nLab
fusion ring

Contents

Context

Algebra

Monoidal categories

monoidal categories

With symmetry

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

Definition

Definition

A unital \mathbb{N}-ring is a ring such that

  1. the underlying abelian group is free abelian group;

  2. there exists a finite \mathbb{N}-basis: a finite set II of elements X iRX_i \in R, iIi \in I, such that

    X iX j= kIc ij kX k X_i X_j = \sum_{k \in I} c_{i j}^k X_k

    for c ij kc_{i j}^k \in \mathbb{N}

  3. the ring unit 1 is among these basis elements.

Let 𝒞\mathcal{C} be a fusion category, i.e. a tensor category which is finite and semisimple category (i.e. it has a finite number of isomorphism classes [X i][X_i] of simple objects, all finite direct sums of these exist, and every object is isomorphic to such).

Definition

The isomorphism classes [X][X] of objects of 𝒞\mathcal{C} form an \mathbb{N}-ring (def. ) under tensor product. This is the fusion ring of 𝒞\mathcal{C}.

Last revised on March 14, 2017 at 13:12:22. See the history of this page for a list of all contributions to it.