Finite periodic fundamental evaluation chain and transfer matrix

Fix $N\ge 2$, a chain length $m\ge 1$, and inhomogeneities $a_1,\dots ,a_m\in \mathbb{C}$. The quantum space is the finite-dimensional vector space $\mathcal{H}=(V_N^{\vee }{)}^{\otimes m}$, where each factor is the dual-natural evaluation module with its own parameter $a_j$ from Evaluation modules of the type-A RTT Yangian; well-definedness is supplied by The gl_N evaluation formula satisfies RTT. Let $V_0=V_N$ be an auxiliary copy. Identify the ordered bases $e_i$ and $f^i$ when defining $P_{0j}(e_p\otimes f^q)=e_q\otimes f^p$, with all other quantum factors unchanged, and put

$L_{0j}(u)=(u-a_j)I+i{P}_{0j}=R_{0j}(u-a_j),M_0(u)=L_{01}(u)L_{02}(u)\cdots L_{0m}(u)\mathrm{.}$

Products are composition with the rightmost factor applied first. The un-twisted periodic transfer matrix is the polynomial $\tau (u)={\mathrm{tr}}_0{M}_0(u)\in {\mathrm{End}}_{\mathbb{C}}(\mathcal{H})[u]$, where ${\mathrm{tr}}_0$ is the ordinary matrix trace over the auxiliary basis and does not act on $\mathcal{H}$. This definition asserts neither Bethe completeness nor Yangian invariance of a periodic Hamiltonian.