Fusion Ring

Definition

There are multiple definitions of the concept fusion ring in the literature, often with subtle differences. On the AnyonWiki the following definition is used.

Definition: Fusion Ring

A fusion ring $(R,+,\times)$ is a ring with unit $\psi_1$ for which the following axioms are fulfilled:

The underlying abelian group $(R,+)$ is a free abelian group.
There exists a finite set $B = \{\psi_i\}_{i \in I} \subset R$ such that $\psi_1\in B$ and $R=\mathbb{Z}B$ .
For all $i,j \in I$ $\psi_i \times \psi_j = \sum_{k\in I} N_{i,j}^{k}\psi_k, \quad N_{i,j}^{k} \in \mathbb{N}$
There exists an involution $i \mapsto i^*$ such that $N_{i,j}^{k} = N_{i^*,k}^{j} = N_{k,j^*}^{i}$ (Frobenius reciprocity).

Immediate consequences:

The fact that $\psi_1$ is a unit reformulates as $N_{i,1}^j = N_{1,i}^j = \delta_i^j$ for all $i,j \in I$ , which reformulates as $N_{i^*,j}^1 = N_{j,i^*}^1 = \delta_{i,j}$ by Frobenius reciprocity.
The associativity of the ring reformulates as: for all $i,j,k \in I$ $\sum_{s \in I} N_{i,j}^s N_{s,k}^t = \sum_{s \in I} N_{j,k}^s N_{i,s}^t$

Frobenius-Perron Dimension

The involution $*$ provides a $*$ -algebra structure on $\mathbb{C}B$ , given by $\psi_i^* = \psi_{i^*}$ .

Theorem: Frobenius-Perron theorem

There is a unique $*$ -homomorphism $d: \mathbb{C}B \to \mathbb{C}$ , with $d(P) \subset \mathbb{R}_{>0}$ .

The number $d(\psi_i)$ is called the Frobenius-Perron dimension of $\psi_i$ , and is noted $\mathrm{FPdim}(\psi_i)$ . The Frobenius-Perron dimension of $R$ is the number $\mathrm{FPdim}(R):= \sum_i \mathrm{FPdim}(\psi_i)^2$ .

General Constructions of Fusion Rings

There are multiple classes of fusion rings that can be constructed according to a fixed set of rules. Some of the more common ones are listed in General Constructions.

Quantum Double Constructions

[Content to be added]