What is quantum algebra?

Question

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

Wikipedia 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) 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 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. 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.)

Answer 1

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.

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.

Answer 2

I think that a modern realistic perception of the term "quantum algebra" has to be understood in its historical context, that is, the algebraic/geometric methods, originating from the study of the quantization problem in its various forms (first and second quantization , QFTs etc):
As far as i know, the term quantum algebra, has been introduced in Dirac's seminal paper "The fundamental equations of quantum mechanics", Proc. Roy. Soc. A, v.109, p.642-653, 1925 (a reprint can be found in Sources of Quantum Mechanics, ed. B.L. van der Waerden, p.307). It was short after Heisenberg had proposed his -revolutionary for the time- idea that quantum observables should correspond to hermitian matrices of -generally- infinite order. However, he considered the non-commutativity of the matrices as an obstacle in the further development of the idea. Heisenberg communicated his ideas to Fowler at Cambridge. Fowler, was by that time, the thesis advisor of Dirac and this is how the latter got involved. Dirac shortly proposed that the non-commutativity of quantum mechanical observables should be treated as a fundamental characteristic of the new theory to be developed. He also proposed that quantum observables $A$ and $B$ should belong in a non-commutative algebra, satisfying the relation $$ [A,B]=i\hbar \{A,B\} $$ as a "measure of departure" from commutativity. ($[.,.]$ stands for the commutator and $\{.,.\}$ for the classical Poisson bracket). A detailed account of the history of the development of the notion of quantum algebra together with references, historical and technical details, can be found at Varadarajan's Reflections on Quanta, Symmetries and Supersymmetries, ch.2.
During the next decades, the term quantum algebra started expanding and embracing new ideas and methods emerging from the studies of different aspects of the various quantization problems. Dirac's commutator was replaced by Moyal bracket (coinciding with Dirac's comm. modulo $\hbar^2$ terms) and this is how the deformation theory (already developed as a separate discipline at the level of assoc. and Lie algebras) entered the picture. Now quantum mechanical algebras of observables were viewed as deformations of the corresponding classical objects. Moshe Flato and his coworkers have been among the pioneers at that direction.
The rise of quantum groups and $q$-mathematics, expanded the term even more. Now whole new families of examples and methods arose, introducing new mathematical ideas and tools into the subject, such as hopf algebras, $q$-analytical tools, representation-theoretic methods, $q$-deformations of Weyl algebras etc.
The continuous development of Quantum Field theories together with the various technical and conceptual problems introduced by them, led to further expansions of the discipline of quantum algebras. Now, algebraic geometric, homological, homotopical and Category theoretical methods and notions got involved. The development of non-commutative geometry, also opened new directions of study. I am far from being an expert into such topics to provide further details but i have the feeling that almost everything inside "quantum algebras" has been in some way connected or at least originating (even in some distant sense) from the study of the quantization problems.

So, concluding, i would say that, although the requirement for a one-sentence definition of the topic of quantum algebra might seem superficial, a rough approximation (modulo my understanding of course) might be:

The study of the algebraic/geometric theories, methods, techniques, notions and questions originating from the study of the various aspects of the quantization problem (broadly interpreted).

P.S.: Inevitably, non-commutativity is a central topic in the frame of quantum algebra. In this sense, the above description, may be viewed to encompass even modern abstract tools and theories on the fundamentals and the properties of algebraic operations and structures. It is just that i feel a little sceptical, as to whether a modern "definition" of the field of quantum algebra should be build around the notion of non-commutativity itself.