Local q=1 deformation proves integrable highest weight universality

The lattice and its freeness. Let $V_R:={\sum }_{w}R{F}_{w}v$ over all ordinary words $w$, including the empty word, so $v\in V_R$ and $V_R$ spans $V$ over $\mathbb{Q}(q)$ because $V=U_q^{-}v$ by [F3] and the words span $U_q^{-}$. For a label $\mu$, the $R$-module $V_R\cap V_{\mu }$ is spanned by the finitely many vectors $F_{w}v$ of label $\mu$ (bounded word length by [F3]), so it is finitely generated; it is torsion-free as a submodule of the $\mathbb{Q}(q)$-space $V_{\mu }$; by [F8] it is free of finite rank, equal to ${\dim }_{\mathbb{Q}(q)}{V}_{\mu }$, and $V_R={⨁}_{\mu }(V_R\cap V_{\mu })$.

Stability under the generators of $U_A$. (i) $K_h(F_{w}v)=q^{⟨\mathrm{w}\mathrm{t}(F_{w}v),h⟩}{F}_{w}v$ is an $R$-unit multiple by [F1], [F2] and [F3]. (ii) $F_j{F}_{w}v=F_{jw}v\in V_R$ trivially. (iii) For $E_i$, use $E_i{F}_{j_1}\cdots F_{j_m}=F_{j_1}\cdots F_{j_m}{E}_i+{\sum }_t{F}_{j_1}\cdots F_{j_{t-1}}[E_i,F_{j_t}]F_{j_{t+1}}\cdots F_{j_m}$ with $[E_i,F_{j_t}]={\delta }_{i{j}_t}{T}_{i,0,1}$ by [F4]; applying to $v$, the first term vanishes, and each surviving term equals $F_{j_1}\cdots \widehat{F_{j_t}}\cdots F_{j_m}\cdot [m_t{]}_{i}v$ where $m_t=⟨\mathrm{w}\mathrm{t}(F_{j_{t+1}}\cdots F_{j_m}v),h_i⟩\in \mathbb{Z}$ is an integer (the weight of the tail is a label by [F3]) and the scalar is $T_{i,0,1}$ on that tail; by [F2], $[m_t{]}_i\in R$. Hence $E_i{V}_R\subseteq V_R$, and by iteration $E_i^r{V}_R\subseteq V_R$. (iv) $T_{i,c,t}$ acts diagonally with scalar ${\prod }_{s=1}^t[m+c-s+1{]}_i\mathrm{/}[s{]}_i={\left(\genfrac{}{}{0px}{}{m+c}{t}\right)}_i$, an element of $R$ by [F2] and [F5]. (v) The divided powers are $E_i^{(r)}=E_i^r\mathrm{/}[r{]}_i!$ and $F_i^{(r)}=F_i^r\mathrm{/}[r{]}_i!$ with $[r{]}_i!\in R^{\times }$ by [F2], so they also preserve $V_R$. Thus $V_R$ is stable under every generator of $U_A$.

The $q=1$ fibre. Let $\bar{V}:=V_R\mathrm{/}(q-1)V_R$, a vector space over $R\mathrm{/}(q-1)=\mathbb{Q}$, with ${\bar{V}}_{\mu }=(V_R\cap V_{\mu })\mathrm{/}(q-1)(V_R\cap V_{\mu })$ of dimension equal to ${\mathrm{rank}}_R(V_R\cap V_{\mu })={\dim }_{\mathbb{Q}(q)}{V}_{\mu }$ by step 1.1; in particular the image $\bar{v}$ of $v$ is nonzero because $Rv\subseteq V_R$ is a direct summand on the top label. By step 1.2 every generator of $U_A$ induces a $\mathbb{Q}$-linear operator on $\bar{V}$, namely ${\bar{E}}_i,{\bar{F}}_i,{\bar{H}}_i:={\bar{T}}_{i,0,1}$ and ${\bar{K}}_h=\mathrm{i}\mathrm{d}$; the values of the toral scalars at $q=1$ are the ordinary integer binomials $\left(\genfrac{}{}{0px}{}{m+c}{t}\right)\in \mathbb{Z}$ and $m\in \mathbb{Z}$ by [F2] (no division by a vanishing denominator is performed: only the integral element $T_{i,c,t}$ is reduced). The defining relations of $U_q$ hold in $U_A$ as integral identities and therefore hold on $\bar{V}$: ${\bar{E}}_i{\bar{F}}_j-{\bar{F}}_j{\bar{E}}_i={\delta }_{ij}{\bar{H}}_i$; the Serre relations with Gaussian binomials reducing to the ordinary binomials; and, the diagonal scalars give ${\bar{T}}_{i,c,t}=\left(\genfrac{}{}{0px}{}{{\bar{H}}_i+c}{t}\right)$ on each label, hence on all of $\bar{V}$; moreover ${\bar{H}}_i{\bar{E}}_j={\bar{E}}_j({\bar{H}}_i+a_{ij})$ and ${\bar{H}}_i{\bar{F}}_j={\bar{F}}_j({\bar{H}}_i-a_{ij})$ hold by reducing $T_{i,0,1}{E}_j=E_j{T}_{i,a_{ij},1}$ and its $F$-analogue from [F4] and evaluating ${\bar{T}}_{i,a_{ij},1}={\bar{H}}_i+a_{ij}$ on weight vectors, where the scalars agree at every label. Hence $\bar{V}$ is a module for the classical algebra generated by the ${\bar{E}}_i,{\bar{F}}_i,{\bar{H}}_i$ with the classical Serre and Cartan relations.

The fibre is an integrable highest weight module. $\bar{V}$ is generated by $\bar{v}\mathrm{\ne }0$, and ${\bar{E}}_i\bar{v}=0$ because $E_{i}v=0$; it has finite-dimensional label pieces by step 2.1 and support in $\lambda -Q_{+}$ by [F3]; and it is integrable: for any $\bar{x}\in \bar{V}$, choosing a lift $x\in V_R$ and using local nilpotence of $E_i,F_i$ on $x$ in the integrable module $V$ gives some $N$ with $E_i^{N}x=F_i^{N}x=0$, and reducing (legitimate since $E_i^N,F_i^N$ preserve $V_R$ by step 1.2) gives ${\bar{E}}_i^N\bar{x}={\bar{F}}_i^N\bar{x}=0$. Top and support claims are unchanged modulo $q-1$ at every label.

Simplicity of the fibre. Work in the regularized auxiliary datum of [F6], where each label is a rational functional on the auxiliary Cartan. Give each Cartan direction its label-diagonal action; the classical Serre and Cartan relations in step 2.1 yield a representation by Serre presentation of a kac moody algebra. Restrict to the minimal realization in the regularization supplier, and extend scalars to $\mathbb{C}$, obtaining ${\bar{V}}_{\mathbb{C}}$. Distinct labels in $\lambda -Q_{+}$ remain distinct on this minimal Cartan because its simple roots are independent; the complementary central directions have the fixed value of $\lambda$. Thus the resulting complex module has finite-dimensional spectral weight spaces and the finite-cone bound required by [F7] (the regularization preserves the grading and the module; its perfect pairing makes distinct labels distinct Cartan functionals, and the old torus is reconstructed from the labels, so dimensions per label are unaffected). The generator $\bar{v}$ is a highest vector of weight $\lambda$, so $\mathrm{\Omega }$ acts on all of ${\bar{V}}_{\mathbb{C}}$ by $c(\lambda )$ by [F7]. Let $W\subseteq {\bar{V}}_{\mathbb{C}}$ be a proper submodule and suppose $W\mathrm{\ne }0$; since the weights above any fixed weight inside $\lambda -Q_{+}$ form a finite set (the simple roots are independent, so the coefficients in $\lambda -\nu$ bound $\nu$), $W$ contains a vector $w$ of a weight $\mu$ that is maximal in the support of $W$; then ${\bar{E}}_{i}w=0$ for all $i$ because ${\bar{E}}_{i}w$ has weight $\mu +{\alpha }_i$, which is not in the support of $W$. Its weight $\mu$ is dominant integral by [F7] applied to the integrable highest weight module generated by $w$. Moreover $w$ lies in the generalized eigenspace ${\bar{V}}_{\mathbb{C}}^{(c(\lambda ))}$ of the central Casimir and is a primitive vector there, and $\lambda$ is a dominant integral primitive weight of the same generalized eigenspace; by the antichain statement of [F7], $\lambda$ and $\mu$ are incomparable unless $\mu =\lambda$. Since $\mu \in \lambda -Q_{+}$ and $\mu \mathrm{\ne }\lambda$ would give $\lambda >\mu$ and hence $c(\lambda )\mathrm{\ne }c(\mu )$, a contradiction, we get $\mu =\lambda$. Then $w$ lies in the one-dimensional top label space and generates ${\bar{V}}_{\mathbb{C}}$, contradicting that $W$ is proper. Hence ${\bar{V}}_{\mathbb{C}}$ is simple. A proper nonzero rational submodule would remain proper and nonzero after extension to $\mathbb{C}$ (extend bases), so $\bar{V}$ is simple as well.

Character equality. By steps 2.1 and 4.1, ${\bar{V}}_{\mathbb{C}}$ is a simple integrable highest weight module of highest weight $\lambda$, hence a simple quotient of the classical Verma module of highest weight $\lambda$; by the uniqueness of that simple quotient [F7], $\bar{V}$ is the rational form of the classical simple integrable highest weight module of weight $\lambda$ in the regularized realization. Comparing dimensions label by label, $\dim V_{\mu }={\mathrm{rank}}_R(V_R\cap V_{\mu })=\dim {\bar{V}}_{\mu }$ by step 2.1, so the character of $V$ equals the classical simple character, independent of the chosen quotient $V$; this is (a).

Uniqueness and universality. Apply step 5.1 to $V=\tilde{L}(\lambda )$ and to its simple graded quotient $L(\lambda )$ of Quantum Verma modules $M(\lambda )$, highest weight vectors, and the simple highest weight quotient $L(\lambda )$: both are nonzero graded quotients with nonzero highest line, so both have the classical character; the quotient map $\tilde{L}(\lambda )\to L(\lambda )$ is thus a surjection whose restriction to every finite-dimensional label space is a surjection of vector spaces of the same dimension, hence an isomorphism on each label; therefore $\tilde{L}(\lambda )\cong L(\lambda )$, and $\tilde{L}(\lambda )$ is graded-simple. The nondegenerate homogeneous form on $L(\lambda )$ supplied by Cartan projection constructs the Verma contravariant form and identifies its radical(ii), together with its ordinary-simplicity argument (iv), then proves ordinary simplicity as well. The universality statement is The dominant quantum Serre quotient is nonzero and integrable(iii) combined with this identification. This is (b) and completes the proof of (c); no $\mathbb{Z}[q,q^{-1}]$-freeness, no global basis and no quantum semisimplicity assumption is used, and all scalars produced are $R$-integral.