Evaluation modules of the type-A RTT Yangian

Let $N\ge 2$, let ${\mathfrak{g}\mathfrak{l}}_N(\mathbb{C})=M_N(\mathbb{C})$ with commutator bracket, and use the $1\le i,j\le N$ convention of The type-A RTT Yangian in the rational i-normalization: $E_{ij}$ is the matrix unit at zero-based position $(i-1,j-1)$ in Matrix units $E_{ij}$ and the Kronecker delta. For a left ${\mathfrak{g}\mathfrak{l}}_N$-module $W$ with action $\rho$ and a parameter $a\in \mathbb{C}$, define the candidate evaluation module $W(a)$ by the generating-series action

$t_{ij}(u)⟼{\delta }_{ij}{I}_W-\frac{i\rho (E_{ij})}{u-a},$

where $(u-a{)}^{-1}=u^{-1}{\sum }_{r\ge 0}(a\mathrm{/}u{)}^r$ is expanded at $u=\mathrm{\infty }$. The map defines a Yangian module by The gl_N evaluation formula satisfies RTT ↗. The rational expression has at most a simple pole at $u=a$; the formal series is always defined.

The dual-natural vector space $V_N^{\vee }$ has basis $f^1,\dots ,f^N$ and the candidate action ${\rho }^{\vee }(E_{ij})f^k=-{\delta }_{ik}{f}^j$. The following well-definedness theorem verifies that this is a ${\mathfrak{g}\mathfrak{l}}_N$-module. On it, the auxiliary matrix of the evaluation series is $I+i{P}_N\mathrm{/}(u-a)$, so the unnormalized Lax matrix $(u-a)T_0(u)$ is $R_N(u-a)=(u-a)I+i{P}_N$.