Module algebra structures of nonstandard quantum group $X_q(A_1)$ on ${\text{C}}_q[x,y,z]$

In this paper, the module algebra structures of $X_q(A_1)$ on quantum polynomial algebra ${\text{C}}_q[x,y,z]$ are investigated, and a complete classification of $X_q(A_1)$-module algebra structures on ${\text{C}}_q[x,y,z]$ is given