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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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    $\begingroup$ What you'll find is variations on the theme of Hopf algebra, where the comultiplication is the “group law”, the antipode is the “group inverse”, and the counit is “evaluation at the group identity.” Different variations correspond to working with quantum “group objects” in different “categories”, as it were, though I can’t help but point out the $C^\ast$-algebraic theory of locally compact topological quantum groups as particularly well-developed from an axiomatic standpoint. $\endgroup$
    – Branimir Ćaćić
    Commented Aug 1, 2020 at 18:46
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    $\begingroup$ A quasi triangular Hopf algebra (en.m.wikipedia.org/wiki/Quasitriangular_Hopf_algebra) is a precisely defined mathematical object that comes close to giving an axiomatic framework for a quantum group. Unfortunately, it doesn’t quite capture the main examples, e.g., for q not a root of unity the R-matrix of the quantized universal enveloping algrebra doesn’t live in H tensor H but rather some completion of it. $\endgroup$
    – Sam Hopkins
    Commented Aug 1, 2020 at 19:43
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    $\begingroup$ I think it's much like "what is the field with one element?"—'the' definition is whatever someone who is investigating the subject needs it to be to prove interesting theorems, while still being related to the common (informal) theme, as articulated by @BranimirĆaćić. $\endgroup$
    – LSpice
    Commented Aug 1, 2020 at 22:08
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    $\begingroup$ @Branimir to be consistent with what you said above I would say that the counit is the inclusion of the group identity. In the finite group case, applying the free functor and then dual functor to the group law and inverse maps gives the comultiplication and antipode. The same functor composition applied to the map from the one element set to G, mapping to the identity element, gives the counit. $\endgroup$
    – JP McCarthy
    Commented Aug 1, 2020 at 22:45
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    $\begingroup$ @JPMcCarthy you’re absolutely right—it’s then telling that the counit is the convolution identity for appropriate convolution algebras of functions on the Hopf algebra. $\endgroup$
    – Branimir Ćaćić
    Commented Aug 2, 2020 at 17:15

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I would say that if you are looking for a concrete definition then it's better to adopt the Tannakian point of view and to focus on the category of representations of the quantum group rather than on the algebra itself. So take as your fundamental object a tensor category (a special type of rigid abelian monoidal category - see here for details). A "quantum group" is then some way of realising the category as a category of representations or corepresentations. There can be different algebras which do this job, and they can come in different flavours, such as Hopf algebras or compact quantum groups in the sense of Woronowicz. This allows one to view the various quantum groups floating around as tools to study the category itself, removing the need for any axiomatic definition.

If the object is really deserving of the name quantum group, then the tensor category should be braided, as is the case for quasi-triangular Hopf algebras and their category of modules. (see the comment of Sam Hopkins above.)

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I would have liked to write this as a comment, but with my points tally I can not. So writing this as an answer.

In quantum groups, we are probably at a stage group theory was, say in the first half of the 19th century (see here and here), where we have several important classes of objects that we more or less agree should qualify to be called quantum groups, but it is not clear yet if we are anywhere close to having a single set of axioms that will cover all these classes. It is in fact far from clear whether the union of all these `subclasses' will be part of one single meaningful class.

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  • $\begingroup$ I think the voting (in which I participated) indicates that this should be an answer after all. $\endgroup$
    – LSpice
    Commented Aug 11, 2020 at 11:21
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A cheap, soft, quick but objectionable meta-definition:

An algebra of functions on a quantum group is an algebra that satisfies some specific axioms such that whenever an algebra satisfying those same axioms is commutative, it is an algebra of functions on a group (and e.g. the group multiplication given as the transpose of the comultiplication); and whenever two commutative algebras satisfy those axioms are isomorphic as objects satisfying those axioms, their underlying groups are isomorphic.

Not satisfactory but a start.

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    $\begingroup$ I don't think that this is a great answer, but I think it's a shame silently to downvote it. Since a downvote means "This answer is not useful", why not explain why it's not useful? $\endgroup$
    – LSpice
    Commented Aug 1, 2020 at 22:10

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