Braid-stable PBW decompositions of the small ordinary image

For each root vector the actual integral identity $X_{\beta }^{\mathrm{\ell }}=[\mathrm{\ell }{]}_{\zeta }!X_{\beta }^{(\mathrm{\ell })}=0$ follows from [F1]; its divided root vector exists before specializing. Ordinary monomial spanning therefore truncates every exponent to less than $\mathrm{\ell }$. Each remaining ordinary monomial is a nonzero scalar multiple of a distinct specialized divided basis vector, since every $[r{]}_{\zeta }!$ with $r<\mathrm{\ell }$ is nonzero. They are consequently independent and form the asserted residue bases of the two image halves. This proves both spanning and independence without assuming a presentation of the small algebra.

Every $K_i^L$ is central at the root. By [F2], its central ideal has triangular vector-space form $U^{-}\otimes ⟨K_i^L-1⟩U^0\otimes U^{+}$. In the toral quotient the powers ${\prod }_i{K}_i^{b_i}$, $0\le b_i<L$, are independent: the formal integral labels have independently varying simple-coroot coordinates, and evaluating a linear relation on these toral characters gives every character of $(\mathbb{Z}\mathrm{/}L\mathbb{Z}{)}^I$. Multiply by each inverse character and sum; nontrivial character sums vanish by translation and the trivial sum is $L^{\mathrm{\mid }I\mathrm{\mid }}$, so every coefficient is zero. Mixed ordinary reordering uses only $(\zeta -{\zeta }^{-1}{)}^{-1}$ and puts all words in negative-torus-positive order. Combining the toral independence with [F2] and step 1.1 proves the stated full triangular basis and dimension. Integral type-1 labels satisfy $K_i^L=1$, so passing to this quotient loses none of the selected module actions.

The symmetries preserve the ordinary form and its inverse before specializing and preserve $U_A$ itself by [F1]. They therefore induce inverse maps on the actual image, rather than merely on an abstract presentation. Their toral images are $K_i^{-1}$ on color $i$ and $K_j{K}_i$ on a neighbor, so they preserve the ideal generated by $K_j^L-1$. Consequently they are automorphisms of $u$. Applying any such automorphism to the triangular basis gives the asserted transformed basis. A module transported by such a symmetry has its formal labels reflected along with the torus; formal labels are retained even where torus characters coincide.

Choose a negative convex ordering ending in ${\alpha }_i$. Mixed ordinary reordering shows that $p_i$ has basis $F_i^r{K}^b{E}^{\mathbf{c}}$ with $r<\mathrm{\ell }$, $0\le b_j<L$, and all positive residue exponents $\mathbf{c}$. Indeed other $E_j$ commute with $F_i$ and the repeated-color commutator introduces only a toral element, so these words span; they are part of the basis of step 2.1, hence independent. The full triangular basis is the product of a monomial in the complementary negative root factors and one such $p_i$ basis vector. This is precisely a free right-module decomposition of $u$ over $p_i$. Its inverse antipode preserves $u$ and $p_i$ by the ordinary formulas in [F2], converting it into a free left-module decomposition as well. Applying the algebra automorphisms of step 3.1 proves the transformed parabolic statements. No inference from equality of characters to exact induction has been used.

The ordinary coproduct, counit and antipode preserve the negative Borel $u^{-}{u}^0$: $\mathrm{\Delta }(F_i)=F_i\otimes 1+K_i\otimes F_i$, $\mathrm{\Delta }(K_i)=K_i\otimes K_i$, $S(F_i)=-K_i^{-1}{F}_i$, and $S(K_i)=K_i^{-1}$. The toral relations are Hopf relations, so these descend from [F2]. The Borel is finite by step 2.1, hence is a finite Hopf algebra. Its negative augmentation ideal times the Borel is nilpotent by positive root height and has semisimple finite-torus quotient. These facts are available for the finite-Hopf Frobenius socle argument. General transformed Borels are obtained as algebras, which is all the parabolic induction argument requires. This proves all assertions.