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

Let $k_q[x,x^{-1},y]$ be the localization of the quantum plane $k_q[x,y]$ over a field $k$, where $0\mathrm{\ne }q\in k$. 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 $U_q({\mathfrak{s}l}_2)$. 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.