Quantum plane
= Quantum plane
{title2=$k_q[X,Y]$}
For a <field> $k$ and $q\in k^\times$, the quantum plane is $k_q[X,Y]=k\langle X,Y\rangle/(YX-qXY)$. Its ordered <monomials> $X^iY^j$, $i,j\geq0$, are a <basis>, and their multiplication is $X^iY^jX^uY^v=q^{ju}X^{i+u}Y^{j+v}$. Leading <monomials> show it is a <noncommutative domain>. Some sources interchange $q$ with $q^{-1}$ in the defining relation.