nLab Z+ ring

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

• Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, Victor Ostrik, section 3.1 in Tensor categories, Mathematical Surveys and Monographs, Volume 205, American Mathematical Society, 2015 (pdf

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