This might be a very naive question. But what is quantum algebra, really?


[Wikipedia](https://en.wikipedia.org/wiki/Quantum_algebra) defines quantum algebra as "one of the top-level mathematics categories used by the arXiv". Surely this cannot be a satisfying definition. The arXiv admins didn't create a field of mathematics by choosing a name out of nowhere.


Wikipedia (and, in fact, [the MathOverflow tag wiki](https://mathoverflow.net/tags/qa.quantum-algebra/info)) also lists some subjects: quantum groups, skein theories, operadic and diagrammatic algebra, quantum field theory. But again, I don't find this very satisfying, as I feel this doesn't tell me what the overarching idea of quantum algebra is.


(For example, inspired by the table of of contents of the [Wikipedia article](https://en.wikipedia.org/wiki/Algebraic_topology) I could define algebraic topology to be "homotopy, homology, manifolds, knots and complexes". But first, I have certainly missed many subfields of algebraic topology, and second, this is missing the overarching idea, contained in the introduction of the Wikipedia article: *algebraic topology is the use of "tools from abstract algebra to study topological spaces"*. It immediately makes the link behind all the themes I listed clearer, and if I encounter a new theme I can tell if it is AT or not using this criterion.)


[This MO question is looking for the intuition behind quantum algebra and relations to quantum mechanics.](https://mathoverflow.net/questions/59588/quantum-mathematics) The main thing I gathered from the answers (that I more-or-less knew already) is that "quantum = classical + ħ", or less informally that we are looking at noncommutative deformations of commutative, classical objects. But this doesn't seem to account for all of quantum algebra. For example, a TQFT is a functor from a category of cobordisms to some algebraic category. Where's ħ? Operadic algebra is also listed as one of the components of quantum algebra, but one can study operads a lot without talking about noncommutative deformations. In fact, I've seen and read many papers about operads listed in math.QA that don't seem to have anything to do with this picture.


In brief: **What could be a one-sentence definition of quantum algebra?** (In the spirit of the definition of algebraic topology above.)



What is quantum algebra?
Quantum algebra is an umbrella term used to describe a number of different mathematical ideas, all of which are linked back to the original realisation that in quantum physics, one finds noncommutativity. The areas now encompassed by the term "quantum algebra" are not necessarily directly or obviously related to each other (and this is even more true for publications tagged math.QA on the arXiv, since arXiv classifications are intended to flag work as "of interest to people in area X", not that "this work is in area X"; the Mathematics Subject Classification is better suited to this, but is naturally a much finer classification, and most items have multiple tags).


The original quantum groups (more precisely, deformation quantizations of enveloping and coordinate algebras) are one example, but their study has largely been absorbed into a wider area of noncommutative geometry (usually with qualifiers: algebraic, projective, differential, ...). One also finds Hopf algebra theory and thence categorical approaches to noncommutative geometry (symmetric and braided monoidal categories, for starters). These lead you towards TQFTs, operads, knot invariants and many other things.


There are plenty of good places to read about what different people think the area encompasses, one being Majid's summary in the article "Quantum groups" (p.272-275) in 


*Gowers, Timothy (ed.); Barrow-Green, June (ed.); Leader, Imre (ed.)*, The Princeton companion to mathematics., Princeton, NJ: Princeton University Press (ISBN 978-0-691-11880-2/hbk; 978-1-400-83039-8/ebook). xx, 1034 p. (2008). [ZBL1242.00016](https://zbmath.org/?q=an:1242.00016).


I would say that a one sentence summary that covers even 80% or so of "quantum algebra" is going to be tricky, but the closest I think you'll get is something along the lines of



> 
> The study of noncommutative analogues and generalisations of
>  commutative algebras, especially those arising in Lie theory.
> 
> 
> 


Some might prefer an additional mention of the original link to mathematical physics, but my personal view is that in some directions we have moved very far away from being directly applicable to mathematical physics (my own areas of interest are really purely algebra), so I have chosen not to include this.


