Simplex equations and their solutions (Q1814269)

Let \(R\in \text{End} (V\otimes V)\), where \(V\) is a finite-dimensional vector space, and let \(R_ i\in \text{End} (V^{\otimes n})\) be \(R\) acting in the \(i\)th and \((i+1)\)st places in the tensor product. The simplex equation is \(R_ 1R_ 2\dots R_ n=R_ n\dots R_ 2R_ 1\). If \(n=3\), this is the quantum Yang-Baxter equation. The authors study solutions of the simplex equation, mainly in the case \(\text{dim} (V)=2\), and also its `classical analogue' (which, in the \(n=3\) case, is the classical Yang-Baxter equation). In the classical case, they find that the parameter space of solutions of the simplex equation is the same as in the \(n=3\) case. It is also shown that, unlike the \(n=3\) case, nondegenerate solutions of the classical simplex equations cannot be quantized if \(n>3\).