Quantum Laurent plane ($uv=qvu,u\text{ invertible}$)

For $q\ne 0$, define a vector space with basis $u^i{v}^j$, $i\in \mathbb{Z}$, $j\ge 0$, and product $(u^i{v}^j)(u^k{v}^{\ell })=q^{-jk}{u}^{i+k}{v}^{j+\ell }$. Bilinearity of the exponent verifies associativity. This constructs the algebra and proves the basis independently of its presentation. It localizes the quantum plane at powers of $u$ and gives a quotient witness for noncommutativity and noncocommutativity of the coordinate Hopf algebra of quantum SL2.