The Fock space node formulas satisfy the quantum affine relations

The toral rules. By [F1] the $K_i$ are diagonal in the partition basis and $F_j,E_j$ shift the weight by $\mp {\alpha }_j$; hence $K_i{E}_j{K}_i^{-1}=q^{⟨{\alpha }_j,h_i⟩}{E}_j=q^{a_{ij}}{E}_j$ and $K_i{F}_j{K}_i^{-1}=q^{-a_{ij}}{F}_j$, the $K$'s commute, and the central product acts by $q$ by [F4].

The mixed commutators. For $i\mathrm{\ne }j$, use the genuine finite-wedge module of Normally ordered q-wedges form a basis for every finite wedge length, obtained from the vector module by the coproduct of The Drinfeld–Jimbo formulas define a Hopf algebra and a quantum-group-stable kernel. Its operators satisfy $[E_i^{[N]},F_j^{[N]}]=0$, where $[N]$ denotes the prefix length, not a divided power. For a given partition choose $N$ beyond the nonvacuum rows of that partition and all its finitely many one- and two-step node outputs. The stable-prefix formula gives $F_j=F_j^{[N]}$ after tail attachment and $E_i=q^{-{\delta }_{i\equiv -N}}{E}_i^{[N]}$ after tail attachment, with the same scalar on both intermediate diagrams. Thus $[E_i,F_j]$ is the image of $q^{-{\delta }_{i\equiv -N}}[E_i^{[N]},F_j^{[N]}]$ and is zero. No relation for the infinite module is assumed in this finite calculation. For $i=j$, the exact tensor cubes in [F2] give $[E_i,F_i]=(K_i-K_i^{-1})\mathrm{/}(q-q^{-1})$.

The Serre sums. By steps 1.1 and 2.1 the node operators satisfy every toral rule and every mixed commutator, and by [F2] each single-color action is integrable; the rank-one Serre engine [F3] then forces both quantum Serre sums for every pair $i\mathrm{\ne }j$, the $e=2$ cubic case included. No Serre relation was used before this point.

Conclusion. The node operators satisfy all defining relations of the affine algebra with the fixed coproduct, each color acts on finite cubes (integrability), the weight shifts are $\mp {\alpha }_i$, and ordinary coefficients with negative powers of $q$ are kept separate from the normalized lattice statement.