Isomorphisms and automorphisms of quantum groups

We consider isomorphisms and automorphisms of quantum groups. Let $k$ be a field and suppose $p,q\in k^{\ast }$ are not roots of unity. We prove that the two quantum groups $U_q({\mathfrak{s}\mathfrak{l}}_2)$ and $U_p({\mathfrak{s}\mathfrak{l}}_2)$ over a field $k$ are isomorphic as $k$-algebras if and only if $p=q^{\pm 1}$. We also rediscover the description of the group of all $k$-automorphisms of $U_q({\mathfrak{s}\mathfrak{l}}_2)$ of Alev and Chamarie, and that ${\text{Aut}}_k(U_q({\mathfrak{s}\mathfrak{l}}_2))$ is isomorphic to ${\text{Aut}}_k(U_p({\mathfrak{s}\mathfrak{l}}_2))$.