Citation: Tambara, Daisuke, and Shigeru Yamagami. "Tensor categories with fusion rules of self-duality for finite abelian groups." Journal of Algebra 209.2 (1998): 692-707.
Web: https://www.sciencedirect.com/science/article/pii/S0021869398975585
Tags: Mathematical, Monoidal-categories
In this classic and lovely paper, Tambara and Yamagmi give a classification of tensor categories with so-called "Tambara-Yamagami fusion rules". These fusion rules take as input a finite group A. There are simple objects according to the elements of A realizing A-fusion rules, and there is an additional object m with the fusion rules m*g=m and m*m=(sum of all elements of A). The authors prove solutions to the pentagon equation (i.e. fusion categories with Tambara-Yamagami fusion rules) are classified by non-degenerate bi-homomorphisms A \times A -> U(1), and a square root of 1/|A|. When G is non-abelian any map into U(1) will necessarily have kernel, and thus in particular this result implies that G must be abelian to admit fusion category realizations.
This paper is mostly a big computation - they expand out the pentagons, get a big series of relations, then gauge-fix until the equations are in a specific form where they are clearly determined and satisfied by a square root of 1/|A| and non-degenerate bilinear form on A. Then, they show that these bilinear forms and square roots are gauge invariant. Then then have a lovely application when A=Z2\times Z2.
I'll now describe roughly what the bilinear form and square root of 1/|A| correspond to. The bilinear form chi tells you about the associativity relation in the fusion a,m,b -> m. Namely, this fusion space is one-dimensional and (after appropriate gauge-fixing) the corresponding F-symbol is chi(a,b). The square root of 1/|A| is the Frobenius-Schur indicator of m. Note that no data is needed to characterize the associativity of the group elements in A with themselves. This is because, after appropriate gauge fixing, the associativity between elements in A can be made trivial. This is in contrast to the case where "m" is not present, in which case the associativity relations are classified by a 3rd cohomology group. This can be seen as an obstruction to defectifying the regular G-symmetry on Vec_G whenever the associativity is a non-trivial cohomology class, as discussed in
> Etingof, Pavel, Dmitri Nikshych, and Victor Ostrik. "Fusion categories and homotopy theory." Quantum topology 1.3 (2010): 209-273.
Speaking of "fusion categories and homotopy theory", there's a lovely story here. Tambara-Yamagami's stated motivation for this work was to ask whether or not the representation categories of the dihedral group D8 and the quaternion group Q8 were equivalent. In doing so they discovered this lovely general formalism. The stated motivation for "fusion categories and homotopy theory" was to derive a computation-free derivation of Tambara-Yamagami's result, but they ended up creating a much more general obstruction theory for fusion categories! To me this is an absolutely amazing example of curiosity-driven mathematics having wonderful results, which have clearly been quite physically relevant.
The authors treat the example of A=Z2 \times Z2 nicely. They observe that there are two non-equivalent bilinear forms. Note here that there are in a naive sense 4 bilinear forms, but 3 of them are permuted by the action of Aut(A) and thus give equivalent categories. So, there are 4 non-equivalent fusion categories in this case. The difference between the D8 and Q8 representation categories here is that D8 has FS-indicator +1 and Q8 has FS-indicator -1. Additionally, the solution with the other bilinear form and positive FS-indicator is identified as the representation category of the Kac-Paljutkin Hopf algebra.
A natural question is to ask what's going on with the fourth fusion category in this picture. It is a theorem of Masuoka that every eight-dimensional noncommutative semisimple Hopf algebra is isomorphic to either D8, Q8, or the Kac-Paljutkin Hopf algebra. So, this fourth category is not the representation category of any Hopf algebra. It must necessarily be the representation category of a weak Hopf algebra!
> Masuoka, Akira. "Semisimple Hopf algebras of dimension 6, 8." Israel Journal of Mathematics 92 (1995): 361-373.