This is a question I've discussed with a lot of mathematicians, and have read some mathematical texts about, and watched some conference talks about: what is, **axiomatically**, a quantum group?

There are many classes of noncommutative algebras that everybody agrees is a quantum group (or quantum algebra): quantizations of certain coordinate rings, quantizations of enveloping algebras, quantizations of semisimple algebraic groups, multiparameter quantizations of the Weyl algebra, etc; but what is the state of the art of attempts to **give an axiomatic definition** for this class of algebras?

A related MO question is What is quantum algebra?. A nice and leisure discussion, albeit not axiomatic, is Shahn Majid's *'What Is... a Quantum Group'* (here).

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