Quantum group

Remarks

Example: let $k$ be a field and $q \in k$ nonzero. The algebra $\mathcal{O}_q(SL_2)$ is generated by $a, b, c, d$ with the relations $$ab = q\,ba, \quad ac = q\,ca, \quad bd = q\,db, \quad cd = q\,dc, \quad bc = cb,$$ $$ad - da = (q - q^{-1})\,bc, \qquad ad - q\,bc = 1.$$ Its coproduct is matrix multiplication of the generator matrix: $\Delta(a) = a \otimes a + b \otimes c$, $\Delta(b) = a \otimes b + b \otimes d$, $\Delta(c) = c \otimes a + d \otimes c$, $\Delta(d) = c \otimes b + d \otimes d$. At $q = 1$ the relations say that $a, b, c, d$ commute and $ad - bc = 1$, which is the coordinate ring of $SL_2$. Non-example in the broad sense: the group algebra $k[G]$ of a group is a Hopf algebra, but it is cocommutative. So it is a classical object rather than a genuine quantum group.