Your question title says "algebraic" - I think a more common use for that term is to mean that a triangulated category is the stable category of an exact category, which is not what you're asking. I mention it because it might be that if there's a mismatch of terminology, that might not be helping your search.

Indeed. Looking back at 4.3 in my paper, a result whose proof is due to Bazlov, it says: "Of the remaining quasi-minuscule weights, we exclude [because they don't have the multiplicity-free property I wanted] the algebra-weight pairs corresponding to adjoint repre- sentations, namely (Al (l ≥ 2), [1, 0, 0, . . . , 0, 1]), (Dl, ω2), (E6, ω2), (E7, ω1) and (E8, ω8), since in these cases the zero weight occurs with multiplicity l, the rank of g, which is greater than one."

It's not clear to me what a good answer looks like for you: with no honest spaces X and Y, I'm not sure what conclusion will be your target. At the risk of self-promotion, in a paper of mine with Cooney we aggregated a number of results in this area for noncommutative P^n's, which is probably close to the boundary of what is known outside of very particular examples. This is arxiv.org/abs/1807.06383 - if you have any questions, feel free to contact me directly.

Thanks; just feels a bit counterintuitive! Sorry, I don't have an answer to the question - except a gut feeling that this might well be pretty rare. It doesn't happen for linearly oriented type A, right?

Do you really mean all of the $f_i$ are surjective, at the same time?

I should also say that yours, as the others, are very good books and have a definite place and purpose that helps the wider community and brings forward the next generation. I've got several in the genre on my own shelves!

I wonder what would happen to said royalties (and those of several other authors) if UCAS abolished personal statements. It would be nice to think that students read for interest and pleasure but I have seen enough of them to have become a bit cynical and suspect that they think we think just saying "I like Maths and am good at it" isn't enough...

If you'd like an approach that's quite different from the standard ones, try my paper here - tandfonline.com/doi/abs/10.1080/00927872.2010.498394 (expanding on an idea of Majid). If that's not quite what you're after, the references may help. The books by both Jantzen and Brown-Goodearl are particularly highly recommended.

Another take might be that many cluster algebras (but not all) have geometric, combinatorial or categorical models. So this is a widespread phenomenon and part of a bigger picture, this not coincidental. E.g. surface models for other Lie types than just A - type A combinatorics does appear in many places and sometimes the induced bijection is just coincidental (lots of things have associated quadratic forms) and just because two things are counted by Catalan numbers doesn't mean they're in a natural bijection. But if you can also do types D and E, or even BCFG, this does suggest structure.

Without being overly philosophical, and to follow up David's question, what to you distinguishes "any old" bijection from one with some meaning? (As in, what makes something not a coincidence?)

For your second, either you've answered your own question in the previous paragraph or you have in mind a different meaning of "generalises". Away from q=1 (and especially for roots of unity) quantum affine space is quite different from the commutative one, and I don't know a way around that. (And it's a matter of taste whether you think this is a feature or a bug.)

For your first question, I think it's fair to say that the conventional take is "no" - in noncommutative geometry, there is no actual space, just the algebra. So one dualises notions from the geometry side over to the algebra (i.e. coordinate ring) and makes the corresponding construction there.

Taking Spec of the coordinate ring puts you in the world of affine varieties: it would be more common to call Spec of the polynomial ring "affine space" and correspondingly Spec of the quantum one "quantum affine space". This might help with searching in and/or reading the literature.