# nLab Frobenius-Perron dimension

Contents

### Context

#### Algebra

higher algebra

universal algebra

## Theorems

#### Monoidal categories

monoidal categories

# Contents

## Definition

### For elements of $\mathbb{N}$-rings

###### 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.

###### Definition

Let $R$ be a unital $\mathbb{N}$-ring (def. ) with finite $\mathbb{N}$-basis $I$, $\vert I\vert = n \in \mathbb{N}$.

For every $X \in R$, let $(N_X)_{i j}\in \mathbb{N}$ be its matrix of left multiplication, defined by

$X X_j = \sum_{i \in I} (N_X)_{i j} X_i \,.$

By the Frobenius-Perron theorem, these matrices $N_X$ have non-negative eigenvalues. This implies that for each of them there is a maximal eigenvalue. This maximal eigenvalue is called the Frobenius-Perron dimension of $X$, $FPdim(X)$.

### For objects of fusion categories

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

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

The Frobenius-Perron dimension of $X \in \mathcal{C}$ is that of its isomorphism class $[X]$ as an element of the fusion ring, according to def. :

$FPdim(X) \coloneqq FPDim([X]) \,.$