De Concini–Kac, Lusztig, and small quantum-group forms at roots of unity

Let $(I,A,D,P,P^{\vee },v)$ be a finite-type symmetrizable Cartan datum, with $v_i=v^{d_i}$, and let $U_v(\mathfrak{g})$ be the scalar extension of The Drinfeld-Jimbo quantized enveloping algebra by generators and relations from $\mathbb{Q}(v)$ to $\mathbb{C}(v)$. The finite-type Lie algebra $\mathfrak{g}$ is the Serre Lie algebra of the matrix $A$ (Serre Lie algebra of a finite-type Cartan matrix). Write $A_L=\mathbb{C}[v,v^{-1}]$ and $A_{DK}=\mathbb{C}[v,v^{-1},(v_i-v_i^{-1}{)}^{-1}(i\in I)]$.

The Lusztig divided-power form $U_{A_L}^L(\mathfrak{g})$ is the scalar extension to $A_L$ of the subalgebra $U_A$ defined in The Lusztig divided-power integral form $U_A$. Thus it is generated by the divided powers $E_i^{(r)},F_i^{(r)}$, the toral elements $K_h^{\pm 1}$, and the toral quantum binomials $T_{i,c,t}$.

The De Concini–Kac form $U_{A_{DK}}^{DK}(\mathfrak{g})$ is the $A_{DK}$-subalgebra of $U_v(\mathfrak{g})$ generated by $E_i,F_i$ and $K_h^{\pm 1}$. It uses ordinary powers of the root generators. This definition does not assert that it is free over $A_{DK}$ or that its later specialization embeds into the Lusztig specialization.

Let $\zeta \in {\mathbb{C}}^{\times }$ be a root of unity. Require ${\zeta }_i-{\zeta }_i^{-1}\mathrm{\ne }0$ for each $i$ and put ${\mathrm{\ell }}_i=\mathrm{ord}({\zeta }_i^2)>1$. The specialization map sends $v\mapsto \zeta$ and has target $\mathbb{C}$. Define $U_{\zeta }^{DK}(\mathfrak{g}):=U_{A_{DK}}^{DK}(\mathfrak{g}){\otimes }_{A_{DK},v\mapsto \zeta }\mathbb{C},U_{\zeta }^L(\mathfrak{g}):=U_{A_L}^L(\mathfrak{g}){\otimes }_{A_L,v\mapsto \zeta }\mathbb{C}\mathrm{.}$ In the standard good odd-order branch, $\zeta$ has odd order $\mathrm{\ell }$ and $\mathrm{\ell }$ is relatively prime to every $d_i$; then every ${\mathrm{\ell }}_i=\mathrm{\ell }$.

A root-power quotient depends on the chosen toral lattice. In the convention where the torus is generated by the $K_i:=K_{d_i{h}_i}$, define the simple-root-power candidate $u_{\zeta }^{\mathrm{c}\mathrm{a}\mathrm{n}\mathrm{d}}(\mathfrak{g})$ as the quotient of that ordinary-generator De Concini–Kac algebra by the two-sided ideal generated by $E_i^{{\mathrm{\ell }}_i},F_i^{{\mathrm{\ell }}_i},K_i^{{\mathrm{\ell }}_i}-1$ for $i\in I$. This names the quotient construction only. In higher rank it is not the definition of the standard small quantum group: nonsimple positive-root powers need not be killed by imposing only the displayed simple-root powers. No finite dimension, embedding, or higher-rank identification is asserted for this candidate. The rank-one results below use this quotient, where there are no nonsimple positive roots, and denote it by $u_{\zeta }({\mathfrak{s}\mathfrak{l}}_2)$.

For a standard small image form, let $D_{\zeta }^{(i)}$ be the specialization of the ordinary-generator form with torus generated by the $K_i^{\pm 1}$. Define ${\bar{U}}_{\zeta }^L=U_{\zeta }^L\mathrm{/}(K_h^{\mathrm{\ell }}-1:h\in P^{\vee }),{\bar{u}}_{\zeta }=\mathrm{im}(D_{\zeta }^{(i)}⟶{\bar{U}}_{\zeta }^L)$ in the good odd-order branch. The map is the specialization of the generic inclusion into the divided-power form localized by the invertible $v_i-v_i^{-1}$, followed by the toral quotient; it is well defined because those denominators are nonzero at the chosen root. This image definition fixes the actual ordinary-generator form without claiming a presentation by simple root-power relations. The general dimension and root-vector PBW description require separate proofs; no such assertion is implicit here.

The generic inclusion of the ordinary-generator form into the divided-power form does not by itself assert an inclusion after specializing $v$. The rank-one items on this page prove a specific root-power quotient and exhibit the failure of injectivity after specialization. This definition does not assert a quantum Frobenius theorem or any generic-$v$ semisimplicity at a root of unity.