General Constructions

This page describes standard mathematical constructions for building new fusion rings and fusion categories from existing ones. These constructions are fundamental tools in the classification and study of topological phases.

Direct Sum (Disjoint Union)

Fusion Rings

The direct sum $\mathcal{R}_1 \oplus \mathcal{R}_2$ of two fusion rings:

Objects: Disjoint union of simple objects

Multiplication:

• Objects from different rings don't fuse
• $X_i^{(1)} \otimes Y_j^{(2)} = 0$
• Within each ring, original fusion rules apply

Properties:

• Rank: $r_1 + r_2$
• $\mathcal{D}_{\text{FP}}^2 = (\mathcal{D}_1^{\text{FP}})^2 + (\mathcal{D}_2^{\text{FP}})^2$

Fusion Categories

For categories $\mathcal{C}_1$ and $\mathcal{C}_2$:

Objects: Pairs $(X, i)$ where $X \in \mathcal{C}_i$

Morphisms: $\text{Hom}((X,i), (Y,j)) = \begin{cases} \text{Hom}_{\mathcal{C}_i}(X,Y) & i=j \\ 0 & i \neq j \end{cases}$

Note: Direct sum of categories is not a connected fusion category (has multiple units).

Tensor Product (Deligne Product)

Fusion Rings

The tensor product $\mathcal{R}_1 \otimes \mathcal{R}_2$:

Objects: Pairs $(X_i^{(1)}, Y_j^{(2)})$

Multiplication:

Properties:

• Rank: $r_1 \cdot r_2$
• $\mathcal{D}_{\text{FP}}^2 = (\mathcal{D}_1^{\text{FP}})^2 \cdot (\mathcal{D}_2^{\text{FP}})^2$

Fusion Categories (Deligne Product)

The Deligne tensor product $\mathcal{C}_1 \boxtimes \mathcal{C}_2$:

Objects: Pairs $X_1 \boxtimes X_2$

Tensor: $(X_1 \boxtimes X_2) \otimes (Y_1 \boxtimes Y_2) = (X_1 \otimes Y_1) \boxtimes (X_2 \otimes Y_2)$

F-symbols: Product of individual F-symbols

Braiding (if both braided):

Group Rings

Group Fusion Ring

For finite group $G$, the group ring $\mathbb{C}[G]$:

Objects: Group elements $\{g\}_{g \in G}$

Multiplication: Group multiplication

Properties:

• Rank: $|G|$
• All $d_i = 1$ (trivial FP dimensions)
• $\mathcal{D}^2 = |G|$
• Pointed fusion ring

Pointed Categories

A pointed category $\mathcal{C}(G, \omega)$:

Objects: Invertible objects corresponding to $G$

Associator: 3-cocycle $\omega \in H^3(G, \mathbb{C}^*)$

Properties:

• All objects have quantum dimension 1
• Braided if $G$ is abelian
• Modular only if $G = \{e\}$

Representation Categories

$\text{Rep}(G)$

The category of finite-dimensional representations of finite group $G$:

Objects: Irreducible representations $V_\rho$

Morphisms: Intertwining operators

Tensor: Tensor product of representations

Properties:

• Rank = number of conjugacy classes
• Symmetric braiding
• Modular only for trivial group

Quantum Group Representations

For quantum group $U_q(\mathfrak{g})$ at root of unity $q = e^{2\pi i/\ell}$:

$\mathcal{C}(\mathfrak{sl}_2, k)$: Level $k$ category

• Rank: $k+1$
• Non-pointed (except $k=1$)
• Modular for all $k \geq 1$

Drinfeld Center (Quantum Double)

Construction

The Drinfeld center $\mathcal{Z}(\mathcal{C})$ of a fusion category $\mathcal{C}$:

Objects: Pairs $(X, \gamma)$ where:

• $X \in \mathcal{C}$
• $\gamma$: half-braiding $\gamma_Y: X \otimes Y \to Y \otimes X$

Properties:

• Always braided (canonical braiding)
• If $\mathcal{C}$ is modular, then $\mathcal{Z}(\mathcal{C}) \cong \mathcal{C} \boxtimes \mathcal{C}^{\text{rev}}$
• Rank: $\sum_i (N_i^i)^2$ where $N_i^i$ are multiplicities

Quantum Double of Group

For finite group $G$, the Drinfeld center $\mathcal{Z}(\mathcal{C}(G))$:

Corresponds to: $D(G)$ quantum double

Objects: Pairs $(g, \rho)$

• $g$: conjugacy class
• $\rho$: irrep of centralizer $C_G(g)$

Properties:

• Rank: number of pairs $(g, \rho)$
• Always modular
• Special case: $D(S_3)$ has rank 8

Equivariantization

Gauging Symmetry

Given fusion category $\mathcal{C}$ with action of finite group $G$:

Equivariantization: $\mathcal{C}^G$

Objects: Objects of $\mathcal{C}$ with compatible $G$-action

Effect: "Gauging" the $G$-symmetry

De-equivariantization: Inverse process

Quotients and Extensions

Taking Quotients

For fusion ring $\mathcal{R}$ with ideal $I$:

Quotient: $\mathcal{R}/I$

Fewer simple objects, some fusions become zero.

Extensions

Extension problem: Given $\mathcal{R}_1 \subset \mathcal{R}_2$, classify all fusion rings $\mathcal{R}$ with:

Group extension: Extend pointed category by group homomorphism

Near-Group Categories

Definition: Fusion category where all but one simple object is invertible.

Structure:

• Group part: $G$ of invertible objects
• Non-invertible object: $m$
• Fusion: $g \otimes m = m \otimes g = m$ for $g \in G$
• $m \otimes m = \sum_{g \in G} n_g g$ for some coefficients $n_g$

Example: Tambara-Yamagami categories

Module Categories and Induction

Module Category

A module category $\mathcal{M}$ over fusion category $\mathcal{C}$:

Action: $\otimes: \mathcal{C} \times \mathcal{M} \to \mathcal{M}$

Morita Equivalence: $\mathcal{C}$ and $\mathcal{D}$ Morita equivalent if they have equivalent module categories

Induction/Restriction

For subcategory $\mathcal{D} \subset \mathcal{C}$:

Induction: $\text{Ind}_{\mathcal{D}}^{\mathcal{C}}: \mathcal{D} \to \mathcal{C}$

Restriction: $\text{Res}_{\mathcal{D}}^{\mathcal{C}}: \mathcal{C} \to \mathcal{D}$

Frobenius reciprocity holds.

Condensation

Anyon Condensation

Physical process: Bosonic anyons condense

Mathematical: Choose algebra object $A \in \mathcal{C}$, form $\mathcal{C}_A$ (local modules)

Effect:

• Reduces number of anyons
• Some anyons become confined
• New fusion rules emerge

Relevance: Phase transitions in topological phases

Categorical Operations

Opposite Category

$\mathcal{C}^{\text{op}}$: Reverse all morphisms

For braided $\mathcal{C}$:

$\mathcal{C}^{\text{rev}}$: Reverse braiding $c_{X,Y}^{\text{rev}} = c_{Y,X}^{-1}$

Dual Category

$\mathcal{C}^*$: Replace objects with duals

In fusion categories: $\mathcal{C}^* \cong \mathcal{C}$

Series and Families

Level $k$ Series

For Lie algebra $\mathfrak{g}$:

$\mathcal{C}(\mathfrak{g}, k)$: Level $k$ category

Examples:

• $\mathcal{C}(\mathfrak{sl}_2, k)$: Rank $k+1$
• $\mathcal{C}(\mathfrak{sl}_3, k)$: $\frac{(k+1)(k+2)}{2}$ simple objects

Haagerup-Izumi Series

Non-group, non-quantum-group categories:

• Haagerup: Rank 6
• Extended Haagerup: Rank 13
• Asaeda-Haagerup: Rank 7

Computational Tools

Software Packages

Anyonica:

• Implements constructions
• Computes tensor products
• Finds Drinfeld centers

TensorCategories.jl:

• Module categories
• Induction/restriction
• Condensation

Learn more →

Applications

These constructions are used for:

• Classification: Systematic enumeration
• Phase transitions: Condensation
• Duality: Understanding category equivalences
• Quantum computation: Building new computational models

Related Concepts

• Fusion Rings →
• Fusion Categories →
• Categorifiability Criteria →

Browse Examples

See these constructions in the database:

View Fusion Rings →
View Fusion Categories →