TensorCategories.jl

TensorCategories.jl is an open-source software package for computations with tensor categories, especially fusion categories. Built on the Julia programming language and the OSCAR computer algebra system, it follows the standard mathematical framework described by Etingof et al. (2015): objects, morphisms, tensor products, associators, and other categorical structures are represented as such, while concrete combinatorial descriptions, such as $F$-symbols, are also supported. The package supports exact symbolic computations over a range of coefficient fields, including number fields and finite fields of positive characteristic, as well as numerical computations intended for applications in mathematical physics such as anyon models and conformal field theory.

Current highlights include:

A general, extensible category framework together with additive, linear, abelian, monoidal, tensor, and fusion structures.
Support for skeletal fusion categories, including exact and numerical access to $F$-symbols, $R$-symbols, pivotal data, and related invariants.
Integration of fusion-category data from AnyonWiki, providing access to a large collection of fusion categories.
A generic algorithm for computing Drinfeld centers, producing explicit central objects with half-braidings rather than only abstract equivalence classes (Mäurer and Thiel, 2024).
Computation of the Drinfeld centers, including $F$-symbols and $R$-symbols, of the 76 monoidal presentations underlying the 279 multiplicity-free AnyonWiki entries of rank at most 5; the results are stored in our TensorCategoriesDatabase. Some of these centers yield new modular categories with high rank and nontrivial fusion multiplicities, for example rank 21 and fusion multiplicity 2 (Mäurer and Thiel, 2024).
Explicit computation of $F$-symbols, $R$-symbols, and pivotal coefficients for the Drinfeld center of the Haagerup subfactor (Mäurer et al., 2026).

Showcase

Here is a showcase example computing the center $\mathcal{Z}(\mathcal{C})$ of the Ising fusion category $\mathcal{C}$ over the field $\mathbb{Q}(\sqrt{2})$. The computation shows that $\mathcal{Z}(\mathcal{C})$ is not split over $\mathbb{Q}(\sqrt{2})$, i.e. some simple objects will decompose after scalar extension to $\mathbb{C}$. We then compute the multiplication table of its Grothendieck ring and the $S$-matrix of this non-split modular category. Simple-object enumeration uses randomized algebra algorithms, so the order below is one possible output.

julia> using TensorCategories, Oscar

julia> K,r2 = quadratic_field(2)
(Real quadratic field defined by x^2 - 2, sqrt(2))

julia> C = ising_category(K,r2)

julia> simples(C)
3-element Vector{SixJObject}:
 𝟙
 χ
 X

julia> Z = center(C)
Drinfeld center of Ising fusion category

julia> S = simples(Z)
5-element Vector{CenterObject}:
 Central object: 𝟙
 Central object: 𝟙
 Central object: 𝟙 ⊕ χ
 Central object: 2⋅χ
 Central object: 4⋅X

julia> T = only([T for T in S if int_dim(End(T)) == 2]);

julia> End(T)
Vector space of dimension 2 over Real quadratic field defined by x^2 - 2.

julia> print_multiplication_table(S, ["X$i" for i in eachindex(S)])
5×5 Matrix{String}:
 "X1"  "X2"  "X3"            "X4"           "X5"
 "X2"  "X1"  "X3"            "X4"           "X5"
 "X3"  "X3"  "X1 ⊕ X2 ⊕ X4"  "2⋅X3"         "2⋅X5"
 "X4"  "X4"  "2⋅X3"          "2⋅X1 ⊕ 2⋅X2"  "2⋅X5"
 "X5"  "X5"  "2⋅X5"          "2⋅X5"         "4⋅X1 ⊕ 4⋅X2 ⊕ 8⋅X3 ⊕ 4⋅X4"

julia> smatrix(Z)
[        1            1    2    2    4*sqrt(2)]
[        1            1    2    2   -4*sqrt(2)]
[        2            2    0   -4            0]
[        2            2   -4    4            0]
[4*sqrt(2)   -4*sqrt(2)    0    0            0]

The two-dimensional endomorphism algebra of $T$ shows why this center is not split over $\mathbb Q(\sqrt2)$: a split simple would have endomorphism algebra equal to the coefficient field. The five displayed objects become nine simple objects after extension to a splitting field; the center chapter carries out that computation.

Installation

You need to have Julia installed. To install TensorCategories.jl, run:

julia> import Pkg
julia> Pkg.add("TensorCategories")

This also installs all dependencies, including OSCAR.

How to cite

If TensorCategories.jl contributes to your research, please cite the paper that introduced the software:

@misc{MaeurerThiel2024ComputingCenter,
  author        = {M{\"a}urer, Fabian and Thiel, Ulrich},
  title         = {Computing the center of a fusion category},
  year          = {2024},
  eprint        = {2406.13438},
  archivePrefix = {arXiv},
  primaryClass  = {math.RT},
  doi           = {10.48550/arXiv.2406.13438}
}

The software itself is archived on Zenodo and can be cited as follows:

@software{Maeurer2026TensorCategories,
  author    = {M{\"a}urer, Fabian},
  title     = {{TensorCategories.jl}},
  year      = {2026},
  publisher = {Zenodo},
  doi       = {10.5281/zenodo.18760250},
  url       = {https://doi.org/10.5281/zenodo.18760250}
}

License

The TensorCategories.jl package is licensed under the GNU General Public License v3.0 or later.

See LICENSE for the full license text. See COPYRIGHT for copyright information.

Acknowledgements

TensorCategories.jl was initiated by Ulrich Thiel (RPTU University Kaiserslautern-Landau) within his project A20 "Towards unipotent character sheaves associated to Coxeter groups" (2020–2024) of the SFB-TRR 195 "Symbolic Tools in Mathematics and their Application", funded by the German Research Foundation (DFG). The package was created and developed by Fabian Mäurer as part of his Master's and PhD work under Thiel's supervision (2021–2026); see Mäurer (2026). Its development is currently supported by Thiel's project A20 "Categorical representation theory" (2024–2028) in the SFB-TRR 195. Additional support is provided by the Forschungsinitiative "SymbTools" of the state of Rheinland-Pfalz, in which Thiel is one of the project leaders.

Gert Vercleyen contributed to the integration of the data from his AnyonWiki.

Since version 0.7, AI assistance has been used in the development of TensorCategories.jl.