Generic quantum Serre halves have classical PBW ranks and a nondegenerate Hopf pairing

The identification of free generators $e_i\mapsto f_i$ carries the positive Serre presentation onto the negative one, reversing degrees; the same holds classically and generically. Thus [F1] gives all three rank equalities and finite freeness on both sides. In a fixed finite free component, lifts of a classical basis have a coordinate matrix whose reduction is invertible over $\mathbb{C}$, so its determinant has nonzero constant term and is an $R$-unit. The adjugate identity makes this matrix invertible over $R$, proving the formal lift assertion.

Give the free positive tensor algebra the primitive-generator coproduct into its scalar-braided tensor square. Its map to $⟨V⟩$ commutes with the cut coproduct: both are algebra maps by [F2] and agree on each letter. By [F1] its kernel is precisely the positive Serre ideal. Hence the composite of the tensor coproduct with the two quotient maps kills that ideal, and [F5] gives its coideal inclusion and descended coproduct. The same presentation identification gives the negative coproduct. Coassociativity and counit follow on primitive generators and hence on the generated algebras. The color-height grading is connected, so the reduced coproduct of a positive-height homogeneous element has both factors of strictly smaller height. The recursion $S(x)=-x-\sum S(x^{\mathrm{\prime }})x^{\mathrm{\prime }\mathrm{\prime }}$ and its right-handed counterpart provide left and right convolution inverses by height induction; associativity of convolution makes them equal. Thus both quotients are braided Hopf algebras.

The substitution $z\mapsto e^{\mathrm{\hslash }}$ embeds $\mathbb{C}(z)$ into $K$: a nonzero polynomial is $(z-1{)}^{m}p(z)$ with $p(1)\mathrm{\ne }0$, and its value is the nonzero product $(e^{\mathrm{\hslash }}-1{)}^{m}p(e^{\mathrm{\hslash }})$ in the domain $R$; fractions then embed by [F6]. In each color degree the generic and formal quotients after extension to $K$ have the same finite word presentation, since the Serre coefficients specialize as in [F3]. A regular rational lift of a classical component basis therefore gives the formal basis of step 1.1 and, after field extension, a generic basis. Applying this degree by degree to the ordered monomials of [F4] proves the PBW monomial clause.

By [F2], the diagonal pairing restricts to $⟨V⟩\times H^{-}$, and [F1] identifies the first factor with $H^{+}$. The cut adjunction is the wordwise concatenation identity. For the other adjunction, the coefficient of a word $w$ in a shuffle $u\ast v$ equals the coefficient of $u\otimes v$ in the primitive braided tensor coproduct of $w$: both sum over the assignments of letters to the two blocks with inversion scalar $q^{-⟨\mathrm{deg}{u}^{\mathrm{\prime }},\mathrm{deg}v⟩}$. The degree form is symmetric by [F3], and the diagonal generator weights multiply in the same way on both sides. Thus both adjunctions hold on free representatives, and step 1.2 and the descended pairing make them adjunctions on the quotient coproducts themselves. The empty word gives $⟨1,1⟩=1$ and all positive-degree counit pairings vanish.

Fix $\alpha \in Q_{+}$. If an element of $H^{+}[\alpha ]{\otimes }_{R}K$ pairs to zero with every negative class, its realization in the shuffle word space pairs to zero with every negative tensor word, since all those words map to quotient classes. The diagonal word pairing has zero left annihilator, so that realization is zero. The realization remains injective after extension to $K$ by [F1] and [F5]. The opposite components have equal finite dimension by step 1.1; an injective map from one into the other's dual is consequently bijective. Thus both annihilators vanish. For nonhomogeneous elements, separate the finitely many homogeneous components, proving the asserted nondegeneracy.

Define the generic word pairing using shuffle coefficients and matching-word weights ${\prod }_t(z_{i_t}-z_{i_t}^{-1}{)}^{-1}$. These are rational functions, and under $z=e^{\mathrm{\hslash }}$ their ratio to the formal pairing in a fixed color degree $\alpha$ is ${\prod }_i{u}_i^{{\alpha }_i}$, where $u_i=\mathrm{\hslash }{d}_i\mathrm{/}(e^{d_i\mathrm{\hslash }}-e^{-d_i\mathrm{\hslash }})$ is an $R$-unit with constant term $1\mathrm{/}2$. This color-multiplicative rescaling preserves the two adjunctions and nondegeneracy. The specialized generic pairing therefore kills the Serre ideals and is nondegenerate by steps 2.2–2.3; the field embedding of step 2.1 implies the same descent and invertible pairing matrices over $\mathbb{C}(z)$. Likewise, coassociativity, coideal inclusions and the coproduct formulas descend generically: in each finite degree their errors vanish after the injective field extension, and the antipode recursion applies there as well. All generic presentations and word pairing coefficients lie in $\mathbb{Q}(z)$, so the same injective extension to $K$ proves the statements over that field. This establishes the full braided Hopf pairing with the rational normalization needed for the Drinfeld–Jimbo commutator.