Categorifiability Criteria

Not all abstract fusion rings arise as the Grothendieck ring of a fusion category. This page explains the criteria and conditions for when a fusion ring can be categorified to different types of fusion categories.

Notation

In the criteria on this page we use the following notation:

$R$ : a fusion ring with basis $(b_i)$ and structure constants $N_{ij}^k$
$X_i$ : the fusion matrices, i.e. $[X_i]^k_j = N_{ij}^k$
$A := \sum_i X_i X_i^*$ (with $X_i^* := X_{i^*}$ ), and with eigenvalues $(c_j)$ . These are also called the formal codegrees of a fusion ring.
$\lambda$ : the matrix that simultaneously diagonalizes all $X_i$ (if it exists), i.e. $[\lambda]^i_j$ are the characters of $R$ .

Criteria for General Categorification

Criteria for when a fusion ring can be categorified to any type of fusion category.

Criteria for Complex Categorification

d-number Criterion

Definition: d-number

An algebraic integer $\alpha$ is called a $d$ -number if its minimal polynomial $p(x) = x^n + a_1x^{n-1}+\cdots+a_n$ (where $a_i \in \mathbb{Z}$ ) satisfies that $(a_n)^i$ divides $(a_i)^n$ for all $i$ .

Theorem: d-number criterion

Let $R$ be commutative. If $R$ admits a complex fusion category, then the formal codegrees $(c_j)$ of $R$ are $d$ -numbers.

Extended Cyclotomic Criterion

Theorem: Extended Cyclotomic Criterion

Let $R$ be commutative. If there is a fusion matrix such that the splitting field of its minimal polynomial is a non-abelian extension of $\mathbb{Q}$ then $R$ admits no complex categorification.

Criteria for Pivotal Categorification

Pivotal Version of Drinfeld Center Criterion

Theorem: Pivotal version of Drinfeld center criterion

Let $R$ be commutative. If $R$ admits a complex pivotal categorification, then there exists $j$ such that for all $i$ , $c_j / c_i$ is an algebraic integer.

Criteria for Unitary Categorification

Schur Product Criterion

The commutative Schur product criterion (corollary 8.5) is the following:

Theorem: Commutative Schur product criterion

Let $R$ be commutative with $[\lambda]^i_1=\max_j\left(\left|\lambda_{i, j}\right|\right)$ . If $R$ admits a unitary categorification, then for all triples $(j_1, j_2, j_3)$ we have

\sum_i \frac{ [\lambda]^i_{j_1} [\lambda]^i_{j_2} [\lambda]^i_{j_3} }{[\lambda]^i_1} \geq 0

Note that the above theorem is the corollary of a (less tractable) noncommutative version (Proposition 8.3) which states:

Theorem: Non-commutative Schur product criterion

A (possibly non-commutative) fusion ring $R$ is unitarily categorifiable if and only if for all triples of irreducible unital $*$ -representations $(\pi_s, V_s)_{s=1,2,3}$ of $R$ over $\mathbb{C}$ , and for all $v_s \in V_s$ , we have

\sum_i \frac{1}{d(x_i)} \prod_{s=1}^3\left(v_s^* \pi_s(b_i) v_s\right) \geq 0

References

Many of these criteria are listed in Classification of Grothendieck rings of complex fusion categories of multiplicity one up to rank six

Database Notation

In AnyonWiki, categorifiability is indicated by boolean flags:

FC: Fusion Category
PFC: Pivotal Fusion Category
UFC: Unitary Fusion Category
BFC: Braided Fusion Category
MFC: Modular Fusion Category

A value of T (True) means at least one categorification of that type exists.

Note: A ring may have multiple categorifications with different properties!