The Coalgebra Automorphism Group of Hopf Algebra $k_q[x, x^{-1}, y]$

TL;DR

This paper investigates the structure of the coalgebra automorphism group of a graded Hopf algebra derived from the quantum plane, specifically focusing on the case where the parameter q is not a root of unity.

Contribution

It provides a detailed description of the graded coalgebra automorphism group and the coalgebra automorphism group of the localized quantum plane algebra.

Findings

Describes the structure of the graded coalgebra automorphism group.

Describes the structure of the coalgebra automorphism group.

Applicable when q is not a root of unity.

Abstract

Let $k_q[x, x^{-1}, y]$ be the localization of the quantum plane $k_q[x, y]$ over a field $k$, where $0\neq q\in k$. Then $k_q[x, x^{-1}, y]$ is a graded Hopf algebra, which can be regarded as the non-negative part of the quantum enveloping algebra $U_q({\mathfrak sl}_2)$. Under the assumption that $q$ is not a root of unity, we investigate the coalgebra automorphism group of $k_q[x, x^{-1}, y]$. We describe the structures of the graded coalgebra automorphism group and the coalgebra automorphism group of $k_q[x, x^{-1}, y]$, respectively.