)} & v \ar[ur]^{φ(u,v)} &  & w \ar[ull]_{φ(ψ(u,v),w)}}

∥

\xymatrix{φ(u, φ(v, w)) & φ(ψ(u, φ(v, w)), ψ(v, w)) & ψ(ψ(u, φ(v, w)), ψ(v, w)) \\ u \ar[urr]^{ψ(u,φ(v,w))} &  & v \ar[ul]_{ψ(v,w)} & w \ar[ulll]^{φ(v,w)}}

Figure 3

§1. Twisted transposition

Let U be a complex manifold, μ : U × U → U × U be a birational automorphism of U × U. We will use the notations: μ(u, v) = (φ(u, v), ψ(u, v)) where u, v ∈ U. Here φ and ψ are meromorphic functions from U × U to U.

Let us introduce the following birational automorphisms of U ×U ×U: σ_{1} = μ×id and σ_{2} = id × μ. We have: σ_{1}(u, v, w) = (φ(u, v), ψ(u, v), w) and σ_{2}(u, v, w) = (u, φ(v, w), ψ(v, w)).

Definition We call a map μ a twisted transposition if the automorphisms σ_{1} and σ_{2} satisfy the following relations:

σ_{1}^{2} = σ_{2}^{2} = id, σ_{1}σ_{2}σ_{1} = σ_{2}σ_{1}σ_{2}  (1)

If μ is a twisted transposition, then for each N ∈ ℕ we have the birational action of the symmetric group S_{N} on the manifold U^{N} such that the transposition (i, i + 1) acts by automorphism σ_{i} = id^{i−1} × μ × id^{N−i−1}. So we have σ_{i}(u_{1}, …, u_{N}) = (u_{1}, …, φ(u_{i}, u_{i+1}), ψ(u_{i}, u_{i+1}), …, u_{N}). It is clear that the relations (1) are equivalent to the defining relations in the group S_{N}: σ_{i}^{2} = e, σ_{i}σ_{i+1}σ_{i} = σ_{i+1}σ_{i}σ_{i+1}, σ_{i}σ_{j} = σ_{j}σ_{i} for |i − j| > 1.

It is easy to check that the relations (1) are equal to the following functional equations for φ and ψ:

φ(φ(u, v), ψ(u, v)) = u, ψ(φ(u, v), ψ(u, v)) = v

φ(u, φ(v, w)) = φ(φ(u, v), φ(ψ(u, v), w))

φ(ψ(u, φ(v, w)), ψ(v, w)) = ψ(φ(u, v), φ(ψ(u, v), w))  (2)

ψ(ψ(u, v), w) = ψ(ψ(u, φ(v, w)), ψ(v, w))

Remarks 1. From (2) it follows that for each N the functions φ(u_{1}, φ(u_{2}, …, φ(u_{N}, w) …) and ψ(… (ψ(w, u_{1}), u_{2}) …, u_{N}) are invariant with respect to the action of the group S_{N} on the variables u_{1}, …, u_{N}.

2. Informally one can consider μ as an infinite dimensional R-matrix in the space of functions. Namely, if we consider the space of meromorphic functions {f : U × U → ℂ} as an ”extended tensor square” of the space of meromorphic functions {f : U → ℂ}, then the linear operator R_{μ} : f → f ∘μ (that is R_{μ}f(u, v) = f(μ(u, v))) satisfies the usual Yang-Baxter relation.

Examples 1. Let q, q^{−1} : U → U be birational automorphisms such that qq^{−1} = q^{−1}q = id. Then

μ(u, v) = (q(v), q^{−1}(u))

is a twisted transposition.

2. Let U = ℂ, then the following formula gives a twisted transposition:

μ(u, v) = (1 − u + uv, \frac{uv}{1 − u + uv})

3. Let U be a finite dimensional associative algebra with unity 1 ∈ U and such that generic element u ∈ U is invertible, for example U = Mat_{m}. Then the following formula gives a twisted transposition:

μ(u, v) = (1 − u + uv, (1 − u + uv)^{−1}uv)

For further examples see §3,4.

§2. Twisted Yang-Baxter relation and corresponding algebraic structures

Let V be a n-dimensional vector space. For each u ∈ U we denote by V (u) a vector space canonically isomorphic to V. Let R be a meromorphic function from U × U to End(V ⊗ V). We will consider R(u, v) as a linear operator

R(u, v) : V (u) ⊗ V (v) → V (φ(u, v)) ⊗ V (ψ(u, v))

Definition We call R a twisted R-matrix (with respect to the twisted transposition μ) if it satisfies the following properties:

1. The composition

V (u) ⊗ V (v) → V (φ(u, v)) ⊗ V (ψ(u, v)) → V (u) ⊗ V (v)

is equal to the identity, that is R(φ(u, v), ψ(u, v))R(u, v) = 1.

2. The following diagram is commutative:

\xymatrix{V (φ(u, v)) ⊗ V (ψ(u, v)) ⊗ V (w) \ar[r]^{1⊗R(ψ(u,v),w)} & V (φ(u, v)) ⊗ V (φ(ψ(u, v)), w)) ⊗ V (ψ(ψ(u, v), w)) \ar[d]^{R(φ(u,v),φ(ψ(u,v),w))⊗1} \\ V (u) ⊗ V (v) ⊗ V (w) \ar[u]_{R(u,v)⊗1} \ar[d]^{1⊗R(v,w)} & Ṽ \\ V (u) ⊗ V (φ(v, w)) ⊗ V (ψ(v, w)) \ar[r]^{R(u,φ(v,w))⊗1} & V (φ(u, φ(v, w))) ⊗ V (ψ(u, φ(v, w))) ⊗ V (ψ(v, w)) \ar[u]_{1⊗R(ψ(u,φ(v,w)),ψ(v,w))}}

Here Ṽ = V (φ(u, φ(v, w))) ⊗ V (ψ(φ(u, v), φ(ψ(u, v), w))) ⊗ V (ψ(ψ(u, v), w)). In other words,

R^{12}(φ(u, v), φ(ψ(u, v), w)))R^{23}(ψ(u, v), w)R^{12}(u, v) =
R^{23}(ψ(u, φ(v, w)), ψ(v, w))R^{12}(u, φ(v, w))R^{23}(v, w)  (3)

Here R^{12} = R ⊗ 1 and R^{23} = 1 ⊗ R are linear operators in V ⊗ V ⊗ V. We call (3) twisted Yang-Baxter relation. Let {x_{i}, i = 1 …, n} be a basis of the linear space V, {x_{i}(u)} be the correspond-

ing basis of the linear space V (u). It is clear that the following two linear operators are twisted R-matrices for each μ:

x_{i}(u) ⊗ x_{j}(v) → x_{i}(φ(u, v)) ⊗ x_{j}(ψ(u, v))
x_{i}(u) ⊗ x_{j}(v) → x_{j}(φ(u, v)) ⊗ x_{i}(ψ(u, v))

Let R_{ij}^{αβ}(u, v) be a matrix element of twisted R-matrix R, which means

R(u, v) : x_{i}(u) ⊗ x_{j}(v) → R_{ij}^{αβ}(u, v)x_{β}(φ(u, v)) ⊗ x_{α}(ψ(u, v))

We sum over repeating indices.

Definition We call the twisted Zamolodchikov algebra associated with R an associative algebra Z_{R} with generators {x_{i}(u); i = 1 …, n, u ∈ U} and defining relations

x_{i}(u)x_{j}(v) = R_{ij}^{αβ}(u, v)x_{β}(φ(u, v))x_{α}(ψ(u, v))

It is clear that the twisted Zamolodchikov algebra satisfies the usual property: for generic u_{1}, …, u_{N} ∈ U the elements {x_{i_{1}}(u_{1}) … x_{i_{N}} (u_{N}); 1 ⩽ i_{1}, …, i_{N} ⩽ n} are linear independent.

Let W be a vector space, L be a meromorphic function L : U → End(V ⊗ W). We will consider L(u) as a linear operator L(u) : V (u) ⊗ W → W ⊗ V (u).

Definition We call L an L-operator for twisted R-matrix R if the following diagram is commutative:

\xymatrix{V (u) ⊗ W ⊗ V (v) \ar[r]^{L(u)⊗1} & W ⊗ V (u) ⊗ V (v) \ar[d]^{1⊗R(u,v)} \\ V (u) ⊗ V (v) ⊗ W \ar[u]_{1⊗L(v)} \ar[d]^{R(u,v)⊗1} & W ⊗ V (φ(u, v)) ⊗ V (ψ(u, v)) \\ V (φ(u, v)) ⊗ V (ψ(u, v)) ⊗ W \ar[r]^{1⊗L(ψ(u,v))} & V (φ(u, v)) ⊗ W ⊗ V (ψ(u, v)) \ar[u]_{L(φ(u,v))⊗1}}

Which is

R(u, v)L^{1}(u)L^{2}(v) = L^{1}(φ(u, v))L^{2}(ψ(u, v))R(u, v)  (4)

where L^{1} = L ⊗ 1 and L^{2} = 1 ⊗ L.

Let {L_{i}^{j}(u); i, j = 1, …, n} be the elements in End(W) such that L(u) : x_{i}(u) ⊗ W → (L_{i}^{α}(u)W) ⊗ x_{α}(u). We can consider L(u) as a n × n matrix with matrix elements (L_{i}^{j}(u)).

Definition The algebra of monodromy matrices M_{R} for the twisted R-matrix R is an associative algebra with generators {L_{i}^{j}(u); i, j = 1, …, n, u ∈ U} and defining relations (4).

It is clear that representations of the algebra M_{R} in a vector space W coincide with L-operators L(u) ∈ End(V ⊗ W).

One can introduce a coproduct Δ in the algebra M_{R} by the usual way:

ΔL_{i}^{j}(u) = L_{i}^{α}(u) ⊗ L_{α}^{j}(u)

In the language of L-operators, if we have two L-operators L_{1}(u) : V (u) ⊗ W_{1} → W_{1} ⊗ V (u) and L_{2}(u) : V (u) ⊗ W_{2} → W_{2} ⊗ V (u), then we can construct an Loperator in the space W_{1} ⊗ W_{2} as the composition:

V (u) ⊗ W_{1} ⊗ W_{2} → W_{1} ⊗ V (u) ⊗ W_{2} → W_{1} ⊗ W_{2} ⊗ V (u)

Definition Q-operator is the element of the algebra M_{R} given by the formula:

Q(u) = trL(u) = L_{i}^{i}(u)

From relations (4) it follows that Q-operators satisfy the following relations:

Q(u)Q(v) = Q(φ(u, v))Q(ψ(u, v))

Remark One can also define the dynamical twisted Yang-Baxter relation.

§3. Local action of the symmetric group and factorization of matrix polynomials

For the general theory of matrix polynomials and factorization see [7]. For our purposes we state the results, which may be well known to the experts.

We denote by S(a) the set of eigenvalues of a matrix a ∈ Mat_{m}. More generally, we denote by S(a_{1}, …, a_{d}), a_{1}, …, a_{d} ∈ Mat_{m}, the set of roots of the polynomial f(t) = det(t^{d} − a_{1}t^{d−1} + ⋯ + (−1)^{d}a_{d}). We will consider polynomials with generic coefficients only, so #S(a_{1}, …, a_{d}) = md.

Proposition 1. Let

t^{d} − a_{1}t^{d−1} + ⋯ + (−1)^{d}a_{d} = (t − b_{1}) … (t − b_{d})  (5)

for generic matrices a_{1}, …, a_{d} ∈ Mat_{m}, then S(b_{i})∩S(b_{j}) = ∅ for i ≠ j and S(b_{1})∪ ⋯ ∪ S(b_{d}) = S(a_{1}, …, a_{d}). For each decomposition S(a_{1}, …, a_{d}) = A_{1} ∪ ⋯ ∪ A_{d}, such that #A_{i} = m, A_{i} ∩ A_{j} = ∅ (i ≠ j) there exists a unique the factorization (5) with S(b_{i}) = A_{i}.

Proof The first statement follows from the equation det(t^{d} − a_{1}t^{d−1} + ⋯ + (−1)^{d}a_{d}) = det(t − b_{1}) … det(t − b_{d}).

On the other hand, if we know eigenvalues of b_{1}, …, b_{d} then we can calculate eigenvectors of them. For λ ∈ S(b_{d}) the corresponding eigenvector is a vector v_{λ}, such that (λ^{d} − a_{1}λ^{d−1} + ⋯ + (−1)^{d}a_{d})v_{λ} = 0. If we know all eigenvectors of b_{d}, then we can calculate all eigenvectors of b_{d−1} similarly and so on. This implies the uniqueness. By our construction of b_{1}, …, b_{d} the determinants of the matrix polynomials in the right hand side and the left hand side of (5) have the same set of roots. Moreover, for each root λ the operators represented by these matrix polynomials have the same kernel if we set t = λ. It implies that these polynomials are equal.

Proposition 2. Let a_{1}, a_{2} ∈ Mat_{m} be generic matrices. Then there exists a unique pair of matrices b_{1}, b_{2} ∈ Mat_{m} such that (t − a_{1})(t − a_{2}) = (t − b_{1})(t − b_{2}) and S(b_{1}) = S(a_{2}), S(b_{2}) = S(a_{1}). We have b_{1} = a_{1} + Λ^{−1}, b_{2} = a_{2} − Λ^{−1} where a_{2}Λ − Λa_{1} = 1.

Proof If S(b_{2}) ∩ S(a_{2}) ≠ ∅, then det(a_{2} − b_{2}) = 0, because a_{2} and b_{2} have a common eigenvector. Otherwise, we can put Λ = (a_{2} − b_{2})^{−1}.

Let U = Mat_{m}. From the propositions 1 and 2 it follows that the formula μ(a_{1}, a_{2}) = (b_{1}, b_{2}) gives a twisted transposition, where b_{1} + b_{2} = a_{1} + a_{2}, b_{1}b_{2} = a_{1}a_{2}, S(b_{1}) = S(a_{2}), S(b_{2}) = S(a_{1}). We have μ(a_{1}, a_{2}) = (a_{1} + Λ^{−1}, a_{2} − Λ^{−1}), where Λ is the solution of the linear matrix equation a_{2}Λ − Λa_{1} = 1.

Let \overline{U} = \overline{Mat_{m}} be the set of m × m matrices with different eigenvalues and fixed order of eigenvalues. The proposition 1 gives an action of the symmetric group S_{mN} on the space \overline{U}^{N} by birational automorphisms. By definition, for σ ∈ S_{mN}, b_{1}, …, b_{N} ∈ \overline{U} we have σ(b_{1}, …, b_{N}) = (b′_{1}, …, b′_{N}) where (t − b_{1}) … (t − b_{N}) = (t − b′_{1}) … (t − b′_{N}) and \overline{S}(b′_{i}) = σ\overline{S}(b_{i}), \overline{S} stands for the ordered set of eigenvalues.

This action is local in the following sense. The transposition (i, i + 1) for αm < i < (α + 1)m acts only inside the α + 1-th factor of Ū^{N} and the transposition (αm, αm + 1) acts only inside the product of the α-th and the α + 1-th factors. We have also the twisted transposition μ : \overline{U} × \overline{U} → \overline{U} × \overline{U} in this case which is the action of the element (1, m + 1)(2, m + 2) … (m − 1, 2m − 1) ∈ S_{2m}.

Remark Let U be the set of matrix polynomials of the form at + b, where a, b ∈ Mat_{m}, a = (a_{ij}), b = (b_{ij}) and a_{ij} = 0 for i < j, b_{ij} = 0 for i > j. It is possible to define the twisted transposition μ such that for μ(f(t), g(t)) = (f_{1}(t), g_{1}(t)) we have f(t)g(t) = f_{1}(t)g_{1}(t), det f(t) and det g_{1}(t) have the same set of roots and the first coefficients of f(t) and g_{1}(t) have the same diagonal elements. In [4] we found the solutions of the corresponding twisted Yang-Baxter relation (for m = 2), which is generalization of the R-matrix from chiral Potts model.

§4. Local action of the symmetric group and factorization on matrix θ-functions

Let Γ ⊂ ℂ be the lattice generated by 1 and τ where Imτ > 0. We have Γ = {α + βτ; α, β ∈ ℤ}. Let ε ∈ ℂ be primitive root of unity of degree m. Let γ_{1}, γ_{2} ∈ Mat_{m} be m × m matrices such that γ_{1}^{m} = γ_{2}^{m} = 1, γ_{2}γ_{1} = εγ_{1}γ_{2}. We have γ_{1}v_{α} = ε^{α}v_{α}, γ_{2}v_{α} = v_{α+1} in some basis {v_{α}; α ∈ ℤ/mℤ} of ℂ^{m}. Let us assume that {v_{1}, …, v_{m}} is the standard basis of ℂ^{m}.

We denote by MΘ_{n,m,c}(Γ) for n, m ∈ ℕ, c ∈ ℂ the space of everywhere holomorphic functions f : ℂ → Mat_{m}, which satisfy the following equations:

f(z + \frac{1}{m}) = γ_{1}^{−1}f(z)γ_{1}

f(z + \frac{1}{m}τ) = e^{−2πi(mnz−c)}γ_{2}^{−1}f(z)γ_{2}  (6)

Proposition 3. dim MΘ_{n,m,c}(Γ) = m^{2}n and for each element f ∈ MΘ_{n,m,c}(Γ) the equation det f(z) = 0 has exactly mn zeros modulo \frac{1}{m}Γ. The sum of these zeros is equal to mc + \frac{mn}{2} modulo Γ.

Proof For m = 1 we have the usual θ-functions Θ_{n,c}(Γ) = MΘ_{n,1,c}(Γ) and all these statements are well known in this case ([8]). One have a basis {θ_{α}(z); α ∈ ℤ/nℤ\} in the space Θ_{n,c}(Γ) such that θ_{α}(z + \frac{1}{n}) = e^{2πi \frac{α}{n}} θ_{α}(z), θ_{α}(z + \frac{1}{n}τ) = e^{−2πi(z− \frac{n−1}{2n} τ− \frac{1}{n} c)}θ_{α+1}(z) ([8]). From (6) it follows that f(z + 1) = f(z) and f(z + τ) = e^{−2πi(m^{2}nz−c_{1})}f(z) for some c_{1} ∈ ℂ. So the matrix elements of f(z) are θ-functions from the space Θ_{m^{2}n,c_{1}}(Γ). We have decomposition f(z) = ∑_{α} φ_{α}θ_{α}(z), where φ_{α} ∈ Mat_{m} are constant matrices, {θ_{α}} is a basis in the space Θ_{m^{2}n,c_{1}}(Γ). Substituting this decomposition in (6) one can calculate the dimension of the space MΘ_{n,m,c}(Γ). We have also det f(z + \frac{1}{m}) = det f(z) and det f(z + \frac{1}{m}τ) = e^{−2πi(m^{2}nz−mc)} det f(z). From this follows the statement about zeros of the equation det f(z) = 0.

Proposition 4. For generic complex numbers λ_{1}, …, λ_{mn} such that λ_{1} + ⋯ + λ_{mn} ≡ mc + \frac{mn}{2} mod\frac{1}{m}Γ and nonzero vectors v_{1}, …, v_{mn} ∈ ℂ^{m} there exist a unique up to proportionality element f(z) ∈ MΘ_{n,m,c}(Γ) such that det f(λ_{α}) = 0, f(λ_{α})v_{α} = 0 for 1 ⩽ α ⩽ mn.

Proof Considering the decomposition f(z) = ∑_{α} φ_{α}θ_{α}(z) one has the system of linear equations {∑_{α} θ_{α}(λ_{β})φ_{α}v_{β} = 0; β = 1, …, mn} for matrix elements of {φ_{α}}. One can see that this system defines {φ_{α}} uniquely up to proportionality for generic λ_{1}, …, λ_{mn}, v_{1}, …, v_{mn}.

We denote by S(f) the set of zeros of the equation det f(z) = 0 modulo \frac{1}{m}Γ. Proposition 5. Let f(z) ∈ MΘ_{n,m,c}(Γ) be generic element and we have factor-

ization f(z) = f_{1}(z) … f_{n}(z) where f_{α}(z) ∈ MΘ_{1,m,c_{α}}(Γ), c_{1} + ⋯ + c_{n} = c. Then S(f_{α})∩S(f_{β}) = ∅ for α ≠ β and S(f) = S(f_{1})∪⋯∪S(f_{n}). For each decomposition S(f) = A_{1} ∪ ⋯ ∪ A_{n} such that A_{α} ∩ A_{β} = ∅ for α ≠ β and #A_{α} = m there exists a unique factorization f(z) = f_{1}(z) … f_{n}(z) up to proportionality of f_{α} such that S(f_{α}) = A_{α} for α = 1, …, n.

Proof is similar to the proof of proposition 1, we just change polynomials by θ-functions.

Let U_{c} be the projectivisation of the linear space MΘ_{1,m,c}(Γ) and U = ⋃_{c∈ℂ} U_{c}. We have the following twisted transposition μ : U × U → U × U. By definition μ(f, g) = (f_{1}, g_{1}), where f(z)g(z) = f_{1}(z)g_{1}(z) and S(f_{1}) = S(g), S(g_{1}) = S(f).

Let \overline{U} be the set of elements f from U with fixed order on S(f). For f ∈ \overline{U} let \overline{S}(f) be the set S(f) with corresponding order. We have a local action of the symmetric group S_{mN} on the space \overline{U}^{N}. By definition, for σ ∈ S_{mN} we have σ(f_{1}, …, f_{N}) = (f_{1}^{σ}, …, f_{N}^{σ}) where f_{1}(z) … f_{N}(z) = f_{1}^{σ}(z) … f_{N}^{σ}(z) and \overline{S}(f_{α}^{σ}) = σ\overline{S}(f_{α}) for 1 ⩽ α ⩽ N.

Remark It is possible to construct twisted R-matrices for this twisted transposition μ as intertwiners of tensor products of cyclic representations of the algebra of monodromy matrices for elliptic Belavin R-matrix [5] at the point of finite order (see also [6]). It will be the subject of another paper.

§5. Generalization of the star-triangle relation

Let U be a complex manifold with an action of the symmetric group S_{m} by birational automorphisms. Let g_{i} be a birational automorphism of U corresponding to the transposition (i, i+1), where 1 ⩽ i < m. We assume also that the group S_{2m} acts on the manifold U×U by birational automorphisms, such that the transposition (i, i + 1) for i < m acts by g_{i} × id and the transposition (i, i + 1) for i > m acts by id × g_{i−m}. Let f be the birational automorphism of U × U corresponding to the transposition (m, m + 1). We will use the notations f(u, v) = (λ(u, v), μ(u, v)). We have: g_{i}^{2} = f^{2} = id, g_{i}g_{i+1}g_{i} = g_{i+1}g_{i}g_{i+1}, fg_{m}^{l}f = g_{m}^{l}fg_{m}^{l}, fg_{1}^{r}f = g_{1}^{r}fg_{1}^{r}, where g_{m}^{l} = g_{m} × id and g_{1}^{r} = id × g_{1}.

It is clear that we have the local action of the symmetric group S_{mN} on the manifold U^{N} for each N.

Let V be a n-dimensional vector space. For each u ∈ U we denote by V (u) a vector space canonically isomorphic to V. Let G_{i} (1 ⩽ i < m) be a meromorphic function from U to End(V) and F be a meromorphic function from U × U to End(V ⊗ V). We will consider G_{i}(u) and F (u, v) as linear operators

G_{i}(u) : V (u) → V (g_{i}(u))

F (u, v) : V (u) ⊗ V (v) → V (λ(u, v)) ⊗ V (μ(u, v))

Definition We call G_{i}, F a twisted GF -matrices if the following properties hold:

1. The compositions

V (u) → V (g_{i}(u)) → V (u)

V (u) ⊗ V (v) → V (λ(u, v)) ⊗ V (μ(u, v)) → V (u) ⊗ V (v)

are equal to the identity.

2. The following diagrams are commutative:

\xymatrix{V (g_{i}(u)) \ar[r]^{G_{i+1}(g_{i}(u))} & V (g_{i+1}g_{i}(u)) \ar[d]^{G_{i}(g_{i+1}g_{i}(u))} \\ V (u) \ar[u]_{G_{i}(u)} \ar[d]^{G_{i+1}(u)} & V (g_{i}g_{i+1}g_{i}(u)) \\ V (g_{i+1}(u)) \ar[r]^{G_{i}(g_{i+1}(u))} & V (g_{i}g_{i+1}(u)) \ar[u]_{G_{i+1}(g_{i}g_{i+1}(u))}}

\xymatrix{V (λ(u, v)) ⊗ V (μ(u, v)) \ar[r]^{G_{m}(λ(u,v))⊗1} & V (g_{m}(λ(u, v))) ⊗ V (μ(u, v)) \ar[d]^{F (g_{m}(λ(u,v))),μ(u,v))} \\ V (u) ⊗ V (v) \ar[u]_{F (u,v)} \ar[d]^{G_{m}(u)⊗1} & V (λ(g_{m}(λ(u, v)), μ(u, v))) ⊗ V (μ(g_{m}(λ(u, v)), μ(u, v))) \\ V (g_{m}(u)) ⊗ V (v) \ar[r]^{F (g_{m}(u),v)} & V (λ(g_{m}(u), v)) ⊗ V (μ(g_{m}(u), v)) \ar[u]_{G_{m}(λ(g_{m}(u),v))⊗1}}

\xymatrix{V (λ(u, v)) ⊗ V (μ(u, v)) \ar[r]^{1⊗G_{1}(μ(u,v))} & V (λ(u, v)) ⊗ V (g_{1}(μ(u, v))) \ar[d]^{F (λ(u,v)),g_{1}(μ(u,v)))} \\ V (u) ⊗ V (v) \ar[u]_{F (u,v)} \ar[d]^{1⊗G_{1}(v)} & V (λ(λ(u, v), g_{1}(μ(u, v)))) ⊗ V (μ(λ(u, v), g_{1}(μ(u, v)))) \\ V (u) ⊗ V (g_{1}(v)) \ar[r]^{F (u,g_{1}(v))} & V (λ(u, g_{1}(v))) ⊗ V (μ(u, g_{1}(v))) \ar[u]_{1⊗G_{1}(μ(u,g_{1}(v)))}}

It is clear that we have also twisted transposition μ in this situation. By definition μ is the birational automorphism of U × U corresponding to the element (1, m + 1)(2, m + 2) … (m − 1, 2m − 1) ∈ S_{2m}. We have

μ = ∏_{1⩽α⩽m} g_{m−α+1}^{l} … g_{m−1}^{l}fg_{m−1}^{r} … g_{α}^{r}

where g_{α}^{l} = g_{α} × id, g_{α}^{r} = id × g_{α} and the product is ordered from α = 1 to α = m. One can also construct the corresponding twisted R-matrix as a product of m^{2} factors:

R = ∏_{1⩽α⩽m}(G_{m−α+1} ⊗ 1) … (G_{m−1} ⊗ 1)F (1 ⊗ G_{m−1}) … (1 ⊗ G_{α})  (7)

Here we omit arguments of R, G_{α} and F.

Let W be a vector space, L be a meromorphic function from U to End(V ⊗ W). We will consider L(u) as a linear operator L(u) : V (u) ⊗ W → W ⊗ V (u).

Definition We call L an L-operator for twisted GF -matrices G_{i}, F if the following diagrams are commutative:

\xymatrix{V (u) ⊗ W \ar[r]^{L(u)} \ar[d]^{G_{i}(u)⊗1} & W ⊗ V (u) \ar[d]^{1⊗G_{i}(u)} \\ V (g_{i}(u)) ⊗ W \ar[r]^{L(g_{i}(u))} & W ⊗ V (g_{i}(u))}

\xymatrix{V (u) ⊗ W ⊗ V (v) \ar[r]^{L(u)⊗1} & W ⊗ V (u) ⊗ V (v) \ar[d]^{1⊗F (u,v)} \\ V (u) ⊗ V (v) ⊗ W \ar[u]_{1⊗L(v)} \ar[d]^{F (u,v)⊗1} & W ⊗ V (λ(u, v)) ⊗ V (μ(u, v)) \\ V (λ(u, v)) ⊗ V (μ(u, v)) ⊗ W \ar[r]^{1⊗L(μ(u,v))} & V (λ(u, v)) ⊗ W ⊗ V (μ(u, v)) \ar[u]_{L(λ(u,v))⊗1}}

That is

G_{i}(u)L(u) = L(g_{i}(u))G_{i}(u), for 1 ⩽ i < m
F (u, v)L^{1}(u)L^{2}(v) = L^{1}(λ(u, v))L^{2}(μ(u, v))F (u, v)  (8)

We can consider L(u) as a n × n matrix L(u) = (L_{i}^{j}(u)), where L_{i}^{j}(u) are the elements in End(W) such that

L(u) : x_{i}(u) ⊗ W → (L_{i}^{j}(u)W) ⊗ x_{j}(u)

Here {x_{i}} is a basis in V and {x_{i}(u)} is the corresponding basis in V (u).

Definition The algebra of monodromy matrices M_{GF} for the twisted GF -matrices G_{i}, F is an associative algebra with the generators {L_{i}^{j}(u); i, j = 1 … n, u ∈ U} and the defining relations (8).

It is clear that the representations of the algebra M_{GF} in the space W coincide with the L-operators L(u) ∈ End(V ⊗ W).

Since the relations (4) follow from the relations (8), one has a homomorphism M_{GF} → M_{R}, which is identity on the generators, where R is given by (7).

It is clear that the algebra M_{GF} has the usual coproduct Δ : L_{i}^{j}(u) → L_{i}^{α}(u) ⊗ L_{α}^{j}(u).

Remarks 1. One can also construct the Zamolodchikov algebra for GF -matrices.

2. For Q(u) = trL(u) = L_{i}^{i}(u) we have the following relations:

Q(u) = Q(g_{i}(u)) for 1 ⩽ i < m
Q(u)Q(v) = Q(λ(u, v))Q(μ(u, v))

3. It is possible to construct twisted GF -matrices for the elliptic local action of the symmetric group defined in §4 as intertwiners of tensor products of cyclic representations of the algebra of monodromy matrices for elliptic Belavin R-matrix ([5]) at the point of finite order. It will be the subject of another paper.

Acknowledgments I am grateful to V.Bazhanov and A.Belavin for useful discussions. I am grateful to Max-Planck-Institut fur Mathematik, Bonn where this paper

was written for invitation and very stimulating working atmosphere.

The work is supported partially by RFBR 99-01-01169, RFBR 00-15-96579, CRDF RP1-2254 and INTAS-00-00055.

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