The generalized Gaudin algebra
Let us take a vector space spanned by \(\require{physics} \set{S^x_u, S^y_u, S^z_u}\), where \(S^\kappa_u \equiv S^\kappa(u)\) and \(u \in \mathbb{C}\). The generalized Gaudin algebra is characterized by the commutation relations (\(u \neq v\)) \[\begin{align} & \comm{S^\kappa_u}{S^\kappa_v} = 0 \label{eq:gga_commutators_first} \\ & \comm{S^x_u}{S^y_v} = i \qty( Y_{uv} S^z_u - X_{uv} S^z_v ) \\ & \comm{S^y_u}{S^z_v} = i \qty( Z_{uv} S^x_u - Y_{uv} S^x_v ) \\ & \comm{S^z_u}{S^x_v} = i \qty( X_{uv} S^y_u - Z_{uv} S^y_v ) \label{eq:gga_commutators_last} \thinspace , \end{align}\] in which \(\kappa=x,y,z\), \(X_{uv} \equiv X(u,v)\), \(Y_{uv} \equiv Y(u,v)\) and \(Z_{uv} \equiv Z(u,v)\) are antisymmetric, i.e. \[\begin{equation} X(v, u) = - X(u, v) \end{equation}\] Furthermore, for the commutation relations \(\eqref{eq:gga_commutators_first}\) to \(\eqref{eq:gga_commutators_last}\) to be consistent with the Jacobi identity \(\comm{S^x_u}{\comm{S^x_v}{S^y_w}}\), the Yang-Baxter equation \[\begin{equation} X(u, v) Y(v, w) + Y(w, u) Z(u, v) + Z(v, w) X(w, u) = 0 \end{equation}\] has to hold. Gaudin (Gaudin 1976) found solutions to this equation, and we will focus on the so-called \(XXX\)-case, in which \[\begin{equation} X(u, v) = Y(u, v) = Z(u, v) = \frac{1}{u - v} \thinspace , \end{equation}\] such that the \(XXX\)-Gaudin algebra reduces to (\(u \neq v\)) \[\begin{align} & \comm{S^\kappa_u}{S^\kappa_v} = 0 \\ & \comm{S^x_u}{S^y_v} = \frac{i}{u-v} \qty(S^z_u - S^z_v) \\ & \comm{S^y_u}{S^z_v} = \frac{i}{u-v} \qty(S^x_u - S^x_v) \\ & \comm{S^z_u}{S^x_v} = \frac{i}{u-v} \qty(S^y_u - S^y_v) \thinspace , \end{align}\] In this case, it is a good idea to work in the basis \[\begin{align} & S^+_u = S^x_u + i S^y_u \\ & S^-_u = S^x_u - i S^y_u \\ & S^z_u = S^z_u \thinspace , \end{align}\] with the commutators becoming (\(u \neq v\)) \[\begin{align} & \comm{S^+_u}{S^+_v} = \comm{S^-_u}{S^-_v} = \comm{S^z_u}{S^z_v} = 0 \label{eq:gaudin_commutators_pp} \\ & \comm{S^+_u}{S^-_v} = \frac{2}{u-v} \qty(S^z_u - S^z_v) \label{eq:gaudin_commutators_pm} \\ & \comm{S^z_u}{S^\pm_v} = \frac{\pm}{u - v} \qty(S^\pm_u - S^\pm_v) \label{eq:gaudin_commutators_zpm} \thinspace . \end{align}\]
So far, the form of the generators of the Gaudin algebra has not been specified. To make a specification, we will use \(N\) copies of an \(\mathfrak{su}(2)\)-algebra, each carrying a real number \(\varepsilon_i\), and we will choose in particular \[\begin{align} & S^+_u = \sum_i^N \frac{T^+_i}{u - \varepsilon_i} \\ & S^-_u = \sum_i^N \frac{T^-_i}{u - \varepsilon_i} \\ & S^z_u = \frac{1}{g} - \sum_i^N \frac{T^z_i}{u - \varepsilon_i} \thinspace , \end{align}\] in which \(g\) is a real number. Using the \(\mathfrak{su}(2)\)-based realization of the Gaudin algebra, we can indeed show that the commutator relations \(\eqref{eq:gaudin_commutators_pp}\) to \(\eqref{eq:gaudin_commutators_zpm}\) hold. Let us also calculate \(S^2_u\): \[\begin{equation} S^2_u = \frac{1}{g^2} - \frac{2}{g} \sum_i^N \frac{T^z_i}{u - \varepsilon_i} + \frac{1}{2} \sum_{ij}^N \frac{T^+_i T^-_j + T^-_i T^+_j + 2 T^z_i T^z_j}{(u - \varepsilon_i)(u - \varepsilon_j)} \thinspace . \end{equation}\] We can see that \(S^2_u\) has a single pole when \(u \to \varepsilon_k\), in one of the terms, and a double pole when \(u \to \varepsilon_k\) in another term. The corresponding residue is calculated as \[\begin{equation} R_k = - \frac{2}{g} T^z_k + \sum_{i \neq k}^N \frac{T^+_i T^-_k + T^-_i T^+_k + 2 T^z_i T^z_k}{\varepsilon_k - \varepsilon_i} \thinspace . \end{equation}\] Paul Johnson (Paul Andrew Johnson 2014) suggests rescaling \(g\) such that \[\begin{equation} R_k = T^z_k - g \sum_{i \neq k}^N \frac{T^+_i T^-_k + T^-_i T^+_k + 2 T^z_i T^z_k}{\varepsilon_k - \varepsilon_i} \thinspace . \end{equation}\] A specific linear combination (Paul Andrew Johnson 2014) of the residues leading to \[\begin{equation} H_{\text{RG}} = \sum_i^N \varepsilon_i T^z_i - g \sum_{ij}^N T^+_i T^-_j \thinspace , \end{equation}\] up to a constant, where the subscript RG stands for Richardson-Gaudin.