Ordered auxiliary-space monodromy

Fix an integer $N\ge 3$. Let $V={\mathbb{C}}^2$ with ordered basis $e_{\uparrow },e_{\downarrow }$, let $V_N=V^{\otimes N}$ in left-to-right site order, and let $V_a=V$ be an auxiliary copy. For $u\in \mathbb{C}$, define $R_{aj}(u)=u{I}_{aj}+i{P}_{aj}$ on $V_a\otimes V_N$, where $P_{aj}$ swaps the auxiliary factor and site $j$. Define $T_a(u)=R_{aN}(u)\cdots R_{a1}(u),t(u)={\mathrm{tr}}_a{T}_a(u),$ where products mean composition with the rightmost factor acting first, and ${\mathrm{tr}}_a$ is the ordinary finite-dimensional trace over $V_a$. For distinct sites $j,k$, $P_{jk}$ swaps those two tensor factors. Define the oriented cyclic shift by $U(v_1\otimes \cdots \otimes v_N)=v_N\otimes v_1\otimes \cdots \otimes v_{N-1}\mathrm{.}$ The spectral parameter $u$ is dimensionless. All operators in this definition are finite-dimensional complex-linear endomorphisms.

The rational spin-1/2 spectral R-matrix

Faddeev, §3, equations (35)–(47)

Classical and Quantum Integrable Systems, Lecture 14, §§14.3–14.5