Quantum mathematics? "Quantum" as a term/prefix used to be genuinely physical: what was supposed to be physically continuous turned out to be physically quantized. 

What sense does this distinction make inside mathematics?

Especially: Is "quantum algebra" a well-chosen name? (According to Wikipedia, it's one of the top-level mathematics categories used by the arXiv, but it's not explained any further.)
 A: I would hold that the term non-commutative algebra is usually used to refer to the  study of general noncommutative algebras. Quantum algebra involves the study of certain types of non-commutative algebras, not all non-commutative algebras. It's not black and white, but reasonably well-defined subfamily. The algebras quite often involve a parameter $q$ st when $q=1$ or $0$ the algebra is commutative - take for example Drinfeld--Jimbo algebras. The parallels with quantum theory here are obvious.
A: I think that the basic intuition relating quantum algebra and quantum physics is something like:
quantum stuff = classical stuff + $\hbar$ (something complicated)
where $\hbar$ is a "small" formal variable. In other words, the point is to consider that the mathematical objects everybody knows are only approximations of more complicated objects. Hence, quantum mathematics has something to do with perturbation theory, because most of the interesting objects in quantum mathematics are perturbations of trivial solutions of some problems/equations. Here, perturbation means that these objects are formal power series in $\hbar$ whose constant term is a trivial solution (eg: 1 :) ) of some equation (eg: the Yang Baxter equation).
Hence, as John pointed out, quantum algebra involves the study of objects for which classical properties (eg: commutativity) are "almost" true (ie: true modulo $\hbar$).
A: Working in "quantum mathematics" myself, I should tend to defend this teminology a bit ;) The term is clearly motivated by the usage in physics and, nowadays, is typically used in situations where you have a "classical" mathematical object (ring, algebra, group, whatever) which traditionally is viewed in a commutative context. Then the "quantum" version means to transfer things into a noncommutative context and see what happens.
Of course, this is all very vague, but why do you call groups "groups" and fields "fields"?
I guess, it is the intuition which makes this notion useful for the community. The intuition from physics is the transition from commutative to noncommutative, and I think that is really what people usually think if they hear from some "quantum blablabla" in math. So I guess, it is not a completely irritating notion :)
