Extensions of set-theoretic solutions of the Yang-Baxter equation and a conjecture of Gateva-Ivanova.
For a vector space \(V\) over a field \(K\), solutions to the quantum Yang-Baxter equation (QYBE) form a class of linear maps \(R\colon V\otimes V\to V\otimes V\), including the identity map. Many solutions are obtained as deformations of the identity map, hence \textit{V. G. Drinfel'd}'s 1990 [Lect. Notes Math. 1510, 1-8 (1992; Zbl 0765.17014)] suggestion to consider set-theoretic solutions, that is, solutions induced by linear extension from a map \(X\times X\to X\times X\), where \(X\) is a basis of \(V\). For finite \(X\), the non-degenerate unitary set-theoretic solutions are equivalent to a binary operation \(\cdot\) on \(X\) such that the left multiplications \(\sigma_x(y):=x\cdot y\) are invertible and the equation \[ (x\cdot y)\cdot(x\cdot z)=(y\cdot x)\cdot(y\cdot z) \] holds in \(X\). Such a set \((X;\cdot)\) is called a cycle set. Every cycle set \(X\) gives rise to a cycle set \(\sigma(X):=\{\sigma_x\mid x\in X\}\) with a well-defined operation \(\sigma_x\cdot\sigma_y:=\sigma_{x\cdot y}\), the retraction of \(X\). A cycle set satisfying \(x\cdot x=x\) is said to be square-free. Such (finite) cycle sets \(X\) are in one-to-one correspondence with an important class of non-commutative Noetherian regular \(K\)-algebras with basis \(X\). \textit{T. Gateva-Ivanova}'s ``strong conjecture'' [J. Math. Phys. 45, No. 10, 3828-3858 (2004; Zbl 1065.16037)] states that for any finite square-free cycle set \(X\), an iterated retraction \(\sigma^n(X)\) must be a singleton. The paper under review provides the first counterexample to this conjecture, a cycle set of order 8. This example is contained in a computer list of cycle sets up to order eight, obtained in connection with the paper of \textit{P. Etingof} et al. [Duke Math. J. 100, No. 2, 169-209 (1999; Zbl 0969.81030)] on set-theoretic solutions of the QYBE. Incidentally, it is the unique counterexample in this list. It can be obtained as an iterated ``double'' of the two-element square-free cycle set. From a given cycle set \(X\), the author considers cycle sets \(Y\) with a surjective morphism \(Y\twoheadrightarrow X\) and fibers of equal size. Such ``covering maps'' are obtained by ``dynamical cocycles''. Semidirect products of cycle sets form a special case. Another class of dynamical cocycles is related to the homology theory of \textit{J. S. Carter} et al. [Fundam. Math. 184, 31-54 (2004; Zbl 1067.57006)].
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