TITLE: Categorifications of the Fibonacci Fusion Ring arising from Conformal Field Theory
QUESTION [6 upvotes]: I was reading about realizations of the "Fibonacci" fusion ring $X \otimes X = X \oplus 1$ in Fusion Categories of Rank 2 by Victor Ostrik.  Apparently, there are two of them and they arise in various ways:

integer-spin representations of integrable $\widehat{sl}_2$-modules of level 3
the minimal model $\mathcal{M}(2,5)$ of the Virasoro algebra (central charge c = -22/5)
representations of $U_q(sl_2)$ with $q = e^{\pi i /5}, e^{3\pi i / 5}$.

In any of these cases, how is the Fibonacci category realized?
I would like to understand each of these specific categorifications of the Fibonacci fusion ring.  Can someone here explain the basics of integrable $\widehat{sl}_2$-modules or about the $\mathcal{M}(2,5)$ minimal model from Conformal Field Theory?  I would also like to learn about $U_q(sl_2)$, though it's probably written in many texts.

REPLY [5 votes]: Unfortunately all three of those realizations are the sort of thing you need to read a book about not a MO post.  I agree with Greg that Kassel's book is a great place to start for the quantum group construction (I don't know the other two constructions well, presumably for the affine algebra construction you'd want to start with Kac's book?).
On the other hand there is an easier to explain elementary diagrammatic description.  As usual with diagram categories you only construct a full subcategory and then you'll need to take the additive and idempotent completions to get an abelian category.
Consider the Temperley-Lieb subcategory, whose objects are indexed by integers and whose morphisms m->n are given by linear combinations of planar diagrams of nonintersecting arcs with m boundary points at the bottom and n boundary points at the top modulo a single relation that a circle can be removed for a multiplicative factor of either the golden ratio or its conjugate.  Composition is stacking, tensor product is disjoint union.  There's an explicit 4-strand projection (called a Jones-Wenzl idempotent) here that has the property that any way you close it off you get zero.  Kill that idempotent.  Now look at the "even part" i.e. the full subcategory whose objects are even integers.  This is your category.  Its simple objects are the 0 and 2-strand Jones-Wenzl idempotents.
There's another way to think of this example.  First checkboard shade the regions of all your even diagrams so that they're unshaded on the outside.  Then collapse all the dark regions to lines.  What you end up with now has half as many boundary points and is allowed to have internal 3-valent and 1-valent vertices.  It's easy to see that they satisfy an I=H relation and a relation allowing absorbing vertices.  This gives a construction of the Fibonacci category using the Yamada polynomial relations (I think to get the usual Yamada polynomial on the nose here you want to actually throw in a bunch of JW2s everywhere but its six of one half dozen the other).
Finally there's a slightly different diagram description given in the appendix of one of my papers with Emily Peters and Scott Morrison.  In our notation there the Fibonacci category is (the additive and idempotent completion of) the tadpole planar algebra T_2.