Finite and semisimple categories

An abelian category $\mathcal C$ is semisimple if every object is a direct sum of simple objects (Etingof et al., 2015; Definition 1.5.1). If $\mathcal C$ is locally finite, these direct sums are finite. It is then a Krull–Schmidt category, and its simple objects are precisely its indecomposable objects. Thus every object $X$ admits a decomposition

\[\label{eq:semisimple-decomposition} X\cong\bigoplus_{S\in\operatorname{Irr}(\mathcal C)}S^{\oplus m_S}.\]

If $D_S=\operatorname{End}_{\mathcal C}(S)$, then $\operatorname{Hom}_{\mathcal C}(S,X)$ is a right $D_S$-module by precomposition and

\[\label{eq:nonsplit-hom-multiplicity} m_S=\dim_{D_S}\operatorname{Hom}_{\mathcal C}(S,X).\]

Schur's lemma says that $D_S$ is a division algebra (Etingof et al., 2015; Lemma 1.5.2). Outside a semisimple category, the converse in Schur's lemma fails: an object with division endomorphism algebra need not be simple, and an indecomposable object need not be simple.

The interface

The predicate is_semisimple(C) records that this property is known for the implemented category and consequently implies is_abelian(C). Semisimplicity alone does not assert local finiteness: the Krull–Schmidt conclusion above uses that additional hypothesis. When the backend can enumerate the simple objects, simples(C) returns chosen representatives of their isomorphism classes. The order of this list belongs to the implementation and must not be treated as a mathematical invariant.

For an object $X$, decompose(X) returns pairs (S,m) representing the decomposition in equation \eqref{eq:semisimple-decomposition}. The generic implementation of decompose(X,S) computes the multiplicity using equation \eqref{eq:nonsplit-hom-multiplicity}; it does not replace the $D_S$-dimension by the dimension over $k$.

The generic is_simple(X) uses indecomposability only when the parent category is semisimple. Otherwise a category-specific test is required.

Example: Finite-group representations

For a finite group $G$, Maschke's theorem gives

\[\label{eq:maschke-condition} \operatorname{Rep}_k(G)\text{ is semisimple} \quad\Longleftrightarrow\quad \operatorname{char}(k)\nmid |G|.\]

using TensorCategories, Oscar
G = cyclic_group(5)
C0 = representation_category(QQ, G)
C5 = representation_category(GF(5), G)
@assert is_semisimple(C0)
@assert !is_semisimple(C5)
(is_semisimple(C0), is_semisimple(C5))

(true, false)

Equation \eqref{eq:maschke-condition} concerns semisimplicity only. Even when it holds, the simple representations need not remain simple after extending the coefficient field.

When a locally finite semisimple category has only finitely many simple isomorphism classes, it is finite: every object is projective, so the direct sum of representatives of the simples is a projective generator.

Continue with scalar extension and splitting.