The nondegenerate integral specialization has a quantum Casimir

First the specialization has the usual triangular factors. Since $t\in \{\nu ,{\nu }^{-1}\}$, we have $t>0$ and $t\mathrm{\ne }1$, so every quantum integer factorial and every toral-binomial denominator ${\prod }_{r=1}^n(t^r-t^{-r})$ is nonzero. Hence every specialized divided power and toral binomial is a scalar multiple of a polynomial in the undivided root generators and the group-like torus. Conversely the torus remains precisely its group algebra: for one simple direction the integral toral basis $K^{\delta }{T}_{0,r}$, $\delta \in \{0,1\}$, is Laurent triangular, since $T_{0,r}$ has powers $K^r,K^{r-2},\dots ,K^{-r}$ with nonzero extreme coefficients; ordered by the new extreme powers, this is an invertible triangular change of basis with all coefficients evaluable at $t$. Tensor the finitely many simple directions and the unchanged lattice-coset representatives. Together with [F1], this proves $U_t\cong U_t^{-}\otimes K[P^{\vee }]\otimes U_t^{+}$ as a vector space. In particular $M_{U_t}(\lambda )=U_t^{-}{v}_{\lambda }$ has the expected free half coordinates for every integral label $\lambda$. This uses only actual integral specialization.

Transport the generic library algebra to the displayed source convention by $q\mapsto t^{-1}$, $E_i^{\mathrm{l}\mathrm{i}\mathrm{b}}\mapsto F_i$, $F_i^{\mathrm{l}\mathrm{i}\mathrm{b}}\mapsto E_i$, and $K_h^{\mathrm{l}\mathrm{i}\mathrm{b}}\mapsto K_h$. The toral actions hold because the root signs and parameter are both inverted. The mixed relation changes sign in both its commutator and its denominator. Symmetric quantum integers are invariant under parameter inversion, so every Serre relation is preserved. The library coproducts become exactly those in the statement. Thus [F2] transports to a generic tensor $\mathrm{\Theta }={\sum }_{\gamma }{\mathrm{\Theta }}_{\gamma }$ with ${\mathrm{\Theta }}_{\gamma }\in U^{-}[-\gamma ]\otimes U^{+}[\gamma ]$ and source identity $\mathrm{\Delta }(u)\mathrm{\Theta }=\mathrm{\Theta }\bar{\mathrm{\Delta }}(u)$, where $\bar{\mathrm{\Delta }}{E}_i=E_i\otimes 1+K_i^{-1}\otimes E_i$ and $\bar{\mathrm{\Delta }}{F}_i=F_i\otimes K_i+1\otimes F_i$. This transported tensor is normalized by ${\mathrm{\Theta }}_0=1$.

Each transported coefficient is regular at the chosen parameter $t$: in global bases its construction in [F2] uses inverses of homogeneous Gram matrices, quantum factorials and nonzero generator normalization scalars. By [F1] the Gram matrices at $q=t^{-1}\in \{\nu ,{\nu }^{-1}\}$ are invertible, and every other displayed denominator is nonzero. Thus every finite-degree tensor identity of step 1.2 can be evaluated, yielding a well-defined specialized $\mathrm{\Theta }$ satisfying the same identities on the actual $U_t$. This is an evaluation of established identities at a separately proved nondegenerate value, not an assumption that a generic nonzero determinant remains nonzero.

Write ${\mathrm{\Theta }}_{\gamma }={\sum }_a{a}_{\gamma ,a}\otimes b_{\gamma ,a}$, in negative and positive degrees respectively, and define $\mathrm{\Omega }={\sum }_{\gamma ,a}S(a_{\gamma ,a})b_{\gamma ,a}$. On any vector of label $\lambda$ in the stipulated module this is a finite sum: a nonzero $b_{\gamma ,a}v$ requires $\lambda +\gamma$ in the bounded-above support, bounding the height of $\gamma$. Apply $m(S\otimes 1)$ to the source $E_i$ intertwining identity from step 2.1. The two terms on its left cancel, because $S(E_i)=-K_i^{-1}{E}_i$. The terms on its right are $-K_i^{-1}{E}_i\mathrm{\Omega }+K_i\mathrm{\Omega }{E}_i$. Hence $K_i^{-1}{E}_i\mathrm{\Omega }=K_i\mathrm{\Omega }{E}_i$. For $F_i$, the two left terms cancel because $S(F_i)=-F_i{K}_i$, leaving $-F_i{K}_i\mathrm{\Omega }{K}_i+\mathrm{\Omega }{F}_i=0$. Toral weight zero gives $K_h\mathrm{\Omega }=\mathrm{\Omega }{K}_h$. These computations can first be truncated in root height; all boundary terms contain a positive-degree factor of height tending to infinity acting on $v$, $E_{i}v$ or $F_{i}v$ and vanish for a sufficiently large truncation. Therefore the operator identities are valid on every vector.

For a homogeneous vector $v$ of label $\lambda \in Q$, set ${\mathcal{C}}_t(v)=t^{f(\lambda )}\mathrm{\Omega }v$. Since $({\alpha }_i,{\alpha }_i)=2$, $f(\lambda )-f(\lambda -{\alpha }_i)=2(\lambda ,{\alpha }_i)$. The $E_i$ identity of step 3.1 says $E_i\mathrm{\Omega }v=t^{2(\lambda +{\alpha }_i,{\alpha }_i)}\mathrm{\Omega }{E}_{i}v$, which, with $f(\lambda +{\alpha }_i)-f(\lambda )=2(\lambda +{\alpha }_i,{\alpha }_i)$, gives ${\mathcal{C}}_t{E}_i=E_i{\mathcal{C}}_t$. The $F_i$ identity says $F_i\mathrm{\Omega }v=t^{-2(\lambda ,{\alpha }_i)}\mathrm{\Omega }{F}_{i}v$, giving the other commutation by the first difference formula. The toral identity handles all group-likes. Thus ${\mathcal{C}}_t$ commutes with $U_t$, is natural for graded module maps and descends to subquotients. If $v_{\lambda }$ is highest, all positive-height $b_{\gamma ,a}$ kill it, and ${\mathrm{\Theta }}_0=1$ gives $\mathrm{\Omega }{v}_{\lambda }=v_{\lambda }$. Hence ${\mathcal{C}}_t$ is $t^{f(\lambda )}$ on its cyclic module.

For $\lambda =-{\alpha }_i$, $f(\lambda )=2-2=0$, so ${\mathcal{C}}_t=1$ on $M_{U_t}(-{\alpha }_i)$, its finite direct sum over $I$, and every graded subquotient. A nonzero highest vector of label $-\beta$ in such a subquotient also has eigenvalue $t^{f(-\beta )}$ by step 4.1. Since $t>0$ and $t\mathrm{\ne }1$, equality with $1$ implies $f(-\beta )=0$, namely $(\beta ,\beta )=2\mathrm{ht}\beta$. This proves all claims.