nLab Tambara-Yamagami category

Contents

Context

Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

Contents

Idea

A Tambara-Yamagami (TY) category is a fusion category that may be regarded as a non-trivial extension of a category of GG-graded vector spaces, for GG a finite group. By nontrivial extension we mean that, even though it is constructed from a category of GG-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)TY(G) is defined as follows.

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

  • U gU g=U ggU_{g}\otimes U_{g'}=U_{gg'}
  • U gU m=U mU_{g}\otimes U_m = U_m
  • U mU g=U mU_m\otimes U_g = U_m
  • U mU m= gGU gU_m\otimes U_m = \sum_{g\in G} U_g

Unless stated otherwise, we assume the ground field of TY(G)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 (χ,τ)(\chi,\tau) for χ:G×G *\chi:G\times G\to \mathbb{C}^* a bicharacter, and τ\tau a square root of 1|G|\frac{1}{|G|}.

Remark

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

Proposition

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

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

Proposition

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

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

Examples

  • TY( 2)TY(\mathbb{Z}_{2}) is realised by two distinct unitary fusion categories (distinguished by the Frobenius-Schur indicator ϰ m{+1,1}\varkappa_{m}\in\{+1,-1\} of mm), each of which admit 44 distinct braidings. All 88 of these unitary braided fusion categories (UBFCs) are modular. In particular, the 44 UBFCs with ϰ m=+1\varkappa_{m}=+1 are also known as the Ising categories (which describe SU(2)SU(2)-anyons at level k=2k=2).

  • As worked out in Tambara & Yamagami (1998), for TY( 2× 2)TY(\mathbb{Z}_2\times\mathbb{Z}_2), there are two possible choices of roots (τ=±12\tau=\pm\frac{1}{2}), and two choices of classes of bicharacters. For τ=12\tau=\frac{1}{2} and trivial bicharacter, TYTY is the category of representations of the dihedral group D 8D_8. For τ=12\tau=-\frac{1}{2} and trivial character, this is the representation category of the quaternion group Q 8Q_8. For τ=12\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

General

Original articles:

Last revised on August 21, 2023 at 02:10:48. See the history of this page for a list of all contributions to it.