Isomorphisms and automorphisms of quantum groups

TL;DR

This paper characterizes when quantum groups of the form U_q(sl_2) are isomorphic over a field, showing they are isomorphic iff their parameters are inverses, and describes their automorphism groups.

Contribution

It establishes a precise criterion for isomorphisms between U_q(sl_2) and U_p(sl_2) and describes their automorphism groups, extending previous results.

Findings

U_q(sl_2) and U_p(sl_2) are isomorphic iff p=q^{±1}

The automorphism group of U_q(sl_2) is explicitly described

Automorphism groups of U_q(sl_2) and U_p(sl_2) are isomorphic

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 $\text{Aut}_k(U_p(\mathfrak {sl}_2))$.