abelian

Non-abelian just in general means non-commutative, which just means that it's different if you do things in a different order.

The problem is that the trick that produce the apparent mass don't work with the U(1) group because it's abelian and with dimension=1.

[!] So far, only Abelian anyons have been detected in fractional quantum Hall systems, but studies suggest that non-Abelian anyons might also be realized.

It's the color of the quarks inside the nucleus it's not abelian so it's more complicated.

Yes, from my (limited but growing) understanding of it, they are indeed similar to abelian groups.

For non-abelian Yang-Mills Theory (t'Hooft Veltman had published their landmark paper in 1972) this is more or less infeasible beyond first order.

The coproduct of two abelian groups is also called the direct sum [edited], and the category of general groups does not have coproducts.

Showing that time values on a clock forms an abelian group under addition is a textbook example everyone uses when introducing group theory.

At the surface they sound like something vaguely resembling an abelian group (+/- inverses), but the conflict resolution stuff is the heart of it I'd guess.

If type-safety is a purely syntactical notion, then type is no longer a tool to enforce the distinction between different kinds of groups, such as abelian groups, torsion-free groups, etc.

That associativity does not hold is a much more unpleasant than if associativity would hold but commutativity wouldn't (the mathematics of common associative, but non-commutative structures is understood quite well, say (non-abelian) groups and monoids).

(2) Abelian group; Zorn's lemma; line bundle.

Associated with other abelian groups there are other L-series-like things called "automorphic L-functions".

I'm not sure "non-Abelian braiding statistics" is the only known robust architecture for QC.

For instance, why use "abelian group", when "commutative group" will do?

The idea of an Abelian category (basically, a "nice" category) requires the existence of coproducts, which I think is widely used enough in universal algebra to be considered a key concept.

Non-Abelian braiding statistics are so far the only known robust architecture for QC, so this will be interesting to watch.

I'm fascinated by Fibonacci #s and phi *, aka the Golden Ratio, but Anyons, Abelian-ness and braids are entirely foreign to me.

Arithmetic modulo 86400 (or 12, 24, 60) can be handy, too (especially if there is an 'on overflow do' facility) And people will want complex numbers, quarternions, non-Abelian groups, etc.