nLab Frobenius-Perron dimension




Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products



Internal monoids



In higher category theory



In the theory of fusion rings?/fusion categories, it is often very useful to have a good theory of dimensions of objects. Dimensions are useful linearization tool, as well as for defining invariants.

The most natural way to define the dimension of an object in a fusion category is via the quantum dimension. Namely, one defines the dimension of an object to be the trace of the identity map on the object. In unitary fusion categories this works perfectly, but in higher generality there are some issues which can arise. The main issue is that trace is not defined. Even in spherical categories where there is a good notion of trace, there is no guarantee that the quantum dimension will be positive.

The key observation is that in the unitary case the quantum dimension of an object will always be the Frobenius-Perron eigenvector of its corresponding fusion matrix. In the spherical non-unitary case the quantum dimension will still be an eigenvector of the fusion matrix, just not necessarily the Frobenius-Perron one.

The Frobenius-Perron dimension of an object is thus defined to be the Frobenius-Perron eigenvector of its corresponding fusion matrix. This is defined in higher generality than the quantum dimension, and serves the purpose of linearization equally well. It thus serves as a great tool in the study of fusion rings and fusion categories.


For elements of \mathbb{N}-rings

The most general setting for Frobenius-Perron dimensions is that 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]) \,.


Last revised on July 18, 2023 at 07:10:25. See the history of this page for a list of all contributions to it.