Isomorphisms between quantum groups $U_q(\mathfrak{sl}_{n+1})$ and $U_p(\mathfrak{sl}_{n+1})$

TL;DR

This paper investigates conditions under which two quantum groups of type A are isomorphic, establishing that for even n, isomorphism implies a specific relation between their parameters, thus answering a classical question of Jimbo.

Contribution

It proves that for even n, isomorphism of quantum groups $U_q(rak{sl}_{n+1})$ and $U_p(rak{sl}_{n+1})$ implies $p= ext{±}q^{ ext{±}1}$, clarifying the parameter relation.

Findings

Isomorphism implies $p= ext{±}q^{ ext{±}1}$ for even n.

Answers Jimbo's classical question.

Provides conditions for quantum group isomorphisms.

Abstract

Let $\mathbb K$ be a field and suppose $p, q\in\mathbb K^*$ are not roots of unity. We prove that the two quantum groups $U_q(\mathfrak {sl}_{n+1})$ and $U_p(\mathfrak{sl}_{n+1})$ are isomorphic as $\mathbb K$-algebras implies that $p=\pm q^{\pm 1}$ when $n$ is even. This new result answers a classical question of Jimbo.