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-Ivanova (Trans. Amer. Math. Soc. 343 (1994) 203) is an Artin-Schelter regular domain of global dimension |X|. Moreover, every such ring gives rise to a non-degenerate unitary set-theoretical solution R:X2→X2 of the quantum Yang-Baxter equation which fixes the diagonal of X2. 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 |X|=∞ which shows that an extension to infinite X is false.