Set-theoretical reflection equation: Classification of reflection maps a b a 2 V. Caudrelier , N. Cramp´e and Q.C. Zhang 1 0 2 a Centre for Mathematical Science, City University London, Northampton Square, London EC1V 0HB, UK t c O b CNRS, Universit´e Montpellier II, Laboratoire Charles Coulomb, UMR 5221, 8 Place Eug`ene Bataillon - CC070, F-34095 Montpellier, France 1 ] h Email: [email protected], [email protected], [email protected] p - h t a m [ 1 v 7 0 1 5 . 0 Abstract 1 2 1 The set-theoretical reflection equation and its solutions, the reflection maps, recently in- : troduced by two of the authors, is presented in general and then applied in the context of v i quadrirational Yang-Baxter maps. We provide a method for constructing reflection maps and X weobtainaclassification ofsolutionsassociatedtoallthefamiliesofquadrirationalYang-Baxter r a maps that have been classified recently. Keywords: Set-theoretical Yang-Baxter equation, set-theoretical reflection equation, quadrira- tional maps, reflection maps. Introduction The Yang-Baxter equation, introduced to study a gas of spin with a δ interaction [1] or a 2- dimensional solvable statistical model in [2], has been developed in many directions and found applications in different areas of mathematics and physics. Much of the effort has been concen- trated on the so-called quantum Yang-Baxter equation which is at the heart of the field of quantum integrable systems. Although less well-known, the set-theoretical solutions of the Yang-Baxter equa- tionhave alsoattractedsomeattention. This questiongoesbacktoDrinfeld[3](andevento[4]where a solution was given). Then, different aspects have been developed and numerous connections have been established with, for example, the Poisson-Lie groups and algebras [5, 6, 7, 8, 9], the geometric crystals [10, 11], the soliton cellular automatons [12, 13], the scattering of multicomponent solitons [14, 15, 16], the discrete integrable systems [17, 18, 19] and the Lax or transfer matrices [20, 21]. In view of these applications, there is also a lot of effort put into the possibility of classifying the solutions of the set-theoretical Yang-Baxter equations [22, 23]. In the context of quantum integrable systems with boundaries, there exists, in addition to the Yang-Baxter equation, a second equation: the reflection equation. This equation was introduced to encode the reflection on the boundary of particles in quantum field theory [24, 25] (then also applied to different models: principal chiral model [26], Toda field theory [27, 28, 29, 30], nonlinear Schro¨dinger equation [31, 32]) and to prove the integrability of quantum models with boundaries [33]. This equation has been intensively used in the context of the spin chain models (see e.g. [34, 35, 36, 37, 38, 39]) and of statistical models [40, 41, 42, 43, 44] but appears also as a consistency relation for the coordinate Bethe ansatz applied to the δ gas with boundaries [45, 46] and as the defining relations for subalgebras of quantum groups [47, 48, 49] which are the symmetry of quantum models such as the BC Sutherland models [50, 51]. n Motivated by all these fields of application of the set-theoretical Yang-Baxter equation and of the (quantum) reflection equation, we propose in this article to explore the interplay between these two notions and to study the set-theoretical reflection equation. Surprisingly, this equation was introduced only very recently by two of the authors [52] in the context of factorization of vector soliton interactions on the half-line. There, it appeared naturally as the equation ensuring that the interactions of the polarizations of an N-soliton solution among themselves and with the boundary factorises consistently as a sequence of soliton-soliton and soliton-boundary interactions. In the first section of this paper, after recalling the important facts about the set-theoretical Yang-Baxter (YB) equation, we give the definition of the set-theoretical reflection equation and its solutions, the reflection maps. We also give its parametric version and comment on its graphical representation. We discuss, in this context, the construction of the transfer matrix to compare with [20] in the case of the YB equation. The second section is dedicated to the construction of reflection maps. The argument is based on a folding technique. Using this for the set-theoretical reflection equation associated to the YB maps classified in [22, 23], we obtain a classification of solutions. Finally, we conclude with the numerous open problems and perspectives related to our study. 1 Set-theoretical reflection equation We present some elements and facts of the theory of reflection maps and set-theoretical reflection equationfirst introduced at theendofthepaper [52]. Todo so, wefirst need torecall somedefinitions 1 related to YB maps (see e.g. [20]). 1.1 Yang-Baxter maps Let S be a set and R : S ×S → S ×S a map from the Cartesian product of S onto itself, R(X,Y) = (U,V) ≡ (f(X,Y),g(X,Y)). (1.1) Define R : SN → SN as the map acting as R on the ith and jth copies of S in the N-fold Cartesian ij product SN and identically on the others. More precisely, for i < j, Rij(X1,...,Xn) = (X1,...,Xi−1,f(Xi,Xj),...,g(Xi,Xj),Xj+1,...,Xn), (1.2) Rji(X1,...,Xn) = (X1,...,Xi−1,g(Xj,Xi),...,f(Xj,Xi),Xj+1,...,Xn). (1.3) In particular, for N = 2, R ≡ R. If R satisfies the following Yang-Baxter relation 12 R R R = R R R , (1.4) 12 13 23 23 13 12 as an identity on S ×S ×S, then R is called a Yang-Baxter map. In addition, if R satisfies R R = Id, (1.5) 21 12 then R is called a reversible Yang-Baxter map. It is useful to introduce the so-called parametric Yang-Baxter map R(a,b) which is an important special case obtained by considering S ×Σ, where Σ is some parameter set (usually C), instead of only S as before. The corresponding map R : (X,a;Y,b) 7→ (f(X,a;Y,b),a;g(X,a;Y,b),b) ≡ (f (X,Y),a;g (X,Y),b), (1.6) ab ab is conveniently written as R(a,b) and seen as acting only on S ×S: R(a,b) : (X,Y) 7→ (f (X,Y),g (X,Y)). (1.7) ab ab It satisfies the parametric YB equation R (a,b)R (a,c)R (b,c) = R (b,c)R (a,c)R (a,b), (1.8) 12 13 23 23 13 12 and the corresponding reversibility condition reads R (b,a)R (a,b) = Id. (1.9) 21 12 The notion of parametric maps is important, for example, for YB maps arising from the study of vector or matrix soliton solutions of certain classical integrable nonlinear equations [14, 15, 16]. There is a well-known pictorial description of YB maps (see e.g. [22]). A map R defined by (1.6) is associated to a quadrilateral as in Fig. 1. Then, the parametric YB equation (1.8) can be understood as the consistency condition ensuring that the two possible orders of action of the map R on (X,a;Y,b;Z,c), described by both sides of the equation shown in Fig.2, provide the same result. Let us also remark the similarities between this graphical representation of the set-theoretical YB equation and the so-called face representation of the YB equation [40, 41, 42, 43]. 2 (X,a) R (V,b) (Y,b) (U,a) Figure 1: Map R (X,a) (X,a) R 13 R 12 R 23 (Y,b) (Y,b) R R 12 23 R 13 (Z,c) (Z,c) Figure 2: Graphical representation of the set-theoretical Yang-Baxter equation 1.2 Reflection maps Based on the form of the quantum reflection equation [24, 33] and on the important role played by its set-theoretical version in [52], we now introduce the following general definition of a reflection map. Definition 1.1 Given a Yang-Baxter map R, a reflection map B is a solution of the set-theoretical reflection equation B R B R = R B R B , (1.10) 1 21 2 12 21 2 12 1 as an identity on S ×S. The reflection map is called involutive if BB = Id. (1.11) To introduce the notion of parametric reflection map, we need some extension of the above discussion for parametric YB maps in order to encompass known situations. Indeed, a parametric YB map has a trivial action on the parameter set Σ but it is known from the solutions found in [52] for reflection maps that a parametric reflection map can in general have a nontrivial action on the parameter set. 3 In general this reads B : (X,a) 7→ (h (X),σ(a)) , X ∈ S , a ∈ Σ, (1.12) a for some maps h and σ. One can then use the convenient notation B(a) for a parametric reflection a map but one has to keep in mind the nontrivial action of σ when composing a reflection map with other maps. To illustrate this, we give the definition of a parametric reflection map as follows: Definition 1.2 Given a parametric Yang-Baxter map R (a,b), a,b ∈ Σ and a map σ : Σ → Σ, a 12 parametric reflection map B(a) is a solution of the parametric set-theoretical reflection equation B (a)R (σ(b),a)B (b)R (a,b) = R (σ(b),σ(a))B (b)R (σ(a),b)B (a) (1.13) 1 21 2 12 21 2 12 1 as an identity on S ×S. The reflection map is called involutive if σ is an involution in Σ and B(σ(a))B(a) = Id. (1.14) Note that classes of parametric reflection maps were constructed in [52] for S = CPn−1, Σ = C and σ(k) = −k∗ with R (k ,k ) being the parametric YB map corresponding to the vector nonlinear 12 1 2 Schro¨dinger equation. Analogous to the quadrilateral description of a YB map, we introduce a pictorial description of a reflection map as a half-quadrilateral shown in Fig. 3. The set-theoretical reflection equation (X,a) B (h (X),σ(a)) a Figure 3: Map B can then be depicted as in Fig. 4. Let us remark that this visualization is reminiscent of the face representation of the reflection equation in [40, 41, 42, 43]. There are also similitudes between Fig. 4 and half of the figure representing the tetrahedron equation for interaction round-a-cube [53, 54]. Explicitly, the left-hand side represents the following chain of maps R B R B (X,a;Y,b) −−→12 (X ,a;Y ,b) −−→2 (X ,a;Y ,σ(b)) −−→21 (X ,a;Y ,σ(b)) −−→1 (X ,σ(a);Y ,σ(b)) 1 1 1 2 2 3 3 3 with X = f (X,Y) , Y = g (X,Y) , Y = h (Y ), 1 ab 1 ab 2 b 1 X = g (Y ,X ) , Y = f (Y ,X ) , X = h (X ). 2 σ(b)a 2 1 3 σ(b)a 2 1 3 a 2 Similarly, the right-hand side means B R B R (X,a;Y,b) −−→1 (X′,σ(a);Y,b) −−→12 (X′,σ(a);Y′,b) −−→2 (X′,σ(a);Y′,σ(b)) −−→21 (X ,σ(a);Y ,σ(b)) 1 2 1 2 2 3 3 4 (X,a) (X,a) (Y ,b) 1 (X’, σ (a)) 1 (Y,b) (Y,b) (Y’,b) 1 (X ,a) 1 (X’, σ (a)) 2 (Y , σ (b)) 2 (Y , σ (b)) 3 (Y , σ (b)) 3 (X 2,a) (Y’,σ (b)) 2 (X , σ (a)) (X , σ (a)) 3 3 Figure 4: Graphical representation of the reflection equation with ′ ′ ′ ′ ′ X = h (X) , X = f (X ,Y) , Y = g (X ,Y), 1 a 2 σ(a)b 1 1 σ(a)b 1 ′ ′ ′ ′ ′ ′ Y = h (Y ) , X = g (Y ,X ) , Y = f (Y ,X ). 2 b 1 3 σ(b)σ(a) 2 2 3 σ(b)σ(a) 2 2 So, Eq. (1.13) ensures that the two ways of obtaining (X ,Y ) from (X,Y) lead to the same result. 3 3 1.3 Transfer maps We conclude this presentation on the theory of reflection maps by defining the notion of transfer maps in analogy with that introduced in [20]. Fix N ≥ 2 and define for j = 1,...,N the following maps of SN into itself, Tj = Rj+1j...RNjB−j RjN ...Rjj+1Rjj−1...Rj1B+j R1j...Rj−1j, (1.15) where B+ is a solution of B R B R = R B R B , (1.16) 1 21 2 12 21 2 12 1 and B− a solution of B R B R = R B R B . (1.17) 1 12 2 21 12 2 21 1 Then one proves by direct (but long) calculation the following result 5 Proposition 1.3 For any reversible Yang-Baxter map R, the transfer maps (1.15) commute with each other T T = T T , j,ℓ = 1,...,N . (1.18) j ℓ ℓ j Of course, this result transfers immediately to the case of parametric maps. 2 Reflection maps for quadrirational Yang-Baxter maps In this section, we provide a classification of solutions for reflection maps associated to the 10 fam- ilies of quadrirational YB maps found in [22, 23]. To achieve this, we consider a certain class of functions of rational type and use an adaptation of the folding argument used in [52]. This is further combined with a duality property which we derive from the particular properties of quadrirational maps exhibited in [22]. The overall amount of work required is reduced by applying the notion of symmetry as explained below. 2.1 Symmetry and quadrirational Yang-Baxter maps We first need the following proposition shown in [23]. Proposition 2.1 Let s(a) be an involutive symmetry of the parametric Yang-Baxter map R(a,b) i.e. s(a)2 = Id and s(a)×s(b) R(a,b) = R(a,b) s(a)×s(b) . (2.1) The map defined by Rs(a,b) = (cid:0)s(a)×Id(cid:1) R(a,b) (cid:0)Id×s(b)(cid:1) (2.2) is a Yang-Baxter map. Such symmetry maps s(a) allow one to relate YB maps that look different a priori. This is in particular the case forquadrirational YBmaps i.e.maps of a certain rationalformacting onS = CP1 with parameter set Σ = C: the F families and the H families (in the notations of [22, 23]) are related by such symmetry maps. In general, in the context of YB maps, different maps may indeed be related to actual different underlying systems. But in the context of reflection maps, the following proposition shows that symmetries leave the set of solutions invariant. Proposition 2.2 The map B(a) is a parametric reflection map for the parametric Yang-Baxter map R(a,b) if and only if the map B(a) is a parametric reflection map for the parametric Yang-Baxter map Rs(a,b). Proof: The proof of this proposition consists in writing the set-theoretical reflection equation for Rs(a,b) expressed thanks to (2.2). Then, remarking for example that (cid:0)Id×s(b)(cid:1)K (a) = K (a)(cid:0)Id× 1 1 s(b)(cid:1) and using that s(a)2 = Id, one proves that this equation is equivalent to the set-theoretical reflection equation for R(a,b). OurpropositionthereforeimpliesthatthereflectionmapsfortheF toF familiesarethesameas I V the reflection maps for the corresponding families H −H 1. We can therefore restrict our attention I V 6 Type f (X,Y) g (X,Y) P ab ab F aYP bXP (1−b)X+b−a+(a−1)Y I b(1−a)X+(a−b)XY+a(b−1)Y F Y P XP aX−bY+b−a II a b X−Y F Y P XP aX−bY III a b X−Y F YP XP 1+ b−a IV X−Y F Y +P X +P a−b V X−Y Table 1: Families F for the quadrirational map to the F families, which we reproduce in Table 1 for completeness. Finally, since these families involve parameters, we will work directly with parametric maps. 2.2 Method We propose an adaptation of the folding method that was used in [52] to obtain families of reflection maps from the YB map associated to the vector nonlinear Schro¨dinger equation. Roughly speaking, the idea is to see a reflection map as a YB map modulo some folding applied on the variables on which the YB map acts. More precisely, here we start from Fig. 1 and impose that b = σ(a) for some map σ and that Y = ϕ (X) and V = ϕ (U) for some map ϕ depending on a. For convenience, we a a a assume that σ is an involution as well as ϕ , the latter meaning that a ϕ (ϕ (X)) = X. (2.3) σ(a) a Now, since (U,V) is related to (X,Y) by the YB map R(a,b), we obtain two possibilities for ex- pressing V in terms of X V = ϕ (cid:0)f (X,ϕ (X))(cid:1) = g (X,ϕ (X)). (2.4) a aσ(a) a aσ(a) a This provides a functional equation constraining admissible functions ϕ and σ. So, for each such a admissible pair (ϕ ,σ), there is a well-defined map between V and X which we denote h a a V = h (X), (2.5) a and which can be computed using (2.4). The map h together with σ are our candidates to get a a reflection map B as in (1.12). 1Withthe understandingthatFIII, HIAII andHIBII sharethesamesolutionsandrecallingthatHIV doesnotexist. 7 To complete our set of functional equations, we can also look at the folding method from the point of view of the inverse map R−1(a,b) which maps (U,V) to (X,Y) = (g (V,U),f (V,U)) ba ba (due to the reversibility of R) and the companion map R(b,a) which maps (V,X) to (Y,U) ≡ (f¯ (V,X),g¯ (V,X))2. From the folding with the inverse map, we get the constraints ba ba Y = ϕ (cid:0)g (ϕ (U),U)(cid:1) = f (ϕ (U),U). (2.6) a σ(a)a a σ(a)a a Notice that, at first glance, this provides a new constraint for ϕ and σ. However, for the quadri- rational maps we consider in this paper, listed in Table 1, we have R (a,b) = R (b,a) i.e. 12 21 f (X,Y) = g (Y,X). Therefore, the constraint given by (2.6) is equivalent to (2.4). Now, pro- ab ba vided that admissible functions (ϕ ,σ) and the related h are found, we deduce that Y = h (U). a a a Using this last result and the folding for the companion map R(b,a), we get Y = h (cid:0)g¯ (h (X),X)(cid:1) = f¯ (h (X),X). (2.7) a σ(a)a a σ(a)a a The quadrirational maps considered in this paper satisfy the following additional property f¯ (X,Y) = g (Y,X), g¯ (X,Y) = f (Y,X). (2.8) ab ba ab ba Taking account of these properties, one gets Y = h (cid:0)f (X,h (X))(cid:1) = g (X,h (X)). (2.9) a aσ(a) a aσ(a) a Comparing (2.4) and (2.9), we observe that ϕ and h play completely dual roles. Therefore, if ϕ is a a a chosen to be of a certain form, then h should be of the same form and vice versa. This is reminiscent a of the properties derived in [22] concerning the quadrirationality of a YB map, its companion map and their inverses. Also, the involution property (2.3) should hold for h . a Combining the ideas of the folding of YB map and the duality between ϕ and h , we are able a a to classify solutions of the set-theoretical reflection equation for the 5 families of quadrirational YB maps presented in [22]. In our context, it is natural to choose ϕ and h to be both M¨obius maps a a with coefficients being first order polynomials in a: p (a)X +p (a) q (a)X +q (a) ϕ (X) = 1 2 , h (X) = 1 2 , (2.10) a a p (a)X +p (a) q (a)X +q (a) 3 4 3 4 where p (a) = p0 +p1a and p (a) = q0 +q1a. This corresponds to 16 free parameters a priori. We j j j j j j also choose σ to be an involutive M¨obius map, which gives 3 additional parameters: c a+c σ(a) = 1 2 . (2.11) c a−c 3 1 We then solve for these 19 parameters by inserting into the following constraints as explained above h (X) = g (X,ϕ (X)), (2.12) a aσ(a) a ϕ (ϕ (X)) = X, h (h (X)) = X . (2.13) σ(a) a σ(a) a The resulting solutions are inserted in the set-theoretical reflection equation to check their validity. 2 Notethatconventionsforthesemapsmayvaryinthe literaturebutresultsareunaffectedbysuchvariationssince all four maps R, R−1, R, R−1 coincide as shown in [22]. 8 2.3 Classification It is straightforward to check that the identity map (i.e. both h and σ are the identity map) is a a solution for all families so in the following, we only present nontrivial solutions, by which we mean that either σ or h or both are not the identity map. With the exception of one solution for the a F solution, all the maps satisfying (2.12) and (2.13) are reflection maps as we found by inspection. I Note that these operations have been performed with the help of symbolic computation softwares. The final results are reported in Table 2. Note that we did not find any solution for the F family. V The duality property is explicitly shown in the table: the roles of h and ϕ can be swapped. All a a solutions depend on an arbitrary parameter µ. In view of the results in [52], this could be interpreted as the parameter controlling the boundary and, accordingly, the integrable boundary conditions of the underlying integrable systems associated to the quadrirational YB maps of the F families. Type σ(a) ϕ (X) (resp. h (X) ) h (X) (resp. ϕ (X) ) a a a a µ2 µX (X(1+µ)−µ−a)µ a a X(a+µ)−(1+µ)a F a+µ2−1 (a+µ2−1)X a+µ−Xµ−1 I a−1 X(a−µ−1)+aµ a−1 1−a (a−1)X µa+1−Xµ−a a(µ2−1)+1 X(a+µa−1)−aµ 1+a(µ2−1) µ2 a(X −1)+1 −aX a µ µ F II −a+2µ X a−µ−aX 2X−1 a−2µ F µ2 aX −aX III a µ µ F −a+2µ −X X −a+µ IV Table 2: Reflection maps for quadrirational map F −F I IV In the geometric construction of the quadrirational maps F −F in [22], the so-called singular I V points play an important role. It turns out that we can make use of these points in our construction to obtain other reflection maps which we call degenerate. This is achieved by allowing ϕ (X) in (2.4) a to take on the singular values of the corresponding family. In general, this can be 0,1,∞,σ(a). Each time a degenerate case is solution, the reflection equation is satisfied for any map σ. The duality 9