semisimple object




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.


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.


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


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