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 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 \(i,j,k \in I\)
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 here.