$\mathcal R$-Matrix and Baxter $\mathcal Q$-Operators for the Noncompact $\mathrm{SL}(N,\mathbb C)$ Invariant Spin Chain

The problem of constructing the $SL(N,\mathbb C)$ invariant solutions to the Yang–Baxter equation is considered. The solutions ($\mathcal R$-operators) for arbitrarily principal series representations of $\mathrm{SL}(N,\mathbb C)$ are obtained in an explicit form. We construct the commutative family of the operators $\mathcal Q_k(u)$ which can be identified with the Baxter operators for the noncompact $\mathrm{SL}(N,\mathbb C)$ spin magnet.