nLab Tambara-Yamagami category



Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products



Internal monoids



In higher category theory



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.


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).


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|}.


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.


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

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


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)).


  • 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).



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.