The Drinfeld pairing between the positive and negative quantum halves

Let $B_K$ be the normalized Kashiwara half form of The negative-half bilinear form is symmetric, nondegenerate and integral on $U_q^{-}$, and let $\iota :U_q^{+}\to U_q^{-}$ be the algebra isomorphism $E_i\mapsto F_i$ (equivalently, the library's signed Chevalley involution composed with the compensating degree sign, The Chevalley involution, bar involution, and contravariant anti-involution preserve the Drinfeld–Jimbo ideal). For $x\in U_q^{+}$ of degree $\nu \in Q_{+}$ and $y\in U_q^{-}$ of degree $-\nu$ define $D(x,y):=d_{\nu }^{-1}{B}_K(\iota x,y),d_{\nu }:={\prod }_i(q_i-q_i^{-1}{)}^{{\nu }_i},$ and set $D(x,y):=0$ when the degrees are not opposite. Then $D$ is homogeneous and bilinear, $D(E_i,F_j)={\delta }_{ij}(q_i-q_i^{-1}{)}^{-1}$, and $D(E_i^{(n)},F_i^{(n)})=\frac{q_i^{-n(n-1)\mathrm{/}2}}{[n{]}_i!(q_i-q_i^{-1}{)}^n},$ by The Lusztig–Kashiwara form, parameter and coproduct dictionary, with $E_i^{(n)}\mapsto F_i^{(n)}$ under $\iota$ and $q_i^{-n(n-1)\mathrm{/}2}\mathrm{/}[n{]}_i!=1\mathrm{/}[n{]}_{q_i^2}!$ for the geometric factorial of that item. The pairing with generator value $1$ is the $B_K$-transport $B_K(\iota x,y)$ (The Lusztig-Kashiwara bilinear form on the negative half: the recursion); it is not this quasi-$R$/Drinfeld normalization. The completed normalized tensor of The root-height completion for weightwise quasi-$R$ intertwiners uses the $D$-dual bases.

On the simple-root Borel extensions the toral normalization is $D(K_{\nu },K_{\mu })=q^{(\nu ,\mu )}$ for $\nu ,\mu \in {\mathbb{Z}}_{\ge 0}I$ (the embedded simple-root toral directions), with mixed toral/root generator pairings $0$ and counit normalization $D(1,1)=1$; the Hopf adjunction identities use $\mathrm{\Delta }$ on the negative Borel and ${\mathrm{\Delta }}^{\mathrm{o}\mathrm{p}}$ on the positive Borel (The Drinfeld–Jimbo formulas define a Hopf algebra). No pairing on arbitrary extra toral directions of $P^{\vee }$ is asserted. The divided-power integral form constrained here is that of The Lusztig divided-power integral form $U_A$; no integrality of $D$ is claimed.

The sign of $\iota$ is fixed by the assignment $E_i\mapsto F_i$: the alternative $E_i\mapsto -F_i$ would change the generator value of $D$. The degree scalar $d_{\nu }$ is the reciprocal of the "generator value $1$" normalization and is exactly the color-degree factor of the normalization dictionary; this is why the $D$-coevaluation carries the coefficient $q_i-q_i^{-1}$ on the $i$-th axis.