Let $V$ be a finite dimensional vector space over a field $K$. Then a map $L:V\otimes V\rightarrow V\otimes V$ is said to satisfy the Yang-Baxter equation if $(L\otimes I)(I\otimes L)(L\otimes I)=(I\otimes L)(L\otimes I)(I\otimes L)$ where $I:V\rightarrow V$ is the identity mapping.
Let $T:V\otimes V\rightarrow V\otimes V$ be the mapping defined by $T(x\otimes y)=y\otimes x$.
We say that a linear mapping $L:V\otimes V\rightarrow V\otimes V$ is partially permutative if there exists an $n$ such that for all $m$ we have $L^{n+m}=T^{m}L^{n}$.
What are some examples of partially permutative mappings $L:V\otimes V\rightarrow V\otimes V$ that satisfy the Yang-Baxter equation?
In this question, we are concerned with linear algebraic solutions to the Yang-Baxter equation rather than solutions obtained from “set-theoretic solutions” to the Yang-Baxter equation.
