The Bethe Ansatz

\[ R(u-v) = (u-v) \boldsymbol{1} + c \boldsymbol{P} = \left( \begin{array}{cccc} u - v + c & 0 & 0 & 0 \\ 0 & u - v & c & 0 \\ 0 & c & u - v & 0 \\ 0 & 0 & 0 & u - v + c \end{array} \right) \]

The correspondence with h.r is

\[ R(\lambda) = \left. \frac{1}{u-v+c} R (u-v) \right|^{Slavnov2018-2-2.3}_{u-v = \lambda, ~c \rightarrow i} \]

\[ L_n(u) = R_{0n}(u - \frac{c}{2}) = \left( \begin{array}{cccc} u + \frac{c}{2} & 0 & 0 & 0 \\ 0 & u - \frac{c}{2} & c & 0 \\ 0 & c & u - \frac{c}{2} & 0 \\ 0 & 0 & 0 & u + \frac{c}{2} \end{array} \right) = \left( \begin{array}{cc} u + \frac{c}{2} \sigma^z & c \sigma^- \\ c \sigma^+ & u - \frac{c}{2} \sigma^z \end{array} \right) \]