Semisimple category - Knowlpedia

In the finite-length convention used here, a semisimple category is an abelian category in which each object is isomorphic to a finite direct sum of simple objects:

X\cong\bigoplus_{i=1}^{r} S_i^{\oplus n_i},\qquad n_i\in\mathbb Z_{\ge0}.

The zero object is the empty sum. The finite list of simples occurring in this expression may depend on $X$ ; semisimplicity alone does not require finitely many simple isomorphism classes in the whole category.

Splitting exact sequences

For a finite-length abelian category, this is equivalent to every short exact sequence splitting. In a sequence $0\to A\xrightarrow{i}B\xrightarrow{p}C\to0$ , a section $s:C\to B$ with $ps=\operatorname{id}_C$ gives an isomorphism $B\cong A\oplus C$ compatible with the sequence:

Splitting a composition series successively gives the direct-sum description. The resulting choice of summands need not be canonical.

Examples and conventions

Finite-dimensional vector spaces are semisimple. Finite-dimensional representations of a finite group over a characteristic-zero field are semisimple by averaging a linear projection onto an invariant subspace. Finite-dimensional $\mathbb Z$ -graded vector spaces are also semisimple, but have one simple isomorphism class for every integer degree, so they are not a fusion category.

Some authors allow arbitrary direct sums of simples in “semisimple category.” The finite-sum convention here is the one relevant to fusion categories; it excludes infinite-dimensional vector spaces as objects.

References

P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik, Tensor Categories (2015), §§1.5 and 4.8; author-hosted text.