Contents

# Contents

## Idea

$\mathbb{Z}_+$ rings (also known as $\mathbb{N}$-rings or fusion rings) are a sort of generalization of finite groups. A $\mathbb{Z}_+$ ring consists of a finite set of objects, and a set “fusion rules” between them. This generalizes groups in the sense that two objects don’t necessarily fuse to create a third object from the set. Instead, they will fuse into a “direct sum” of the other elements. The compatibility conditions on these fusion rules can be concisely phrased as saying ring-theoretically. Namely, extending the fusion rules $\mathbb{Z}$-linearly one arrives at a ring whose underlying abelian group is a freely generated $\mathbb{Z}$-module.

$\mathbb{Z}_+$ rings naturally appear in the context of fusion categories. This is because in a fusion categories there is a notion of tensor product, direct sum, and simple object. The tensor product of simple objects will be the direct sum of other simple objects. This gives a “fusion rule” on the finite set of isomorphism classes of simple objects, which in turn induces the structure of a $\mathbb{Z}_+$.

Fusion rings appear very naturally in the algebraic theory of non-abelian anyons. Namely, the fusion of two non-abelian anyons has a non-deterministic result. The spectrum of possible results of the fusion and their relative probabilities gives a fusion rule.

## Definition

There are various definitions of $\mathbb{Z}_+$ ring. Our treatment here follows closely EGNO15.

###### Definition

Let $A$ be a ring whose underlying abelian group is a finitely generated free $\mathbb{Z}$-module. A $\mathbb{Z}_+$ basis for $A$ is a basis for $B=\{b_i\}_{i\in I}$ for $A$ as a $\mathbb{Z}$-module such that $b_ib_j$ is a non-negative linear combination of elements of $B$ for all $i,j\in I$. That is, there exists $c^{k}_{i,j}\in \mathbb{Z}_+$ such that

$b_i b_j=\sum_{k\in I}c^{k}_{i,j}\cdot b_k.$

Simple put, a $\mathbb{Z}_+$-ring is a ring equipped with a $\mathbb{Z}_+$-basis, with the extra condition that the multiplicative identity element $1$ be a non-negative linear combination of basis elements.

The relation of the multiplicative identity to the $\mathbb{Z}_+$-basis is a very subtle one. In particular, the fact that the the identity is not part of the basis can cause troubles. For this reason, we define a unital $\mathbb{Z}_+$-ring to be a $\mathbb{Z}_+$ ring whose identity element is part of the distinguished $\mathbb{Z}_+$-basis.

A $\mathbb{Z}_+$ ring is uniquely defined by its fusion rules. Hence, the definition of $\mathbb{Z}_+$-ring can be restated in terms of a collection of explicit properties that the fusion rules must satisfy. Seeing as the fusion rules are often what is of most interest, this restatement can be very useful.

###### Proposition

Let $B=\{b_i\}_{i\in I}$ be a finite set, equipped with integers $(c^{k}_{i,j})_{i,j,k\in I}$. Let $A$ be the free $\mathbb{Z}$-module generated by $B$. Define a binary operation $A\times A\to A$ by linearly extending the rule

$b_i\cdot b_j=\sum_{k\in I}c^{k}_{i,j}\cdot b_k,$

for all $i,j\in I$.

• (Associativity) Given $i,j,k\in I$, $b_i \cdot (b_j\cdot b_k)=(b_i\cdot b_j)\cdot b_k$ if and only if

$\sum_{s\in I}c^{s}_{i,j}c^t_{s,k}=\sum_{s\in I}c^{s}_{j,k}c^{t}_{i,s}$

for all $t\in I$.

• (Identity) Given $i \in I$, we have that $b_1\cdot b_{i}=b_{i}\cdot b_1=b_{i}$ if and only if

$c^{j}_{1,i}=c^{j}_{i,1}= \begin{cases} 1 & i=j\\ 0 & \text{otherwise} \end{cases}$

for all $j\in I$.

Thus, the fusion rule $(c^{k}_{i,j})$ induces a $\mathbb{Z}_+$ ring structure if and only if the properties listed are satisfied.

## Fusion rings

Most of the subtlety and power in the theory of finite groups comes from the existence of inverses. Similarly, a good theory of $\mathbb{Z}_+$-rings should incorporate some idea of inverses. On the level of fusion categories this corresponds to rigidity, and on the level of anyons this corresponds to the existence of antiparticles.

###### Definition

Let $A$ be a unital $\mathbb{Z}_+$-ring with $\mathbb{Z}_+$-basis $B$. We call $A$ a fusion ring if for all $i\in I$ there exists a unique $i^*\in I$ such that

$c^{1}_{i,j}= \begin{cases} 1 & j=i^*\\ 0 & \text{otherwise} \end{cases}$

and the quantity $c^{k^*}_{i,j}$ is invariant under cyclic permutations of $i,j,k$.

In this sense, $b_{i^*}$ is the “dual” or “inverse” of $b_i$. This induces an involution $*: A\to A$ by sending

$a:=\sum_{i\in I}a_i\cdot b_i\mapsto a^*:=\sum_{i\in I}a_i\cdot b_{i^*}.$

## References

Created on July 18, 2023 at 07:39:43. See the history of this page for a list of all contributions to it.