nLab Frobenius-Perron dimension

Idea

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.

Definition

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

The most general setting for Frobenius-Perron dimensions is that 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. :

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

References

• Damien Calaque, Pavel Etingof, section 4 of Lectures on tensor categories (arXiv:0401246)

• Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, Victor Ostrik, section 3.3 in Tensor categories, Mathematical Surveys and Monographs, Volume 205, American Mathematical Society, 2015 (pdf

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• Pavel Etingof, Dmitri Nikshych, Viktor Ostrik, “On Fusion Categories”, Annals of mathematics, 2005 (pdf)