= Quantum Laurent plane
{title2=$uv=qvu,\quad u\text{ invertible}$}
For $q\ne0$, define a vector space with basis $u^iv^j$, $i\in\mathbb Z$, $j\ge0$, and product $(u^iv^j)(u^kv^\ell)=q^{-jk}u^{i+k}v^{j+\ell}$. 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>.
Back to article page
