A family of solutions of the Yang-Baxter equation. (Q403608)

The authors study involutive non-degenerate set-theoretic solutions of the Yang-Baxter equation (YBE). To each such solution \(R\colon X^2\to X^2\), there is an associated structure group and a certain factor group \(G(X)=G(X,R)\) isomorphic to the adjoint group of the associated brace. The finite groups of the form \(G(X)\) are solvable and called involutive Yang-Baxter groups, abbreviated IYB groups. The problem to determine the IYB groups and the corresponding solutions of the YBE is widely open and a challenge in finite group theory. In the paper under review, the authors present a construction from a given solution \(R\) on \(X\) to a solution on \(R^n\) on \(X^n\) such that \(G(X^n,R^n)\) embeds as a subgroup into \(G(X,R)\). They provide sufficient criteria for this embedding to be an isomorphism.