nLab semisimple object

Definition

Definition

An object $X$ in an abelian category is said to be semisimple or completely reducible if it is a coproduct (direct sum) of simple objects.

Remark

Sometimes this notion is considered in a bit more general setup than abelian (where it is most often used); sometimes with subtle variants. For Lie algebras, it is a bit different by convention: a Lie algebra is a semisimple Lie algebra if it is a direct sum of nonabelian simple Lie algebras; though a 1-dimensional abelian Lie algebra is simple in the categorical (and in common) sense.

Definition

A semisimple object is isotypic if it is a direct sum of isomorphic simple objects (the isomorphism class of a simple object is called its type).

Properties

Proposition

An object in an abelian category is semisimple if and only if it is the coproduct of all its simple subobjects.

Proposition

If an object $X$ in an abelian category is semisimple then every short exact sequence $0 \to A \to X \to B \to 0$ splits.

An abelian category in which all short exact sequences split is called a spectral category, and not every spectral category is semisimple (see MathOverflow).

Special classes

• semisimple left $R$-modules where $R$ is a ring;

• completely reducible representations of a group

Related concepts

• reductive group

• semisimple category

References

• Wikipedia, Semisimplicity.

• Wikipedia, Semisimple module.