Contents

category theory

# Contents

## 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