= Coordinate Hopf algebra of quantum SL2
{title2=$\mathcal O_q(\operatorname{SL}_2)$}
The generators $a,b,c,d$ obey $ab=qba$, $ac=qca$, $bd=qdb$, $cd=qdc$, $bc=cb$, $ad-da=(q-q^{-1})bc$ and $ad-qbc=1$. Matrix multiplication gives the coproduct, and the antipode matrix is $\left(\begin{smallmatrix}d&-q^{-1}b\\-qc&a\end{smallmatrix}\right)$. For $q\ne0,\pm1$, its quotient with $c=0$, $a=u$, $d=u^{-1}$ and $b=v$ is the quantum Laurent-plane algebra $uv=qvu$, proving noncommutativity. The same quotient distinguishes $\Delta b$ from its flipped coproduct.
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