# nLab fusion ring

Contents

### Context

#### Algebra

higher algebra

universal algebra

## Theorems

#### Monoidal categories

monoidal categories

# 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 $I$ of elements $X_i \in R$, $i \in I$, such that

$X_i X_j = \sum_{k \in I} c_{i j}^k X_k$

for $c_{i j}^k \in \mathbb{N}$

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

Let $\mathcal{C}$ 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 $[X_i]$ of simple objects, all finite direct sums of these exist, and every object is isomorphic to such).

###### Definition

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

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