Isomorphisms and automorphisms of quantum groups

Abstract

We consider isomorphisms and automorphisms of quantum groups. Let $k$ be a field and suppose $p,q\in k^{\ast }$ 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 ${\text{Aut}}_k(U_p({\mathfrak{sl}}_2))$.

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Cited by in corpus (2)

• Isomorphisms between quantum groups $U_q({\mathfrak{sl}}_{n+1})$ and $U_p({\mathfrak{sl}}_{n+1})$
• The Coalgebra Automorphism Group of Hopf Algebra $k_q[x,x^{-1},y]$