nLab Tambara-Yamagami category

Idea

A Tambara-Yamagami (TY) category is a fusion category that may be regarded as a non-trivial extension of a category of $G$-graded vector spaces, for $G$ a finite group. By nontrivial extension we mean that, even though it is constructed from a category of $G$-graded vector spaces, it itself cannot be presented as such.

In 2-dimensional quantum field theory (QFT), a TY category is thought to encode as a symmetry category what is nowadays known as a non-invertible symmetry, where the objects of the category correspond to line defects.

Definition

A Tambara-Yamagami category $TY(G)$ is defined as follows.

Starting from a finite group $G$, consider the set $I=G\coprod \{m\}$. We construct $TY(G)$ by defining the simple objects as $\{U_g\}_{g\in G}\coprod \{U_m\}$. These have fusion rules:

• $U_{g}\otimes U_{g'}=U_{gg'}$
• $U_{g}\otimes U_m = U_m$
• $U_m\otimes U_g = U_m$
• $U_m\otimes U_m = \sum_{g\in G} U_g$

Unless stated otherwise, we assume the ground field of $TY(G)$ to be $\mathbb{C}$. It has been shown that the above fusion rules also admit categorification over the reals (see Plavnik & Sanford & Sconce 2023).

Properties

It was shown by Tambara & Yamagami (1998) that these categories are classified by pairs $(\chi,\tau)$ for $\chi:G\times G\to \mathbb{C}^*$ a bicharacter, and $\tau$ a square root of $\frac{1}{|G|}$.

Remark

Since $\chi$ is symmetric and nondegenerate, $TY(G)$ exists only if $G$ is abelian. $TY(G)$ is not necessarily unique for a choice of $G$.

Proposition

$TY(G)$ is a unitary fusion category.

This follows from (Galindo & Hong & Rowell 2013, theorem 5.20).

Proposition

$TY(G)$ is a braided fusion category if and only if $G\cong\mathbb{Z}_{2}^{n}$.

The above is due to (Siehler 2000, theorem 1.2(1)).

Examples

• $TY(\mathbb{Z}_{2})$ is realised by two distinct unitary fusion categories (distinguished by the Frobenius-Schur indicator $\varkappa_{m}\in\{+1,-1\}$ of $m$), each of which admit $4$ distinct braidings. All $8$ of these unitary braided fusion categories (UBFCs) are modular. Two of the UBFCs with $\varkappa_{m}=+1$ are known as the Ising categories, while two of the UBFCs with $\varkappa_{m}=-1$ describe $SU(2)$-anyons at level $k=2$.

• As worked out in Tambara & Yamagami (1998), for $TY(\mathbb{Z}_2\times\mathbb{Z}_2)$, there are two possible choices of roots ($\tau=\pm\frac{1}{2}$), and two choices of classes of bicharacters. For $\tau=\frac{1}{2}$ and trivial bicharacter, $TY$ is the category of representations of the dihedral group $D_8$. For $\tau=-\frac{1}{2}$ and trivial character, this is the representation category of the quaternion group $Q_8$. For $\tau=\frac{1}{2}$ and nontrivial character, this is the representation category of the Kac-Paljutkin 8-dimensional Hopf algebra. The remaining choice does not correspond to the category of representations of a Hopf algebra (see also Tannaka duality).

References

• Julia Plavnik, Sean Sanford, Dalton Sconce, Tambara-Yamagami Categories over the Reals: The Non-Split Case, preprint (2023) [arXiv:2303.17843]

• Cesar Galindo, Seung-Moon Hong, Eric Rowell, Generalized and Quasi-Localizations of Braid Group Representations, International Mathematics Research Notices 2013(3) (2013) 693-731 [doi:10.1093/imrn/rnr269]

• Jacob Siehler, Braided Near-Group Categories, preprint (2000) [arXiv:0011037]

• AnyonWiki, List of small multiplicity-free fusion rings

General

Original articles:

• Daisuke Tambara, Shigeru Yamagami, Tensor categories with fusion rules of self-duality for finite abelian groups, Journal of Algebra 209 (1998) 692-707 [doi:10.1006/jabr.1998.7558]