  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.
