 Quantum plane

For a field $k$ and $q\in k^{\times }$, the quantum plane is $k_q[X,Y]=k⟨X,Y⟩/(YX-qXY)$. Its ordered monomials $X^i{Y}^j$, $i,j\ge 0$, are a basis, and their multiplication is $X^i{Y}^j{X}^u{Y}^v=q^{ju}{X}^{i+u}{Y}^{j+v}$. Leading monomials show it is a noncommutative domain. Some sources interchange $q$ with $q^{-1}$ in the defining relation.