Recall that the Tambara-Yamagami categories are those with fusion ring $\mathbb{Z}[A \cup m]$ where $A$ is an abelian group and $m$ is a non-invertible (simple) object such that $ma = am = m$ for all $a \in A$ and $m^2 = \sum_{a \in A} a$.

Tambara and Yamagami showed that the non-identity associativity isomorphisms are given by a symmetric pairing (bicharacter) $\langle, \rangle: A \times A \to \mathbb{C}^\times$ and a choice of sign for $\sqrt{|A|}$ (and, furthermore, that this data determines the category up to tensor equivalence) in [TY]. For example, the associativity isomorphism $\mathbf{a}_{a,m,b}: (a \otimes m) \otimes b \to a \otimes (m \otimes b)$ is given by $\langle a, b \rangle \mathbf{1}_m$. (Note that it is in $\mathrm{Hom}(m,m)=\mathbb{C}$.)

Gelaki, Naidu, and Nikshych compute the Drinfel'd center of the Tambara-Yamagami categories in section 4 of [GNN]. They give a list of the possible half-braidings for the simple objects in Proposition 4.1. It should be noted that, given the associativity isomorphisms found in [TY] for the category and properties of the half-braiding, one is able to write down commutative diagrams which give a system of linear equations with the half-braidings as unknowns that yields the [GNN] calculation (This method is less general and more computational than that in their paper; the resulting half-braidings are what is important.)

For example, the half-braiding $e_a(-): a \otimes (-) \to (-) \otimes a$ for an invertible object $a \in A$ is given by:

$e_a(b) = \langle a,b \rangle$ where $b \in A$ and the morphism is in $\mathrm{Hom}(ab,ab)=\mathbb{C}$ (recall A is abelian)

$e_a(m) = \epsilon$ where $\epsilon^2 = \langle a,a \rangle$

Now, on the other hand, in section 3 of his second paper on the structure of sectors associated with Longo-Rehren inclusions [I2], Izumi realizes the T-Y fusion categories as systems of endomorphisms of a (weak closure of a) Cuntz algebra $\mathcal{O}_n$ where $n=|A|$ . This happens in the following way: Let $\{S_g\}_{g\in A}$ be the generators of $\mathcal{O}_n$ and define the following endomorphisms of $\mathcal{O}_n$:

$a(S_g) = S_{ag}$ ($a \in A$)

$m(S_g) = U(g)\left(\frac{1}{\sqrt{|A|}}\sum_{h\in A} S_h\right)U(g)^*$

where $U$ is the unitary representation of $A$ in $\mathcal{O}_n$ given by $U(h)=\sum_{k\in A} \langle h, k \rangle S_kS_k^*$ (where $\langle , \rangle$ is the symmetric pairing giving the T-Y category).

Clearly the unitary equivalence classes for these endomorphisms have the T-Y fusion rule (i.e. $[ma]=[am]=[m]$ and $[m^2] = \oplus_{a \in A}[a]$). The big difference here is that these categories are *strict*.

Izumi now also computes the center of the T-Y categories using this realization. This is where my problem arises. In the C* category language the half-braiding is given as a family of operators in $\mathcal{O}_n$ all belonging to the obvious corresponding intertwiner space, i.e. a half braiding for the object (endomorphism) $X$ of C* category $\mathcal{C}$ is given as a collection

$\{ E_X(Y) \in (XY,YX) : Y$ object of $\mathcal{C}\}\subseteq \mathcal{O}_n$

satisfying some properties.

Izumi computes the half-braidings for the simple objects (i.e. the group objects and $m$) in Lemma 3.2. For example, a half braiding for object $a \in A$ is given by:

$E_a(b) = \langle a, b \rangle \in (ab,ba)$ where $b \in A$

$E_a(m) = \alpha(a)U(a) \in (am,ma)$ where $\alpha: A \to \mathbb{C}^\times$ such that $\partial \alpha = \langle , \rangle$

There are clear parallels in the two results here, specifically highlighted in the half-braidings for a group object offered as examples, but I am stuck when it comes to writing down a rigorous relationship between the two results. In particular, they become much more "different" looking when we look at the half braidings involving the non-invertible element $m$.

**Questions:**

It seems to me that the information from the non-trivial associativities is now "hiding out" in the unitary representation $U(g)$ and the intertwiner spaces (e.g., $(m, ma) = \mathbb{C}U(a)$ and $(a, m^2) = \mathbb{C}S_a$); rather than having non-trivial associativities we now have non-trivial unitary equivalences, e.g. we have $ma$ is unitarily equivalent to $m$ via $U(a)$. My first attempt to understand this was to look at the intertwiner spaces corresponding to associativities, e.g. $((am)b, a(mb))$, however my calculations haven't gone anywhere (I am unfortunately quite unfamiliar with C* algebras at this point). Taking the other two basic intertwiner spaces as trivial (i.e. $(g \cdot h,gh) = \mathbb{C}$ and $(m, am) = \mathbb{C}$) can someone tell me what the aforementioned associativity intertwiner space is? Is this even a good way to try to work out the relationship between the two results?

How might one begin to write down a tensor equivalence between the two ways of realizing the categories? Would this be a good way to recognize the relationship between the two computations of the half-braidings?

My main end goal here is to be able to take C*-categorical computations of the centers of fusion categories (such as those for other categories in Izumi's paper [I2]) and rewrite them in the more abstract language of [GNN] and [TY]. Is this something that is already known?

PS -- Is there a resource out there establishing some kind of "dictionary" between C* fusion categories and abstract fusion categories?

Thanks in advance for any kind of response!

References:

[TY] = MR1659954

[GNN] = arXiv:0905.3117

[I2] = MR1832764