Yang-Baxter maps associated to elliptic curves
aa r X i v : . [ m a t h . QA ] J un Yang–Baxter maps associated to elliptic curves
Vassilios G. Papageorgiou , ∗ and Anastasios G. Tongas , † Department of Mathematics, University of Patras, 265 00 Patras, Greece Department of Applied Mathematics, University of Crete, 714 09 Heraklion, Greece
November 12, 2018
Abstract
We present Yang–Baxter maps associated to elliptic curves. They are related to discrete ver-sions of the Krichever-Novikov and the Landau-Lifshits equations. A lifting of scalar integrablequad–graph equations to two–field equations is also shown.
The importance of the Yang-Baxter (YB) equation to a variety of branches in physics and mathe-matics is well known. Its solutions are intimately related to exactly solvable statistical mechanicalmodels, link polynomials in knot theory, quantum and classical integrable models, conformal fieldtheories, representations of groups and algebras, quantum groups and many others. More interest-ingly, YB equation provides various connections among the aforementioned disciplines.Historically, YB equation has its roots in the theory of exactly solvable models in statisticalmechanics [35, 8] and the quantum inverse scattering method [31]. For an extensive account of earlywork on the YB equation see [14]. In its original form the quantum YB equation is the relation R (2 , R (1 , R (1 , = R (1 , R (1 , R (2 , , (1)in End ( V ⊗ ), for a k -linear operator R : V ⊗ V ⊗ , where V is a vector space over a field k .Here, R (1 , is meant as R acting on the first and third factors of the tensor product V ⊗ andas identity on the second, and similarly for R (1 , and R (2 , . Drinfel’d suggested to study thesimplest possible solutions of the YB equation by replacing ( k –Vect , ⊗ ) with (Set , × ), where theYB equation is regarded as an equality of maps in X for a finite set X . As it was pointed outin [11], this setting provides potentially new interesting solutions of the original YB equation byconsidering the free module generated by the set X . Various interesting examples of YB maps suchas those arising from geometric crystalls [12], have revealed a richer structure of the underlyingset; X is an algebraic variety and R is a birational isomorphism. As in [33] we refer to solutions ofthe YB equation in (Set , × ) simply as YB maps. One has to have in mind though that YB mapsthrough a natural coupling can be regarded as equations on the edges of graphs.One of the most distinguished properties of integrable partial differential equations is theirinvariance under Darboux and B¨acklund transformations [21], [29]. The nonlinear superpositionformulae of the solutions generated by the B¨acklund-Darboux transformations provide naturaldiscrete versions of the continuous equations. In turn, the key transformation properties of thediscrete equations are intimately related to the YB property for maps or its proper generalization ∗ [email protected] † [email protected]
1n higher dimensions namely the functional tetrahedron equation. A prime example of the latterare the star-triangle transformations in electric networks and the Ising model and their connectionwith discrete equations in the three-dimensional lattice associated with the Kadomtsev–Petviashvili(KP) hierarchy and its modifications [15].Recent interest on solutions of the YB equation for maps has appeared in the literature. Thisis mainly due to the development of the dynamical theory of YB maps [33], and classificationresults of YB maps for X = CP , in connection to analogous results for two–dimensional integrablediscrete equations on the square lattice [5, 6]. The latter correspondence was further investigatedin [26, 27] by exploiting the local groups of symmetry transformations of the discrete equations. Itwas demonstrated that an integrable quadrilateral equation with a sufficient r -parameter symmetrygroup gives rise to a YB map. However, there exist discrete equations which do not admit any localsymmetry group. Such an equation is a discrete version of the Krichever-Novikov (KN) equation[18, 19], introduced by V. E. Adler in [3].A characteristic feature of the discrete KN equation is that the lattice parameters lay on an ellip-tic curve. Remarkably, from the very early times exactly solvable two-dimensional models appearedin statistical mechanics it was observed that the solution of many of these models ultimately leadsto the introduction of elliptic functions, such as the eight-vertex model on the square lattice [7].Thus, it would be interesting to investigate whether there exist also solutions of the YB equationfor maps related to elliptic curves. Already this problem was addressed in [24] where a theoreticalframework was introduced for deriving YB maps from factorization of matrix polynomials and θ –functions.The main aim of the present work is to exhibit YB maps with parameters living on elliptic curvesand which are associated to integrable partial differential equations (PDE). In Section 2 we presentbackground material on the YB maps. In Section 3, we present key transformation propertiesof integrable lattice equations, encoded into braid type equations, and give a brief account onsymmetry aspects of discrete integrable equations and their usage in deriving YB maps. Finally,we present a way for deriving a YB map from integrable lattice equations of certain type, withoutusing a local symmetry group. The latter method is applied to generic integrable lattice equationsassociated to elliptic curves such as discrete KN equation and discrete Landau–Lifshits equationand the results are presented in Sections 4 and 5, respectively. The paper concludes in Section 6with various comments and perspectives. Braid and YB maps
In the following we use the notation and terminology introduced in [33] (for a recent review see[34]). Let X be an algebraic variety, and R : X × X → X × X a birational isomorphism. Let R ( i,j ) : X n → X n denote the map acting as R on the components ( i, j ) of the n -fold Cartesianproduct X × X × · · · × X and as the identity on all others. More explicitly, for x, y ∈ X let us write R ( x, y ) = (cid:0) f ( x, y ) , g ( x, y ) (cid:1) . (2)Then, for n ≥ ≤ i, j ≤ n , i = j the map R ( i,j ) is given by R ( i,j ) ( x , . . . , x n ) = ( x , . . . , x i − , f ( x i , x j ) , x i +1 , . . . , x j − , g ( x i , x j ) , x j +1 , . . . x n ) , i < j, ( x , . . . , x j − , g ( x i , x j ) , x j +1 , . . . , x i − , f ( x i , x j ) , x i +1 , . . . x n ) , i > j . (3)2n particular, for n = 2 we have R (1 , = R and R (2 , ( x, y ) = (cid:0) g ( y, x ) , f ( y, x ) (cid:1) . The latter map is R conjugated by the permutation map σ , defined by σ ( x, y ) = ( y, x ), i.e. R (2 , = σ R σ . (4) Definition 2.1. (i) The map R is called a YB map if R satisfies the YB equation R (2 , R (1 , R (1 , = R (1 , R (1 , R (2 , , (5) regarded as an equality of maps of X into itself.(ii) R is called reversible, or unitary, if it satisfies the condition R (2 , R = Id X . (6) (iii) R is called non-degenerate if the maps from X into itself defined by s → f ( s, y ) and t → g ( x, t ) are bijective rational maps for any fixed y and x , respectively. A schematic representation of the YB equation is given by the two decompositions of an ele-mentary 3-cube as depicted in figure 1. The composition of maps in the LHS and RHS of the YBequation (5) are given by(a) : ( x , x , x ) R (1 , −→ ( x ′ , x ′ , x ) R (1 , −→ ( x ′′ , x ′ , x ′ ) R (2 , −→ ( x ′′ , x ′′ , x ′′ ) , (7)(b) : ( x , x , x ) R (2 , −→ ( x , x ∗ , x ∗ ) R (1 , −→ ( x ∗ , x ∗ , x ∗∗ ) R (1 , −→ ( x ∗∗ , x ∗∗ , x ∗∗ ) , respectively. The YB equation guarantees that the images of ( x , x , x ) ∈ X under the twocomposition of maps in (7) are identical, thus the two parts of the 3-cube can be glued together.Figure 1: A cubic representation of the YB relation Remark . Defining B = σR the YB equation (5) for R translates to the braid equation B (2 , B (1 , B (2 , = B (1 , B (2 , B (1 , , (8)for B . The unitarity condition (6), in view of (4), takes the form B = Id X , (9)and B is said to be an involution. 3 ax matrices for YB maps Instead of a single map one may consider a whole family of YB maps parametrized by two continuousparameters α , α ∈ C , where C is an algebraic set in C . The YB relation then takes the parameter-dependent form R (2 , α ,α ) R (1 , α ,α ) R (1 , α ,α ) = R (1 , α ,α ) R (1 , α ,α ) R (2 , α ,α ) , (10)and the unitarity (reversibility) condition becomes R (2 , α ,α ) R ( α ,α ) = Id X . (11)In the following we drop the dependence on the parameters and write just R for a two-parameterYB map, since we always consider maps of this type.Let L ( x ; α, λ ) ∈ Mat( r, X ) be a two-parameter family of r × r matrices depending on x ∈ X andpolynomially/rationally on the coordinates of α, λ ∈ C . The following notion of Lax matrix for aYB map was introduced in [30]. Definition 2.3. (i) L ( x ; α, λ ) is called a Lax matrix of the YB map R , if the relation R ( x , x ) =( x ′ , x ′ ) implies that L ( x ; α , λ ) L ( x ; α , λ ) = L ( x ′ ; α , λ ) L ( x ′ ; α , λ ) , (12) for all λ ∈ C . L ( x ; α, λ ) is called a strong Lax matrix of R , if the converse also holds.(ii) L ( x ; α, λ ) satisfies the n -factorization property if the identity L ( x ′ n ; α n , λ ) · · · L ( x ′ ; α , λ ) L ( x ′ ; α , λ ) ≡ L ( x n ; α n , λ ) · · · L ( x ; α , λ ) L ( x ; α , λ ) , (13) over C , implies that x i ′ = x i , i = 1 , . . . , n .Remark . The 2-factorization property of L corresponds to the unitarity property of R , whilethe 3-factorization property to the YB property. Indeed, the composition of maps( x , x ) R (1 , −→ ( x ′ , x ′ ) R (2 , −→ ( x ′′ , x ′′ ) , (14)is represented by the matrix factorization L ( x ; α , λ ) L ( x ; α , λ ) = L ( x ′ ; α , λ ) L ( x ′ ; α , λ ) = L ( x ′′ ; α , λ ) L ( x ′′ ; α , λ ) . (15)Evidently, the unitary property of R is equivalent to the 2-factorization property of L . On the otherhand, the cubic representation of the YB relation (see figure 1) suggests to consider the product L ( x ; a , λ ) L ( x ; a , λ ) L ( x ; a , λ ) . It can be factorized in two different ways, according to the composition of maps in (7), i.e.(a) L ( x ; a , λ ) L ( x ; a , λ ) L ( x ; a , λ ) = L ( x ; a , λ ) L ( x ′ ; a , λ ) L ( x ′ ; a , λ )= L ( x ′′ ; a , λ ) L ( x ′ ; a , λ ) L ( x ′ ; a , λ )= L ( x ′′ ; a , λ ) L ( x ′′ ; a , λ ) L ( x ′′ ; a , λ ) , (b) L ( x ; a , λ ) L ( x ; a , λ ) L ( x ; a , λ ) = L ( x ∗ ; a , λ ) L ( x ∗ ; a , λ ) L ( x ; a , λ )= L ( x ∗ ; a , λ ) L ( x ∗ ; a , λ ) L ( x ∗∗ ; a , λ )= L ( x ∗∗ ; a , λ ) L ( x ∗∗ ; a , λ ) L ( x ∗∗ ; a , λ ) . where we have used the fact that L is a Lax matrix for R in each face of the cube and the associativityof matrix multiplication. Thus, the YB property of R is equivalent to the 3-factorization propertyof L . 4 xample . Consider the map R : CP × CP CP × CP defined by R ( x, y ) = (cid:18) y α + x yα + x y , x α + x yα + x y (cid:19) , (16)introduced in [27]. The above map admits the Lax matrix L ( x ; α, λ ) = (cid:20) x λ xx α (cid:21) . It is straightforward to check that the discrete zero curvature equation (12), implies the map (16).Conversely, according to [30] a hint for considering the matrix L of the above form, is based onthe YB map itself. Indeed, one notices that the second component of the map can be written as alinear fractional transformation induced by the linear transformation with matrix L ( x ; α , λ ) | λ = α on [ y T . For the above Lax matrix the n -factorization property can be proved as follows. Thenull space of the linear transformation in the LHS of (13), for λ = α , is spanned by the vector[ − α x ′ ] T . Similarly, [ − α x ] T spans the null space of the RHS linear transformation. Becauseof the identity (13), we conclude that x = x ′ , and the rightmost matrices cancel out. Therefore,by induction, L satisfies the n -factorization property. The YB map (16) is simply related to the F III map obtained in the recent classification [6] for the case X = CP . In this section we present first braid transformation properties of integrable discrete equationsdefined on elementary squares. Next, we briefly summarize a method for obtaining YB maps fromintegrable discrete equations on quad-graphs, which is based on the existence of a local group ofsymmetry transformations of the equations. Finally, we present another method to the same endwhich does not prerequisite the existence of a local group of symmetry transformations.
Main properties of integrable discrete equations
We consider discrete equations on quad-graphs given by an algebraic equation Q ( f A , f B , f Γ , f ∆ ; α, β ) = 0 , (17)relating the values of a function f : Z → X assigned on the four vertices of an elementary plaquette.It is assumed that (i) opposite edges on the plaquette carry the same lattice parameter α , β and(ii) equation (17) it can be solved uniquely for each f i , say f Γ , i.e. f Γ = ϕ ( f A , f B , f ∆ ; α, β ) . (18)In order to make contact with the special properties of the integrable discrete equations we interpretequation (17) as a map B : X → X defined by B ( f ∆ , f A , f B ) = ( f ∆ , f Γ , f B ) , (19)where f Γ is given by (18) (see fig. (2) ).Let us now define B j : X n → X n by B j = Id X × · · · × B × · · · × Id X , (20)where B acts on the j − j and the j + 1 factors of X n with parameters ( a j − , a j ). The keyproperties of maps associated to integrable discrete equations on quad-graphs are the relations B j = Id X n , ( B j B j +1 ) = Id X n , B j B i = B i B j , | i − j | > , (21)5igure 2: An elementary quadrilateral for a quad-graph equation. The arrow indicates the flip( f ∆ , f A , f B ) B → ( f ∆ , f Γ , f B ).see [1]. The first one means that each transformation B j is an involution. The second one, in viewof the first, yields the following braid-type relation B j +1 B j B j +1 = B j B j +1 B j . (22)Figure 3: A cubic representation of the braid relation (22) for quad-graph equations.The braid relation (22) guarantees that the three-dimensional consistency property [23], [9],which nowadays is synonymous with the integrability of a quad-equation holds (for a recent accounton the subject we refer to the monograph [10]). From integrable discrete equations to YB maps via symmetry groups
Local symmetry groups of transformations of integrable discrete equations provide a natural way forobtaining YB maps from them. The main observation is that the variables of certain YB maps canbe chosen as invariants of the symmetry group admitted by the corresponding lattice equation. Thesymmetry approach was exploited in [26], where it was also shown that all classified quatrirationalYB maps, for X = CP , found in [6], can be constructed from integrable quadrilateral equations. Definition 3.1.
Let G be a one-parameter group of transformations on X , of the form G : ( x, y, z ) ( X ( x ; ε ) , Y ( y ; ε ) , Z ( z ; ε )) , ε ∈ C . (23) and B a map of X into itself. G is said to be a local (Lie-point) group of symmetry transformationsof the map B if G ◦ B = B ◦ G , for every ε ∈ C . Let X = CP and consider the following map B ( x, y, z ) = ( x, y + a − bx − z , z ) , (24)which is associated with the discrete KdV equation. The corresponding map B j defined by (20) sat-isfies the braid type relation (22). Moreover, the map (24) commutes with the group of translationsgiven by G : ( x, y, z ) ( x + ε , y + ε , z + ε ) . (25)6hus, G is a Lie-point symmetry of the map (24). The action of G on X is regular with one-dimensional orbits, thus local coordinates on the set of orbits of G are provided by the completeset of functionally independent invariants for the group action: u = y − x , v = z − y . Projecting the map (24) to the set of orbits of G we obtain the map B ( u, v ) = (cid:18) u − a − bu + v , v + a − bu + v (cid:19) , (26)which satisfies the parameter braid relation (8). Thus, the map R ( u, v ) = σ B ( u, v ) = (cid:18) v + a − bu + v , u − a − bu + v (cid:19) , (27)is a YB map, known as the Adler map [2]. The most general local group of symmetry transforma-tions of the map (24) is G ∼ = SO (1 , G and the one-parameter subgroups G , G given by the group actions G : ( x, y, z ) ( x − ε , y + ε , z − ε ) ,G : ( x, y, z ) (cid:0) x e − ε , y e ε , z e − ε (cid:1) . By using similar arguments one may consider the set of orbits of the subgroups G , or G , to obtainother YB maps from the map (24). More precisely, we have the following Proposition 3.2.
Let X = C n and a map B : X → X satisfying the braid-type relation (22). If B admits a local group G of symmetry transformations which acts regularly on X with n -dimensionalorbits, then the projection of the map B to the set of orbits of G satisfies the braid relation.Proof. The assumptions for the action of G on X guarantee the existence of a 2 n -dimensionalquotient manifold denoted by X /G , i.e. the set of all orbits of G . Local coordinates ( u, v ) ∈ X can be chosen by a complete set of functionally independent invariants for the group action, seee.g. Theorem 3.18 in [25].Let us denote by B : X X the projection of the map B on X /G . The braid property of themap B is inherited by the braid type relation (22) which satisfies the map B . This can be easilydeduced from the cubic representation of the relation (22) (Figure 3). It should by noted that theinvariants of the group action (YB or braid variables) can be naturally assigned to the edges of theelementary squares instead of the vertices where the variables of the original map B are assignedto. Two–field integrable discrete equations as YB maps
The existence of a symmetry group of the map B provides us a way to obtain a YB map by usingas YB variables the invariants of the group action which are naturally attached on the edges ofthe squares. As it was shown recently [28], [32], generic integrable quad-graph equations, such asthe KN discrete equation, do not admit any local symmetry group of transformations. Thus thequestion arises whether such equations are related to YB maps, as well. This question is answeredin the affirmative in section 4. Proposition 3.3 below shows how to cast two–field quad–graphequations of a certain type into YB map form. Moreover, it motivates a way of lifting an integrablescalar quad–graph equation to a two–field one and consequently to recast the equation into a YBmap. 7pecifically, we consider lattice equations where at each vertex there is a two-field ( u, v ) ∈ X and the defining relations on the quadrilateral are (see figure 2)( u Γ , v Γ ) = (cid:0) F ( u A , u B , v ∆ ; a, b ) , F ( v A , u B , v ∆ ; a, b ) (cid:1) , (28)where F , F take values in X . This scheme of two-field quad-graph equations, although not thegeneric one since it does not involve all eight values of the fields, arises in the superposition formulaeof B¨acklund transformations for two-field integrable PDEs e.g. the nonlinear Schr¨odinger system[16], [1]. The aim is to recast discrete equations of the form (28) into a YB map form. To this endFigure 4: The quadrilateral with “thickened” edges as a parallelepiped and the assignment of theYB variables for the map (33). The variables x and q are identified, respectively y and p ,(6–point scheme).we group the fields appearing in the RHS of equation (28) as follows( x , x ) = ( u B , v A ) , ( y , y ) = ( u A , v ∆ ) . (29)A pictorial representation of the assignment of the YB variables for the map R : (cid:0) ( x , x ) , ( y , y ) (cid:1) → (cid:0) ( p , p ) , ( q , q ) (cid:1) , (30)(equation (33) below) is shown in Figure 4 where( p , p ) = ( u Γ , v ∆ ) , ( q , q ) = ( u B , v Γ ) . (31) Proposition 3.3.
Let the map B : X × X × X → X × X × X , defined by B (cid:0) ( u ∆ , v ∆ ) , ( u A , v A ) , ( u B , v B ) (cid:1) = (cid:0) ( u ∆ , v ∆ ) , ( u Γ , v Γ ) , ( u B , v B ) (cid:1) , (32) where ( u Γ , v Γ ) is given by (28), satisfies the braid relation (22). Then the map R : X × X → X × X defined by R (cid:0) ( x , x ) , ( y , y ) (cid:1) = (cid:0) ( F ( y , x , y ; α, β ) , y ) , ( x , F ( x , x , y ; α, β ) (cid:1) ) , (33) satisfies the YB relation.Proof. By straightforward calculations, we derive first the relations for the functions F , F suchthat B satisfies the braid type relation. The values ( u Γ ′ , v Γ ′ ) and ( u ∆ ′ , v ∆ ′ ) are found in twodifferent ways, according to the left and right hand side of the braid relation and are given by u K = F ( u Γ , u ∆ , v B ; b, a ) u ∆ ′ = F ( u ∆ , u E , v K ; c, a ) u Γ ′ = F ( u K , u ∆ ′ , v B ; c, b ) v K = F ( v Γ , u ∆ , v B ; b, a ) v ∆ ′ = F ( v ∆ , u E , v K ; c, a ) v Γ ′ = F ( v K , u ∆ ′ , v B ; c, b ) u K ′ = F ( u ∆ , u E , v Γ ; c, b ) u Γ ′ = F ( u Γ , u K ′ , v B ; c, a ) u ∆ ′ = F ( u K ′ , u E , v Γ ′ ; b, a ) v K ′ = F ( v ∆ , u E , v Γ ; c, b ) v Γ ′ = F ( v Γ , u K ′ , v B ; c, a ) v ∆ ′ = F ( v K ′ , u E , v Γ ′ ; b, a )8espectively. Thus, we have the following functional relations satisfied by F , F : F ( F ( u Γ , u ∆ , v B ; b, a ) , F ( u ∆ , u E , F ( v Γ , u ∆ , v B ; b, a ); c, a ) , v B ; c, b ) = F ( u Γ , F ( u ∆ , u E , v Γ ; c, b ) , v B ; c, a ) , (34) F ( F ( v Γ , u ∆ , v B ; b, a ) , F ( u ∆ , u E , F ( v Γ , u ∆ , v B ; b, a ); c, a ) , v B ; c, b ) = F ( v Γ , F ( u ∆ , u E , v Γ ; c, b ) , v B ; c, a ) , (35) F ( u ∆ , u E , F ( v Γ , u ∆ , v B ; b, a ); c, a ) = F ( F ( u ∆ , u E , v Γ ; c, b ) , u E , F ( v Γ , F ( u ∆ , u E , v Γ ; c, b ) , v B , c, a ) , b, a ) , (36) F ( v ∆ , u E , F ( v Γ , u ∆ , v B ; b, a ); c, a ) = F ( F ( v ∆ , u E , v Γ ; c, b ) , u E , F ( v Γ , F ( u ∆ , u E , v Γ ; c, b ) , v B ; c, a ); b, a ) . (37)On the other hand, the YB relation for the map (33) gives the following functional relations for F , F F ( F ( z , y , z ; β, γ ) , F ( y , x , F ( y , y , z ; β, γ ); α, γ ) , z ; α, β ) = F ( z , F ( y , x , y ; α, β ) , z ; α, γ ) , (38) F ( F ( y , y , z ; β, γ ) , F ( y , x , F ( y , y , z ; β, γ ); α, γ ) , z ; α, β ) = F ( y , F ( y , x , y ; α, β ) , z ; α, γ ) , (39) F ( y , x , F ( y , y , z ; β, γ ); α, γ ) = F ( F ( y , x , y ; α, β ) , x , F ( y , F ( y , x , y ; α, β ) , z ; α, γ ); β, γ ) , (40) F ( x , x , F ( y , y , z ; β, γ ); α, γ ) = F ( F ( x , x , y ; α, β ) , x , F ( y , F ( y , x , y ; α, β ) , z ; α, γ ); β, γ ) , (41)and two additional equations which are trivially satisfied.Making the following substitutions u Γ z u ∆ y u E x a γ,v B z v Γ y v ∆ x b βc α in equations (34)-(37), the latter become identical to (38)-(41), respectively. Remark . Consider the case F = F = F i.e.( u Γ , v Γ ) = (cid:0) F ( u A , u B , v ∆ ; a, b ) , F ( v A , u B , v ∆ ; a, b ) (cid:1) . (42)If u i = v i , i = A, B, ∆ , then from equation (42) we have u Γ = v Γ and the map (32) essentiallyreduces to a single field map, namely B ↓ (cid:0) u ∆ , u A , u B (cid:1) = (cid:0) u ∆ , u Γ , u B (cid:1) , (43)and B can be thought as a lift of B ↓ . This observation suggests to lift the discrete KN equation toa two-field quad-graph equation and then write it as a YB map. The lifting process can be appliedto all scalar integrable quad-equations listed in [5].9 xample . The simplest equation of the classification in [5] is the discrete (potential) KdVequation, namely f n +1 ,m +1 = f n,m + a − bf n +1 ,m − f n,m +1 , (44)where ( n, m ) ∈ Z . Its lift obtained by equation (42) takes the explicit form u n +1 ,m +1 = u n,m + a − bu n +1 ,m − v n,m +1 , v n +1 ,m +1 = v n,m + a − bu n +1 ,m − v n,m +1 , (45)and satisfies the braid–type relation (22). The corresponding YB map obtained by using Proposition3.3 reads ( p , p ) = (cid:18) y + a − bx − y , y (cid:19) , ( q , q ) = (cid:18) x , x + a − bx − y (cid:19) . (46)The YB map (46) was derived in [17] from matrix factorization and is symplectic with respect toa canonical structure. On the other hand, equations (45) are the Euler-Lagrange equations for thediscrete variational problem associated to the following Lagrangian density L = u n +1 ,m v n,m − u n,m v n,m +1 + ( a − b ) ln ( u n +1 ,m − v n,m +1 ) . (47)The problem of the Lagrangian formulation of the quad–graph equations classified in [5] has beenaddressed recently in [20]. Remark . The YB maps (33) obtained by this method are non–quadrirational. The quadrira-tionality property of maps as was introduced in [6] is equivalent to the nondegeneracy propertywhich is often imposed additionally on the YB maps. This can be seen immediately since the map( x , x ) → ( F ( y , x , y ; α, β ) , y ) is independent of x . The master scalar integrable quad equation listed in [5], in the sense that the rest integrablediscrete equations can be derived from it by proper degenerations of the elliptic curve or limitingprocedures, is discrete KN equation [3]. Using the identification ( f A , f B , f Γ , f ∆ ) = ( x, y, w, z ) onthe quadrilateral (Figure 2) the latter equation reads a ( xy + wz ) − b ( xz + wy ) − aB − bA − a b (cid:0) xw + yz − ab (1 + wxyz ) (cid:1) = 0 . This is the form introduced by Hietarinta in [13], where the parameters a ≡ ( a, A ) and b ≡ ( b, B )lay on Jacobi quartics given by E = (cid:8) ( a, A ) ∈ C : A = a + k a + 1 (cid:9) , and k is the modulus of E . The binary operation ⊕ defined by a ⊕ b = (cid:18) aB + bA − a b , ( A B + k a b )(1 + a b ) + 2 ab ( a + b )(1 − a b ) (cid:19) , endows the set E with an abelian group structure, in which e = (0 ,
1) is the identity element andthe inverse of a point a = ( a, A ) is the point − a = ( − a, A ). In the following we use the notation x = ( x, X ) etc, for points in C . 10 roposition 4.1. The map R ( x , y ) = ( p , q ) defined by ( p, P ) = ( F ( y, x, Y ; a , b ) , Y ) , ( q, Q ) = ( x , F ( X, x, Y ; a , b )) , (48) where F ( x, y, z ; a , b ) = (1 − a b )( b z − a y ) x + ( a B − b A )( y z − a b )( a B − b A )( a b y z − x + (1 − a b )( a z − b y ) , is a unitary YB map, with Lax matrix given by L ( x ; a , λ ) = ρ ( x ; a ) − / W ( x ; a , λ ) , (49) where W ( x ; a , λ ) = − λ X − x a Λ − Aλ − a λ a (cid:18) λ x X a Λ − Aλ − a λ (cid:19) − a (cid:18) x X + λ a Λ − Aλ − a λ (cid:19) λ x + X a Λ − Aλ − a λ , and the scalar function ρ is ρ ( x ; a ) = (cid:0) x X + 1 (cid:1) a − x − X + 2 A x X , a , λ ∈ E .Proof. First we prove that the matrix L ( x ; a , λ ) is a Lax matrix for the map (48) by showing thatthe factorization problem L ( y ; b , λ ) L ( x ; a , λ ) = L ( p ; a , λ ) L ( q ; b , λ ) , (50)is equivalent to equations (48). Taking into account that λ ∈ E we have (cid:0) W ( y ; b , λ ) W ( x ; a , λ (cid:1) ij = P k =0 P ℓ =0 S ijkℓ ( x , y ; a , b ) λ k Λ ℓ ( a λ − b λ − . Equating the different powers of λ , Λ, the matrix equation (50) is equivalent to the following systemof algebraic relations S ijkℓ ( x , y ; a , b ) ρ ( x ; a ) / ρ ( y ; b ) / = S ijkℓ ( q , p ; b , a ) ρ ( p ; a ) / ρ ( q ; b ) / , (51) i, j = 1 , k = 0 , ℓ = 0 , . . . ,
6. We calculate the terms S ( x , y ; a , b ) = a b x ( y − X ) ,S ( x , y ; a , b ) = a b Y ( X − y ) ,S ( x , y ; a , b ) = a b ( X − y ) ,S ( x , y ; a , b ) = a b X y ( b Y − a x ) + b x − a Y + a B y − b A X ,S ( x , y ; a , b ) = Xy ( bY − ax ) + ab ( bx − aY ) + xY ( aBx − bAy ) . From system (51) we get S ijkℓ ( x , y ; a , b ) S ( x , y ; a , b ) = S ijkℓ ( q , p ; b , a ) S ( q , p ; b , a ) . (52)For ( ijkℓ ) = (1100) and ( ijkℓ ) = (1160) equations (52) lead to q = x , P = Y , (53)11espectively and using them, equations (52) for ( ijkℓ ) = (1210) , (2110) lead to a linear systemwhich is uniquely solved for ( p, Q ), yielding p = F ( y, x, Y ; a , b ) , Q = F ( X, x, Y ; a , b ) . (54)With the solution given by (53), (54) by straightforward calculations we find that system (51) issatisfied.Next we prove that the Lax matrix L given by (49) satisfies the n -factorization property. For λ = a we obtain W ( x ; a , a ) = α (cid:20) x (cid:21) [ − X . Thus the kernel of the linear transformation L ( x n ; a n , a ) · · · L ( x ; a , a ) L ( x ; a , a ) , is spanned by the vector [1 X ] T which leads us to conclude that X = X ′ in (13). Likewise, for λ = e we have W ( x n ; a n , e ) · · · W ( x ; a , e ) W ( x ; a , e ) = α n Y i =2 α i ( X i − − x i ) ! (cid:20) X n (cid:21) [ − x . In this case the kernel of the linear transformations in the LHS and RHS of (13) is spanned bythe vectors [ 1 x ′ ] T and [ 1 x ] T , respectively. Thus, x = x ′ and consequently L ( x ′ ; a , λ ) = L ( x ; a , λ ) for all λ ∈ E . Hence, the number of matrices in equation (13) is reduced by one and byinduction the n -factorization property is proved. Finally, by remark 2.4 the Proposition is true. In the literature, there exist several discrete versions of the Landau-Lifshits equation representingthe nonlinear superposition formula for the solutions generated by the B¨acklund auto-transformation[22], [1], [4] . Here, we use the one introduced in [1] and we present the end result, namely thecorresponding YB map derived from the lattice equations by using Proposition 3.3. The map readsthe form R (cid:0) ( x, X ) , ( y, Y ) (cid:1) = (cid:0) ( F ( y, x, Y ; a , b ) , Y ) , ( x, F ( X, x, Y ; a , b ) (cid:1) , (55)where F ( x, y, z ; a , b ) = K ( y, z ) x − L ( y, z ) M ( y, z ) x + N ( y, z ) , F ( x, y, z ; a , b ) = K ( y, z ) x + L ( y, z ) − M ( y, z ) x + N ( y, z ) ,K ( y, z ) − N ( y, z ) = 2 s y z − ( αs + s ) ( y − z ) − αs − βs ,K ( y, z ) + N ( y, z ) = ( y + z ) (cid:0) a b ( αs + 3 s ) + 4 βs + 3 αs + s (cid:1) / ( a − b ) ,L ( y, z ) = s yz + ( αs + 2 βs )( y − z ) + 4 βs − α s ,M ( y, z ) = s yz + s ( y − z ) − s ,s m = B a m − + A b m − . The parameters a = ( a, A ), b = ( b, B ) lay on the Weierstraß elliptic curve E = (cid:8) ( χ, X ) ∈ C : X + χ + α χ + β = 0 (cid:9) , (56)12here α, β are complex constants, the invariants of the curve. It should be noted that in contrastto the previous case the functions F , F are different reflecting the fact that the discrete equationsconstitute a genuine two-field system. The Lax matrix introduced in [1], is also a Lax matrix forthe map (55) and is given by L ( x ; a , λ ) = ρ ( x ; a ) − / W ( x ; a , λ ) . (57)Here, ρ ( x ; a ) = 2 A ( r + x − X + a ) , the matrix components of W ( x ; a , λ ) are given by (cid:0) W ( x ; a , λ ) (cid:1) = (Λ + A ) r + ( λ − a )( λ + a − X ) , (cid:0) W ( x ; a , λ ) (cid:1) = (Λ + A )( λ + x − X ) − ( λ − a )( λ + 2 a ) r − A ( λ − a ) , (cid:0) W ( x ; a , λ ) (cid:1) = Λ + A − ( λ − a ) r , (cid:0) W ( x ; a , λ ) (cid:1) = − (Λ + A ) r − ( λ − a )( λ + a + x ) , and r = 12 A (cid:0) x X + a ( x − X ) + α + 2 a (cid:1) . The Lax matrix (57) satisfies the n -factorization property for n = 2 ,
3. Indeed, first we note thatfor λ = λ , where λ = − a − A , Λ = − A − A ( λ − a ) , A = − a + α A , the matrix W takes the dyadic form W ( x ; a , λ ) = ( λ − a ) − (cid:0) A − A ( a − x ) (cid:1) ( a − x )( a + X ) h a + X A − A ( a + X ) i . Thus, the kernel of the linear transformation with matrix L ( x ; a , λ ) is spanned by the vector (cid:2) A ( a + X ) − A a + X (cid:3) T from which we conclude that X = X ′ . Next, for general λ , the productof two Lax matrices takes the form (cid:0) L ( y ; b , λ ) L ( x ; a , λ (cid:1) ij = 1 ρ ( x ) / ρ ( y ) / X k =0 1 X ℓ =0 S ijkℓ ( x , y ; a , b ) λ k Λ ℓ , where we have used that ( λ, Λ) ∈ E . Equating the different powers of λ , Λ, the matrix equation forthe 2-factorization is equivalent to the system of algebraic relations S ijkℓ ( x ′ , y ′ ; a , b ) ρ ( x ′ ; a ) / ρ ( y ′ ; b ) / = S ijkℓ ( x , y ; a , b ) ρ ( x ; a ) / ρ ( y ; b ) / . For ( ijkℓ ) = (1221) , (1211) , (1120) we find that S = − ( X + y ) , S − S = − x ( X + y ) . Hence, x = x ′ and consequently L satisfies the 2-factorization property.Likewise, the product of three Lax matrices reads (cid:0) L ( z ; c , λ ) L ( y ; b , λ ) L ( x ; a , λ (cid:1) ij = P k =0 P ℓ =0 S ijkℓ ( x , y , z ; a , b , c ) λ k Λ ℓ ρ ( x ; a ) / ρ ( y ; b ) / ρ ( z ; c ) / . ijkℓ ) = (1241) , (1221) , (1130) we find that S = ( X + y )( Y + z ) , S − S = x ( X + y )( Y + z ) . From the corresponding ratios we have that x = x ′ , and the number of matrices is reduced by one.The 3-factorization property is reduced to the 2-factorization property, which is satisfied, and themap (55) is a unitary YB map. Two families of YB maps with parameters living on elliptic curves are presented. Both of themare based on the combinatorics and the geometry of a certain type two–field quad–graph system(6-point scheme) that allows to cast them into YB map form. It is this scheme that suggested thelifting of scalar integrable quad–graph equations to two-field ones and subsequently the derivationof their YB form.We end by giving a rough account on YB maps in C × C arising from two–field integrablequad–graph equations. In [27] such YB maps were derived by exploiting the symmetry groups ofthe equations listed in [1]. In the present work it is shown how all discrete equations listed in [1]are casted in YB map form (Proposition 3.3). This list of YB maps is enhanced by “lifting” allintegrable quad–graph equations listed in [5], as it was demonstrated here for the generic equationof the class, namely the discrete KN equation, denoted by Q in [5]. Moreover, the list of YB mapsis enriched by considering the symmetry groups of the lifted discrete equations. Thus, it turns outthat even in this particular case (corresponding to the 6-point scheme) one has already quite anamount of YB maps in C × C and the problem of their classification becomes interesting in orderto: i) find the representatives up to equivalence with respect to some group of transformations andii) make the list exhaustive. Acknowledgements
This work was completed at the Isaac Newton Institute for Mathematical Sciences in Cambridgeduring the programme Discrete Integrable Systems.
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