Fusion category - Knowlpedia

Fix an algebraically closed field $k$ . A fusion category over $k$ is an essentially small $k$ -linear semisimple abelian category with a rigid monoidal structure, such that:

each vector space $\operatorname{Hom}(X,Y)$ is finite-dimensional;
there are finitely many isomorphism classes of simple objects;
the tensor product is $k$ -bilinear on morphisms;
the tensor unit $\mathbb1$ is simple.

Here semisimplicity means every object is a finite direct sum of simples. Essentially small means that the isomorphism classes of objects form a set. Many treatments additionally assume $\operatorname{char}k=0$ ; that assumption will be stated whenever needed here.

Unpacking the bundle

Ingredient	What it lets you do
Abelian	Form kernels, cokernels, and exact sequences.
$k$ -linear	Add and scale parallel arrows.
Semisimple	Decompose objects into finite sums of simple pieces.
Monoidal	Tensor objects and arrows, with coherent associativity and unit maps.
Rigid	Evaluate and coevaluate left and right duals.
Finiteness and simple unit	Use a finite list of simple building blocks and an indecomposable tensor identity.

Neither braiding nor symmetry is required. Forgetting the simple-unit condition gives a multifusion category. Equivalently, a fusion category is a semisimple finite tensor category.

Fusion of simple objects

Choose simple representatives $S_0=\mathbb1,S_1,\ldots,S_r$ . Their tensor products decompose as

S_i\otimes S_j\cong\bigoplus_{\ell=0}^{r}S_\ell^{\oplus N_{ij}^{\ell}}, \qquad N_{ij}^{\ell}=\dim_k\operatorname{Hom}(S_\ell,S_i\otimes S_j).

The nonnegative integers $N_{ij}^{\ell}$ are the fusion multiplicities. These integers do not specify the associator; the monoidal coherence data remain part of the category.

Examples

Finite-dimensional $k$ -vector spaces form a fusion category with one simple class. For a finite group $G$ and $\operatorname{char}k=0$ , finite-dimensional representations of $G$ give another example.

Finite-dimensional $G$ -graded vector spaces, with $(V\otimes W)_g=\bigoplus_{ab=g}V_a\otimes W_b$ , give an example for any finite group $G$ . Simples are one-dimensional spaces concentrated in a single degree. If $G$ is nonabelian, this category cannot be braided: the simple tensor products in degrees $gh$ and $hg$ would otherwise have to be isomorphic.

The unit and the ground field

Under the hypotheses above, simplicity of $\mathbb1$ is equivalent to $\operatorname{End}(\mathbb1)\cong k$ . In a semisimple category, a decomposition into two nonzero summands gives a nontrivial idempotent endomorphism, so an endomorphism algebra equal to $k$ forces simplicity. Conversely, simplicity, finite-dimensional Hom spaces, and algebraic closure force its endomorphisms to be scalars. The finite-dimensionality and ground-field assumptions should not be dropped from that converse.

References

P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik, Tensor Categories (2015), §4.1 and §4.9; author-hosted text.