The models at the orbifold points, with hamiltonians (7.83), belong to a family of generalized six-vertex models with lattice variables $\sigma$ in either a "singlet" $(\sigma=0)$ or a multiplet $(\sigma=1,2, \ldots, \mathcal{N})$ transforming as the fundamental irrep of $U(\mathscr{N})$. The non-vanishing Boltzmann weights are
$$
\begin{aligned}
\mathscr{R}_{00}^{00}(u) & =\mathscr{R}_{a b}^{b a}(u)
\end{aligned}=\sinh (u+i \gamma) \quad a, b \in\{1,2, \ldots, \mathscr{N}\},
$$
When $\mathcal{N}=1$, this becomes the $X X Z$ model. For $\mathscr{N}=2$ and $\gamma=\pi / 2$, we recover from this $\mathscr{R}$-matrix the hamiltonian (7.83) without the term $\sum \sigma_j^0$, which can be added by hand anyway since it commutes with all the other terms of the hamiltonian. Show that the hamiltonian $H$ associated with the $\mathscr{R}$-matrix (7.206), which acts on the space $\otimes^L \mathbb{C}^{I+N}$, is given by
$$
H_{N \Delta}=\frac{1}{\sin \gamma} \sum_{j=1}^L\left[\sum_{a=1}^N\left(\tau_{a, j}^{+} \tau_{a, j+1}^{-}+\tau_{a, j}^{-} \tau_{a, j+1}^{+}\right)+\Delta\left(\tau_j^z \tau_{j+1}^z-1\right)\right]
$$
where $\tau_a^{ \pm}$are the Pauli matrices $\sigma^{ \pm}$acting on the labels 0 and $a$ :
$$
\left(\tau_a^{+}\right)_{m n}=\delta_m^0 \delta_n^a, \quad \tau_a^{-}=\left(\tau_a^{+}\right)^{\dagger}
$$
and $\tau^z$ is a "global" $\sigma^z$ Pauli matrix
$$
\left(\tau^{\Sigma}\right)_{m n}=2 \delta_m^0 \delta_n^0-\delta_m^n
$$
not to be confused with $\tau_a^z=\left[\tau_a^{+}, \tau_a^{-}\right]$. Show that $\sum \tau_j^z$ commutes with $H_{+}, \Delta$, and that it can be added to the hamiltonian as the contribution from an external homogeneous magnetic field.