On the set-theoretical Yang-Baxter equation
Let \(G\) be a group with a left and a right action on itself. The left (respectively right) action of \(u\in G\) on \(v\in G\) is denoted by \(u\to v\) (respectively \(v\gets u\)). The main result tells that if the compatibility condition \(uv=(u\to v)(u\gets v)\) holds for any \(u,v\in G\), then \(R\colon G\times G\to G\times G\), \(R(u,v)=(u\gets v,u\to v)\) is an invertible solution of the set-theoretical quantum Yang-Baxter equation on the set \(G\). Two conceptual reformulations of the above conditions on \(G\) are given by using braiding operators and bijective 1-cocycles. It is shown that this construction generalizes the ones of \textit{A. Weinstein} and \textit{P. Xu} [Commun. Math. Phys. 148, No. 2, 309-343 (1992; Zbl 0849.17015)] and \textit{P. Etingof, T. Schedler} and \textit{A. Soloviev} [Duke. Math. J. 100, No. 2, 169-209 (1999)]. A universal construction for non-degenerate solutions of the set-theoretical quantum Yang-Baxter equation is presented.
scientific article; zbMATH DE number 1501663