Fusion rings
A fusion ring is an associative ring whose underlying abelian group has a finite distinguished basis B = {b0, …, br−1}, containing the unit b0. Products have nonnegative integer coefficients:
The basis has a duality i ↦ i* that extends to an additive anti-involution, with Nij0 = δi*,j and Frobenius reciprocity Nijk = Ni*kj = Nkj*i. The ring need not be commutative. See [1, §3.1].
The rank is r, the number of basis elements. The fusion multiplicity is max Nijk, including the unit coefficients. A based isomorphism permutes the distinguished basis and preserves multiplication and the unit. The census counts these isomorphism classes, rather than arbitrary ring isomorphism classes.
Fusion categories
Throughout this atlas, a fusion category is a ℂ-linear, semisimple, rigid abelian monoidal category with finite-dimensional morphism spaces, finitely many isomorphism classes of simple objects, and a simple tensor unit. Rigidity supplies dual objects. The tensor product has associativity isomorphisms satisfying the pentagon axiom. [2]
For representatives Xi of the simple objects, the coefficients Nijk record the multiplicities in Xi ⊗ Xj. They describe the fusion rules, but do not specify the associators.
Grothendieck rings and categorification
The Grothendieck ring K0(𝒞) is generated additively by object classes, with [X ⊕ Y] = [X] + [Y] and [X][Y] = [X ⊗ Y]. Its simple-object classes form a distinguished basis, making it a fusion ring. [1, §4.5]
A categorification of a fusion ring R is a fusion category 𝒞 together with a based-ring isomorphism K0(𝒞) ≅ R. In this atlas, a realized Grothendieck ring means a based ring with at least one recorded categorical model. The project classifies ring types; different tensor categories may have the same based ring.
A valid fusion tensor, feasible modular data or a compatible based module is only part of an existence problem. None alone supplies coherent associators or proves categorifiability.
Frobenius–Perron dimensions
A fusion ring has a unique unital ring character d that is strictly positive on its basis. Its value di = FPdim(bi) is the spectral radius of the corresponding fusion matrix; d0 = 1 and di* = di. These values depend only on the fusion ring. [1, §3.3]
These positive algebraic dimensions should be distinguished from the categorical dimensions attached to a pivotal or spherical structure. The displayed decimal dimensions on case pages are approximations; the structural filters below use exact arithmetic.
The ring-property filters
Filters combine by conjunction. Each selector offers “Any”, “Yes” and “No”. Thus weakly integral = Yes together with integral = No selects the strictly weakly integral rings. These conditions concern the underlying based ring, independently of its recorded existence status.
| Property | Meaning in this atlas |
|---|---|
| Integral | Every simple FP dimension di is an integer. |
| Weakly integral | The global FP dimension ∑ di2 is an integer. Every integral ring is weakly integral. [3] |
| Simple | There is no proper nontrivial based fusion subring: no subset of the basis containing b0, other than {b0} and B, that is closed under duality and every constituent of every product. This is simplicity as a fusion ring, not simplicity as an abstract ring or algebra. |
| Perfect | The unit is the only invertible basis element. This terminology is used in the fusion-category literature; it is distinct from simplicity. [4] |
| Pointed | Every basis element is invertible, equivalently every di = 1. Invertibility means bibi* = b0. |
| Commutative | bibj = bjbi for all i, j. A braided category has a commutative Grothendieck ring; the converse does not follow. |
| Self-dual | Every basis element equals its dual: i* = i for all i. |
| Multiplicity-free | Every fusion coefficient is 0 or 1, so the maximum multiplicity is one. |
A fusion category is called integral, weakly integral, pointed or perfect when its Grothendieck ring has the corresponding property. Based fusion subrings correspond to full fusion subcategories, so simplicity means that the only fusion subcategories are the trivial category and the whole category. [1]
These filters apply to the complete census directory, the unresolved directory, the constructions catalogue and the realized-ring catalogue. All eight property filters cover every one of the 1,434,042 released census rings. Each has an individual record; detailed profiles add proofs, constructions and research history where documented. Missing detailed documentation never means unresolved categorification. The rank–multiplicity count tables retain their full recorded populations.
Three useful examples
Finite groups. The pointed category VecGω has based ring ℤG, independently of the associator class ω. It is integral and pointed; its ring is commutative exactly when G is abelian. The representation category Rep(G) is also integral, with dimensions equal to ordinary representation degrees, but its fusion ring is always commutative. [1, §§4.1, 4.5, 4.10]
Fibonacci. The rule X² = 1 + X gives dX = (1 + √5)/2 and global FP dimension (5 + √5)/2. Its ring is simple, perfect, self-dual and multiplicity-free, and is not weakly integral.
Ising. The rules ψ² = 1, ψσ = σψ = σ and σ² = 1 + ψ give dimensions 1, 1, √2 and global FP dimension 4. This ring is weakly integral and nonintegral. Its invertible element ψ makes it nonperfect, and the subring {1, ψ} makes it nonsimple.
Existence status and proof scope
Realized records a construction of at least one category. Excluded over ℂ records a proposed obstruction to every complex fusion categorification. Unresolved means neither conclusion has been established in the current record. All project results retain the independent-verification qualification.
Pivotal, spherical, pseudo-unitary, unitary and braided describe categorical structures or properties. A pivotal structure is a monoidal natural isomorphism from the identity functor to double dual; sphericality equates the two pivotal traces of every endomorphism. Pseudo-unitarity means the intrinsic global categorical dimension equals FPdim; it supplies a canonical positive spherical structure. Unitary fusion categories are C* tensor categories with unitary associativity constraints. A braiding is an additional coherent interchange of tensor factors. [2]
A unitary obstruction alone leaves general complex existence open. A reduction by Galois conjugation is an implication about existence of a related category, and does not endow the original category with a unitary structure. Each case page records only the implications supported for that ring.
Standard references and further reading
- Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych and Victor Ostrik, Tensor Categories, Mathematical Surveys and Monographs 205, American Mathematical Society, 2015. The basic reference for based rings, FP dimensions and fusion categories. Publisher · Author’s final version.
- Pavel Etingof, Dmitri Nikshych and Viktor Ostrik, On fusion categories, Annals of Mathematics 162 (2005), 581–642. Foundations, dimensions and rigidity. Journal · arXiv.
- Shlomo Gelaki and Dmitri Nikshych, Nilpotent fusion categories, Advances in Mathematics 217 (2008), 1053–1071. Based-ring structure and dimensional gradings. arXiv.
- Sebastian Burciu and Sebastien Palcoux, Burnside type results for fusion categories, arXiv:2302.07604. Character methods and the terminology of perfect fusion categories. arXiv.
For the project’s rank-specific arguments, exact tensors and verification instructions, continue to the manuscripts and case directory.