additive and abelian categories
(AB1) pre-abelian category
(AB2) abelian category
(AB5) Grothendieck category
left/right exact functor
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.
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).
An object in an abelian category is semisimple if and only if it is the coproduct of all its simple subobjects.
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).
completely reducible representations of a group
Wikipedia, Semisimplicity.
Wikipedia, Semisimple module.
Last revised on July 1, 2023 at 06:02:58. See the history of this page for a list of all contributions to it.