Isomorphisms and automorphisms of quantum groups

Abstract

We consider isomorphisms and automorphisms of quantum groups. Let $k$ be a field and suppose $p, q\in k^*$ are not roots of unity. We prove that the two quantum groups $U_q(\mathfrak {sl}_2)$ and $U_p(\mathfrak{sl}_2)$ 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 $U_q(\mathfrak{sl}_2)$ of Alev and Chamarie, and that $\text{Aut}_k(U_q(\mathfrak {sl}_2))$ is isomorphic to $\text{Aut}_k(U_p(\mathfrak {sl}_2))$.