Normalized cyclotomic functors give an integrable quantum action

The normalized operators. Projectivity of the cyclotomic bimodules supplies exact functors preserving finite projectives, so the displayed shifts define $A$-linear operators on the graded Grothendieck groups, where $A=\mathbb{Z}[q,q^{-1}]$. The torus acts on the root component $\beta$ by $K_a=q^{⟨a,\mathrm{\Lambda }-\beta ⟩}$, and each Grothendieck element has finite root support.

The mixed relation on $K_0$. Let $h=⟨h_i,\mathrm{\Lambda }-\beta ⟩$ at input root $\beta$. The $E_i$ in $E_i{F}_i$ has output root $\beta$, so its normalization contributes $q_i^{1-h}$. The $E_i$ in $F_i{E}_i$ has output root $\beta -{\alpha }_i$, whose coroot label is $h+2$, so its normalization contributes $q_i^{1-(h+2)}=q_i^{-1-h}$. Multiplying the positive basic commutator of [F1] by $q_i^{1-h}$ therefore turns its $q_i^{-2}{F}_i^{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{c}}{E}_i^{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{c}}$ term into precisely the normalized $F_i{E}_i$, and its identity terms into ${\sum }_{k=0}^{h-1}{q}_i^{1-h+2k}=[h{]}_i$. For $h\le 0$, the negative basic branch similarly gives $F_i{E}_i=E_i{F}_i+{\sum }_{k=0}^{-h-1}{q}_i^{-1-h-2k}\mathrm{I}\mathrm{d}$, whose sum is $[-h{]}_i$, so the same signed relation follows. Thus $E_i{F}_i-F_i{E}_i=[h{]}_i\mathrm{I}\mathrm{d}=(K_i-K_i^{-1})\mathrm{/}(q_i-q_i^{-1})$ on root $\beta$, with $K_i=q_i^h$; the quantum integer is integral, so no division in the coefficient ring is needed.

The unequal-color relation and the module actions. Write $h_i(\beta )=⟨h_i,\mathrm{\Lambda }-\beta ⟩$ and $b_{ij}=d_i{a}_{ij}=({\alpha }_i\mid {\alpha }_j)$ for $i\mathrm{\ne }j$. At input root $\beta$, the normalization of $E_i$ in $E_i{F}_j$ is $q_i^{1-h_i(\beta +{\alpha }_j-{\alpha }_i)}$, whereas that in $F_j{E}_i$ is $q_i^{1-h_i(\beta -{\alpha }_i)}$. Since $h_i(\beta +{\alpha }_j-{\alpha }_i)=h_i(\beta -{\alpha }_i)-a_{ij}$, their ratio is $q_i^{a_{ij}}=q^{b_{ij}}$. This cancels the basic crossing factor $q^{-b_{ij}}$ in [F2], giving the normalized relation $E_i{F}_j=F_j{E}_i$. The canonical $G_0$-action has the same shifts and hence the same computations as $K_0$. For the KK $G_0$-action the shift is instead carried by $F$: its exponent on input root $\beta$ is $1-h_j(\beta )$, so the ratio between $E_i{F}_j$ and $F_j{E}_i$ is $q_j^{h_j(\beta -{\alpha }_i)-h_j(\beta )}=q_j^{a_{ji}}=q^{b_{ij}}$, again cancelling the basic factor. For the KK same-color relation, $E_i{F}_i$ has shift $q_i^{1-h}$ and $F_i{E}_i$ has shift $q_i^{-1-h}$, exactly as in step 2.1. Both module actions therefore satisfy the required mixed relations; restriction lowers root height and the weight labels give the toral conjugation relations.

Local nilpotence and integrability. By [F3], $R^{\mathrm{\Lambda }}(\beta +n{\alpha }_i)=0$ for $n$ large, so the induction operator $F_i$ is locally nilpotent componentwise; restriction lowers height, so $E_i$ is locally nilpotent as well. The toral conjugation, mixed relations and local nilpotence are exactly the hypotheses of the abstract lemma [F3], which then proves both full symmetrizable Serre families and equips the scalar extension to $\mathbb{Q}(q)$ with the unique integrable type-one $U_q(\mathfrak{g})$-module structure. No finite-dimensionality of full weight spaces and no ungraded monic deformation is asserted; no cyclic input from vacuum generation or the integral highest-module identification is used.