The rational spin-1/2 spectral R-matrix

Let $V={\mathbb{C}}^2$ with ordered basis $e_1,e_2$. Tensor products are over $\mathbb{C}$ and tensor powers are left-associated, as in Scalars, tensor powers, the empty tensor, opposite algebras and finite sums. The elementary tensors $e_a\otimes e_b$ form a basis of $V\otimes V$, and the elementary tensors $e_a\otimes e_b\otimes e_c$ form a basis of $V^{\otimes 3}$, by The elementary tensors of two bases form the product basis of the tensor product.

Define the flip $P$ on $V\otimes V$ by $P(e_a\otimes e_b)=e_b\otimes e_a$ and extend $\mathbb{C}$-linearly. For $u\in \mathbb{C}$ set $R(u)=uI+iP$ and $\check{R}(u)=PR(u)=uP+iI$, where $I$ denotes the identity on $V\otimes V$.

On $V^{\otimes 3}$ define $P_{12}$, $P_{13}$, and $P_{23}$ on the displayed basis by swapping the indicated tensor positions and leaving the third position fixed. Put $R_{ij}(u)=uI+i{P}_{ij}$. Put ${\check{R}}_1(u)=P_{12}{R}_{12}(u)$ and ${\check{R}}_2(u)=P_{23}{R}_{23}(u)$. Products of endomorphisms mean composition, with the rightmost factor applied first.

This is a family of finite-dimensional linear maps. The definition asserts neither invertibility for every u nor a Hopf-algebra or categorical braiding structure.