nLab
Frobenius-Perron dimension

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

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 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.

Definition

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

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

XX j= iI(N X) ijX i. X X_j = \sum_{i \in I} (N_X)_{i j} X_i \,.

By the Frobenius-Perron theorem, these matrices N XN_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 XX, FPdim(X)FPdim(X).

For objects of fusion categories

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

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

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

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

References

Created on November 4, 2016 at 06:27:22. See the history of this page for a list of all contributions to it.