Frobenius-Perron dimension




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In higher category theory



For elements of \mathbb{N}-rings


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 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).


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]) \,.


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