Why are Hopf algebras called quantum groups?
I can not comment, and also this need to be a comment ...
Observe that the inquiry in your title and also the inquiry in the body of your inquiry are fairly various!
A non - commutative non - cocommutative Hopf algebra is not the very same point as a non - commutative team, and also quantum groups are generally associative.
One manner in which Hopf algebras show up is as the algebra of (actual or facility) operates on a topological team. The reproduction is commutative given that it is simply pointwise reproduction of features. Nonetheless, in non - commutative geometry you intend to change the algebra of features on a room with a non - commutative algebra, offering a non - commutative Hopf algebra.
This connects to quantum technicians due to the fact that there the analog of the timeless coordinate features of placement and also energy do not commute. Consequently we consider the algebra of features on a quantum "room" as being non - commutative.