How does the Tannaka duality work for weak Hopf algebras and fusion categories?

Question

I'm a physicist and not yet an expert in fusion category. I've heard that it's possible to reconstruct a weak Hopf algebra from its category of representations, and would like to know how this works in detail.

As a specific example, consider the fusion category $\mathcal{C}$ describing Ising anyons, with fusion rules $\sigma\times \sigma=1+\psi, \sigma\times \psi=\sigma, \psi\times\psi=1$, and the $F$-symbol $[F^{abc}_e]_{df}$ describing the associator $(a\times b)\times c\cong a\times(b\times c)$ is also given. Then how do I construct a weak Hopf algebra $H$ whose category of representations $Rep(H)$ is equal to $\mathcal{C}$?

Answer 1

The procedure is more or less the standard Tannakian reconstruction argument. The first thing you need is a "forgetful" fiber functor $F:C\to Vect$, then you consider $R=End(F)$ the natural endomorphisms of the functor $F$ and this is the object you then show has the algebraic structure you require. Then the theorem is that $C$ is a category of representations over $R$.

For fusion categories you get that a generalized fiber functor always exists. You can consider the algebra $A$ to be the endomorphism algebra of the direct sum of (the finitely many isomorphism classes of) the simple objects, that is $A=End(\bigoplus_{S_{i}}S_{i})$ where $S_{i}$ are the (isomorphism classes of the) simple objects.

And then you can construct a functor $F:C\to A-Bimod$, this is simply equivalent to $Hom_{C}(\bigoplus S_{i},\_)$. Then it was shown that in fact $End(F)$ has a weak Hopf algebra structure and $C$ is a category of representations over it.

The proof that this has a weak Hopf algebra structure was originally (I think!) in:

Szlachányi, Kornél, Finite quantum groupoids and inclusions of finite type, Longo, Roberto (ed.), Mathematical physics in mathematics and physics. Quantum and operator algebraic aspects. Proceedings of a conference, Siena, Italy, June 20-24, 2000. Dedicated to Sergio Doplicher and John E. Roberts on the occasion of their 60th birthday. Providence, RI: AMS, American Mathematical Society. Fields Inst. Commun. 30, 393-407 (2001). ZBL1022.18007.

You can find this as Proposition/Exercise 7.23.11 and 7.23.12 of EGNO and around Remark 2.21 of:

Etingof, Pavel; Nikshych, Dmitri; Ostrik, Viktor, On fusion categories., Ann. Math. (2) 162, No. 2, 581-642 (2005). ZBL1125.16025.

I hope this is of help, I am not at all familiar with the physics side of things so I cannot write this in the dialect that perhaps physicist uses so my apologies if this is not entirely clear.