====== 2024-07-21 ======
reading Etingof on quantum group

===== quantum group =====
quantum group is just a fancy name for Hopf algebra. 

A Hopf algebra is just an algebra with $(\Delta, \epsilon, S)$, coproduct, counit and antipode satisfying a bunch of axioms. 

$\epsilon$ and $S$ are determined by $\Delta$. For the algebra of function $O(G)$, the coproduct is interesting. It determines $\epsilon$ to be the restriction to $e$. But then, given a function $f$, we pullback, and restrict to the anti-diagonal. fine, that gives the fiber of the $f(e)$ spread along $G$. 

All these seems very natural. Now we deform to $U_q(sl_2)$.