Evaluation module at a nonzero loop parameter

Let $a\in {\mathbb{C}}^{\times }$ and let $\rho :\mathfrak{g}\to {\mathrm{End}}_{\mathbb{C}}(V)$ be a finite-dimensional representation. The evaluation module at $a$ for Untwisted affine central extension has action ${\rho }_a(x\otimes f+bc)=f(a)\rho (x)\mathrm{.}$ In particular $x_m$ acts as $a^m\rho (x)$ and $c$ acts as zero. Evaluation on the loop algebra is a Lie homomorphism because $(fq)(a)=f(a)q(a)$, so $[f(a)\rho (x),q(a)\rho (y)]=(fq)(a)\rho ([x,y])$. The central term is killed by the stipulated zero action of $c$, establishing the representation identity for the central extension too. This does not mean that the scalar cocycle itself vanishes. Nonzero $a$ is required because $t^{-1}$ must be evaluated and $t{t}^{-1}=1$. The zero representation and the zero-dimensional module are allowed. No action of the degree derivation is part of this definition.