A decomposition theorem for square-free unitary solutions of the quantum Yang-Baxter equation

It is known that every skew-polynomial ring with generating set X and binomial relations in the sense of Gateva-lvanova (Trans. Amer. Math. Soc. 343 (1994) 203) is an Artin-Schelter regular domain of global dimension vertical bar X vertical bar. Moreover, every such ring gives rise to a nondegenerate unitary set-theoretical solution R : X-2 -> X-2 of the quantum Yang-Baxter equation which fixes the diagonal of X-2. Gateva-Ivanova's conjecture (Talk at the International Algebra Conference, Miskolc, Hungary, 1996) states that conversely, every such solution R comes from a skew-polynomial ring with binomial relations. An equivalent conjecture (Duke Math. J. 100 (1999) 169) says that the underlying set X is R-decomposable. We prove these conjectures and construct an indecomposable solution R with vertical bar X vertical bar = infinity which shows that an extension to infinite X is false. (c) 2004 Elsevier Inc. All rights reserved.