The rank-one De Concini–Kac to Lusztig specialization map is not injective

Given: The rank-one forms of De Concini–Kac, Lusztig, and small quantum-group forms at roots of unity, the divided-power integral form The Lusztig divided-power integral form $U_A$, and a primitive odd root $\zeta$ of order $\mathrm{\ell }>2$.

The rank-one De Concini–Kac source is generated by $E,F,K^{\pm 1}$ with $KE{K}^{-1}={\zeta }^{2}E$, $KF{K}^{-1}={\zeta }^{-2}F$, and $EF-FE=(K-K^{-1})\mathrm{/}(\zeta -{\zeta }^{-1})$ (The Drinfeld-Jimbo quantized enveloping algebra by generators and relations, De Concini–Kac, Lusztig, and small quantum-group forms at roots of unity). An assignment satisfying these three-generator relations extends to a homomorphism from the presented rank-one source.

In the Lusztig integral form, $EF-FE=T_{1,0,1}=(K-K^{-1})\mathrm{/}(v-v^{-1})$ before specialization; the toral product defining $T_{1,0,1}$ specializes by substituting $v=\zeta$ (Divided-power multiplication, toral binomials and the $E_i^{(a)}{F}_i^{(b)}$ reordering identity, The Lusztig divided-power integral form $U_A$).

$E^{\mathrm{\ell }}$ is nonzero in the source by the rank-one PBW basis, and the divided power $E^{(\mathrm{\ell })}$ is an element of the Lusztig form (The rank-one root-power quotient is a finite-dimensional Hopf algebra, The Lusztig divided-power integral form $U_A$).

$E^{\mathrm{\ell }}=[\mathrm{\ell }{]}_v!E^{(\mathrm{\ell })}$ in the generic divided-power algebra, by the definition $E^{(\mathrm{\ell })}=E^{\mathrm{\ell }}\mathrm{/}[\mathrm{\ell }{]}_v!$ (The Lusztig divided-power integral form $U_A$, Quantum integers, factorials, Gaussian binomials and divided powers at $q_i$).