Prime-power specializations of the integral quantum half have a nondegenerate Green form

The minimal polynomial of $\nu$ is $X-\nu \in \mathbb{Z}[X]$ if $Q$ is a square, and $X^2-Q$ otherwise. In either case it is monic with constant term of absolute value greater than $1$. If an integer polynomial $P$ with $P(0)=1$ vanished at $\nu$, division by the monic minimal polynomial would express $P$ as its product with an integer polynomial: perform monic polynomial division in $\mathbb{Z}[X]$, and the remainder must be zero because it has smaller degree and vanishes at $\nu$. Comparing constant terms would make an integer of absolute value greater than $1$ divide $1$, a contradiction. Consequently no such $P$ vanishes at $\nu$.

Reversal $R(F_{i_1}\cdots F_{i_n})=F_{i_n}\cdots F_{i_1}$ preserves the generic Serre ideal: reversing a Serre polynomial interchanges its summand indexed by $r$ with the summand indexed by $1-a_{ij}-r$, multiplying the whole polynomial by $(-1{)}^{1-a_{ij}}$. It therefore defines an involutive anti-automorphism of the generic half, and preserves $U_A^{-}$ since it reverses products of divided powers. Its matrix $T(t)$ in the global basis has entries in $A$ and inverse itself. In particular $T(\nu )$ is invertible.

The primitive braided coproduct preserves $U_A^{-}$: on a divided power the quantum binomial formula gives $r(F_i^{(n)})={\sum }_{r+s=n}{t}^{-rs}{F}_i^{(r)}\otimes F_i^{(s)}$, and the divided powers generate the integral half. Thus its coassociativity and counit specialize from the generic identities, as do both form adjunctions of [F3], because their coefficients lie in the localization at nonzero Gaussian factorials. At $t={\nu }^{-1}$ the tensor scalar is ${\nu }^{(\mathrm{deg}y,\mathrm{deg}{x}^{\mathrm{\prime }})}$. The specialized algebra remains connected and root graded, so the usual height recursion $S(x)=-x-\sum S(x^{\mathrm{\prime }})x^{\mathrm{\prime }\mathrm{\prime }}$ for the reduced coproduct gives its antipode: the two convolution recursions give left and right inverses, and associativity makes them equal.

Fix a root degree and its global basis in [F1]. By [F2], the determinant $\mathrm{\Delta }(t)$ of its Gram matrix belongs to the indicated localization, is regular at zero and has $\mathrm{\Delta }(0)=1$. Clear its finitely many Gaussian-factorial denominators, obtaining $\mathrm{\Delta }(t)=L(t)\mathrm{/}D(t)$ with $L\in \mathbb{Z}[t,t^{-1}]$ and $D\in \mathbb{Z}[t]$ with $D(0)=1$. Since $\mathrm{\Delta }$ is regular at zero, $L=\mathrm{\Delta }D$ has no negative Laurent powers, so in fact $L\in \mathbb{Z}[t]$ and $L(0)=1$. Each $D_m(\nu )$ is a product of positive geometric sums and is nonzero; step 1.1 gives $L(\nu )\mathrm{\ne }0$. Hence the specialized Gram determinant is nonzero. The specialized global family is a basis because it was an $A$-basis. Thus $B_{\nu }$ is nondegenerate, and its component dimension is the rank in [F1].

For $\alpha ={\sum }_i{n}_i{\alpha }_i$ put $c_{\alpha }=((\alpha ,\alpha )-{\sum }_i{n}_i({\alpha }_i,{\alpha }_i))\mathrm{/}2$, the integer sum of the pairings over all unordered pairs of letter positions in any word of degree $-\alpha$. Reversal of the first word complements the inversion set in the matching-permutation formula of [F3]; the color form is symmetric, so $B_{t^{-1}}(u,v)=t^{c_{\alpha }}{B}_t(Ru,v)$ for equal-degree words. For arbitrary generic elements, coefficient inversion consequently gives $\overline{B_t(x,y)}=t^{c_{\alpha }}{B}_t(R\bar{x},\bar{y})\mathrm{.}$ The global basis is bar fixed by [F1]. Therefore its Gram matrix $M$ satisfies $M(t^{-1})=t^{c_{\alpha }}T(t{)}^{\mathsf{T}}M(t)$. Evaluating at $t=\nu$ and using steps 1.2 and 2.1 proves that $M({\nu }^{-1})$ is invertible. Its entries are evaluable since all Gaussian-factorial denominators are positive at ${\nu }^{-1}$. This proves nondegeneracy on $H_{-}$, with the same ranks from [F1].

On a homogeneous degree $-\alpha$ multiply the form by $(Q-1{)}^{-\mathrm{ht}\alpha }$. In a nonzero summand of either adjunction the factor degrees add to $\alpha$, and heights add; hence the scaling factors multiply and both adjunctions remain true. Its simple-generator values are ${\delta }_{ij}\mathrm{/}(Q-1)$ by [F3], and its nondegeneracy follows from step 3.1 since its scaling factors are nonzero. Together with steps 1.1–3.1 this proves every assertion.