nLab
semisimple object

Context

Category theory

Concepts

Universal constructions

    • /

    • /

Theorems

Extensions

Applications

Additive and abelian categories

Context and background

Categories

  • ,

  • (AB1)

  • (AB2)

  • (AB5)

Functors

  • left/right

Derived categories

Contents

Definition

Definition

An object XX in an abelian category AA 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

References

Last revised on October 29, 2013 at 22:22:43. See the history of this page for a list of all contributions to it.