# Tag Info

• 89.7k

### Why are fusion categories interesting?

(Unitary) fusion categories are interesting in physics because they classify gapped phases on the boundary of 2+1D quantum states of matter. Similarly, unitary modular tensor categories are ...
• 4,388
Accepted

### Does an equivalence of fusion categories depend on choice of simple objects within isomorphism classes?

Let's forget about the monoidal structure for a moment. $C$ is semisimple, and semisimplicity means that to give a functor out of $C$ is the same thing as specifying where it sends simple objects. ...
• 26.6k
Accepted

Such tensor categories are called "near-group categories": these are semisimple tensor categories whose fusion rule includes exactly one noninvertible simple object. In your case, the group is $\{1,a,... • 40.4k 7 votes Accepted ### What are the necessary conditions for a real number to be a cyclotomic integers？ I suspect you might be looking for the following fact: If$x$is a cyclotomic integer, and$p$a prime does not divide the discriminant, then the minimal polynomial of$x$factors modulo$p$into ... • 7,308 7 votes ### When modular tensor categories are equivalent? A tensor category includes the information of a tensor product, which is something that takes objects and returns objects. This means that a tensor functor can't just "preserve tensor product" it ... • 26.6k 6 votes ### When is the endofunctor category of a monoidal category braided? When is it ribbon? Fusion? Modular? This is in some sense an unnatural question to ask. Endofunctors only form a monoidal category in general, and if you want a braiding, that's not just an extra property: it's extra structure. Where ... • 108k 6 votes Accepted ### For what$G$is$Rep(D(S_3))_{ad}$Grothendieck equivalent to$Rep(G)\$?

Despite my love of the finite group game, let me give an argument that doesn't use the classification of groups of order 18. The 1-dimensional objects correspond to representations of the ...
• 26.6k
Accepted

### No basis change in a fusion ring allowed?

That's right, fusion rings have more data than just being a ring, they also have a preferred basis corresponding to the simple objects. So two fusion rings can be isomorphic as rings but not as ...
• 26.6k