From the text: Let \(X\) be a topological space. We let \(T\) denote the transformation of the space \(X\times X\) to itself that transposes coordinates (that is \(T(x,y)=(y,x))\). We say that the transformation \(R:X\times X\to X\times X\) satisfies the quantum Yang-Baxter equation if \[ R_{12} R_{13} R_{23} = R_{23} R_{13} R_{12} : X \times X \times X\to X \times X \times X,\tag{1} \] where \(R_{12} = R\times \text{id}\), \(R_{23} = \text{id}\times R\) and \(R_{13} = T_{12} R_{23} T_{12}\).
Definition. A transformation \(F: X\times X\to X\times X\) is a Yang-Baxter transformation if it satisfies the equation \[ F_{12} F_{23} F_{12} = F_{23} F_{12} F_{23}: X\times X\times X\to X\times X\times X.\tag{2} \] The author considers in more detail the case when \(X\) is the Euclidean space \(\mathbb{R}^n\) and \(F\) is a linear transformation mapping \(\mathbb{R}^n\times \mathbb{R}^n\) into itself. Without loss of generality one can fix a basis in \(\mathbb{R}^n\) and express \(F\) as a \((2\times 2)\) block matrix \(\binom{A\quad B}{C\quad D}\), where the blocks are \((n\times n)\) matrices. Let \(M(n)\) denote the algebra of all such matrices. These will be written in the form \((A,D;B,C)\). Let YB\((n)\) denote the subset of matrices in \(M(n)\) that define Yang-Baxter transformations.
Theorem 1. Let \((A, D; B, C)\in \text{YB}(n)\). Then in the algebra \(M(n)\) the following relation holds: \((A,D;B,C)(A,D;DB,AC)=(A,D; DB,AC)\).
Corollary. Let \(F = (A, D; B,C)\in\text{YB}(n)\). If \((A, D;B, C)\neq (E, E; 0,0)\), then the determinant of the matrix \((A, D; DB, AC)\) vanishes. Thus, the identity transformation \(I\in \text{YB}(n)\) is an isolated point in the topology induced by the embedding \(\text{YB}(n)\subset M(n)\). Let \(V = \{(x,y)\in \mathbb{R}^n\times \mathbb{R}^n\mid F(x,y) = (x,y)\}\) be the subspace of fixed points. Then \(\dim V\geq \max(\text{rank} A, \text{rank }D)\).