P. Etingof, T. Schedler, A. Soloviev, Set-theoretical solutions to the quantum Yang-Baxter equation
In this interesting and nicely written paper the authors study solutions R:V⊗V→V⊗V of the quantum Yang-Baxter equation which come from a map S:X×X→X×X, X a basis of V. Constructions of such solutions were given by A. D. Weinstein and P. Xu [Comm. Math. Phys. 148 (1992), no. 2, 309–343; MR1178147 (93k:58102)] and by J.-H. Lu, M. Yan and Y.-C. Zhu [“Positive Hopf algebra and set-theoretical Yang-Baxter equation”, to appear]. In the paper under review, the authors study solutions with additional conditions of nondegeneracy of S and invertibility (S2= identity map of X2≡X×X). For purposes of this review, such an (X,S) will be called a solution. In this case, the symmetric group Sn acts on Xn by a twisted action. This action is in general different from the usual action, but the two actions are conjugate. A structure group GX is constructed, which has two actions on Xwhich are conjugate to each other. This group is extensively studied, in particular as a subgroup of Aut(X)×ZX. It is shown that for X=Z/pZ, p a prime, there is a unique (up to isomorphism) indecomposable solution. In general, GX is a solvable group. The quantum algebras arising from R are studied. One section of the paper is devoted to various sources for constructions of solutions. They are, roughly speaking, linear, affine, multipermutational, twisted unions and generalized twisted unions. Such solutions are classified and their properties studied. A computer calculation found all solutions of order at most eight. A final section considers generalizing linear solutions of the quantum Yang-Baxter equation to power series solutions.
For more information: http://projecteuclid.org/euclid.dmj/1077227351