. Q A ] 9 A pr 2 00 3 LAX MATRICES FOR YANG - BAXTER MAPS

It is shown that for a certain class of Yang-Baxter maps (or set-theoretical solutions to the quantum Yang-Baxter equation) the Lax representation can be derived straight from the map itself. A similar phenomenon for 3D consistent equations on quadgraphs has been recently discovered by A. Bobenko and one of the authors, and by F. Nijhoff. Introduction. In 1990 V.G. Drinfeld suggested the problem of studying the solutions of the quantum Yang-Baxter equation in the case when the vector space V is replaced by an arbitrary set X and tensor product by the direct product of the sets (“set-theoretical solutions to the quantum Yang-Baxter equation”) [1]. In the paper [2] one of the authors investigated the dynamical aspects of this problem and suggested a shorter term “Yang-Baxter map” for such solutions. For each Yang-Baxter map one can introduce the hierarchy of commuting transfermaps which are believed to be integrable (see [2]). In this note we explain how to find Lax representations for a certain class of Yang-Baxter maps thus giving another justification for this conjecture. We were motivated by the explicit examples of the Yang-Baxter maps from [2] and recent results on the equations on quad-graphs, satisfying the so-called “3D consistency condition” [3, 4]. Yang-Baxter maps and their Lax representations. Let X be any set and R be a map: R : X ×X → X ×X. Let Rij : X n → X, X = X × X × ..... × X be the map which acts as R on i-th and j-th factors and identically on the others. Let R21 = PRP , where P : X → X is the permutation: P (x, y) = (y, x). Following [2], we call R the Yang-Baxter map if it satisfies the Yang-Baxter relation (1) R23R13R12 = R12R13R23, considered as the equality of the maps of X ×X ×X into itself. If additionally R satisfies the relation (2) R21R = Id, it is called reversible Yang-Baxter map. Reversibility condition will not play an essential role in this note but it is satisfied in all the examples we present. The standard way to represent the Yang-Baxter relation is given by the diagram in Fig. 1. However we would like to use here also an alternative (dual) way to visualize it, which emphasizes the relation with 3D consistency condition for discrete equations on quad-graphs (see [3, 5]). In this representation the fields (elements of X) are assigned to the edges of elementary quadrilaterals, so that Fig.2 encodes the map R : (x, y) 7→ (x̃, ỹ). Then the Yang-Baxter relation is illustrated as in Fig. 3. 1 2 YURI SURIS AND ALEXANDER VESELOV z y x y1 x2 z1 z12 y13