With braiding
With duals for objects
category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
With duals for morphisms
monoidal dagger-category?
With traces
Closed structure
Special sorts of products
Semisimplicity
Morphisms
Internal monoids
Examples
Theorems
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 -graded vector spaces, for a finite group. By nontrivial extension we mean that, even though it is constructed from a category of -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 is defined as follows.
Starting from a finite group , consider the set . We construct by defining the simple objects as . These have fusion rules:
Unless stated otherwise, we assume the ground field of to be . 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 for a bicharacter, and a square root of .
Since is symmetric and nondegenerate, exists only if is abelian. is not necessarily unique for a choice of .
is a unitary fusion category.
This follows from (Galindo & Hong & Rowell 2013, theorem 5.20).
is a braided fusion category if and only if .
The above is due to (Siehler 2000, theorem 1.2(1)).
is realised by two distinct unitary fusion categories (distinguished by the Frobenius-Schur indicator of ), each of which admit distinct braidings. All of these unitary braided fusion categories (UBFCs) are modular. Two of the UBFCs with are known as the Ising categories, while two of the UBFCs with describe -anyons at level .
As worked out in Tambara & Yamagami (1998), for , there are two possible choices of roots (), and two choices of classes of bicharacters. For and trivial bicharacter, is the category of representations of the dihedral group . For and trivial character, this is the representation category of the quaternion group . For 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).
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
Original articles:
Last revised on October 15, 2024 at 08:56:18. See the history of this page for a list of all contributions to it.