I'm trying to get my head around quantum groups, and the basic ideas that (a) they're actually algebras, and (b) they have ''deformed'' relations such as $$ab=ba\rightarrow ab=qba$$ are all fine and good. But I'm struggling to see the point of them. I mean, I keep seeing claims that they're important and have lots of interesting uses in quantum computing, etc. And I'm fairly willing to take those claims on faith - after all I wouldn't be trying to learn about them otherwise. But I don't see a ``from the ground up'' path to understanding them.
I mean, when we talk about ordinary groups, we can start off discussing symmetries, like reflecting something in a mirror, or rotating a regular object through some angle. And then talk about how some symmetries like reflection are discrete, and some like rotation can be continuous. And then discuss how group elements can be ``generated'' from some basic elements, etc. And so ideas like Lie groups and algebras get built up to serve a purpose, and have a connection to something tangible.
Quantum groups, on the other hand, don't. At least not as far as I can see. They seem like one day someone said ''what if 1 + 2 = 3, but we let 2 + 1 = goldfish?'' I mean that would be fine, if there were situations in which you could have two rocks in your hand, and you picked up another rock and suddenly found yourself holding a goldfish, instead of three rocks. So my question can kind of be restated as ''why would I want to deform an algebra anyway?'' The usual axioms-first approach that mathematicians love to use doesn't answer this question for a newcomer, and it leads to a series of ever-more-frustrating diversions to look up definitions of things like ''antipodes'' and ''non cocommutativity'', which tend to themselves be defined axioms-first, leading to more diversions to look up definitions, ad infinitum. Conversely, more direct descriptions often don't go far enough (e.g. Gauge fields, knots and gravity by Baez and Muniain, which for the most part has plenty of examples and a nice conversational style, simply says that quantum groups ''...are closely related to knot theory, since their axioms are closely related to the Reidemeister moves.'' but doesn't spell out anything more.)
That last point makes me suspect that, at least as a starting point, quantum groups have something to do with crossings and twists. Like maybe if you want to play around with framed knots (links, braids) i.e. ones made of ribbons instead of 1D line segments, you need a deformation parameter $q$ to keep track of how the ribbons twist as you cross and uncross them. If that were so, could you use quantum groups to prove the Dirac scissors trick works? Is that even remotely on the right track? Is there a specific example (or several) of instances where operations are not simply non-commutative so that $ab=ba+k$, but reversing the order of operations introduces a multiplicative factor, so that $ab=qba$ (or equivalently $q^{-1/2}ab=q^{1/2}ba$), and if so can that be extended to other situations, and used to provide examples of stuff like cocommutativity, etc?
