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

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 $$\mathcal{C}$$?

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.

• +1, but what is EGNO? Commented Sep 5, 2023 at 7:52
• Sorry my bad. It is the book Tensor Categories by Etingof, Gelaki, Nikshych and Ostrik,
– AT0
Commented Sep 5, 2023 at 7:58