category theory

# Contents

## Definition

###### Definition

An object $X$ in an abelian category $A$ 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).

## Special classes

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

• completely reducible representations of a group

## References

Revised on October 29, 2013 22:22:43 by Urs Schreiber (82.169.114.243)