semisimple

English

Etymology

From semi- +‎ simple.

Pronunciation

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Adjective

semisimple (not comparable)

1. (mathematics, algebra, of an algebraic structure) In any of several technical senses, decomposable into sub-objects that have a simple structure.
   1. (category theory, most generally, of an abelian category) Containing a collection of simple objects such that all objects in the category are direct sums of these simple objects.
   2. (module theory, of a module) In which each submodule is a direct summand; equivalently, equal to a direct sum of simple submodules.
   3. (ring theory, of an algebra or ring) Semisimple as a module over itself; equivalently, such that all (left) modules are semisimple.
      1. (of a ring, somewhat proscribed) Semiprimitive: having trivial Jacobson radical.

         Synonym: Jacobson semisimple

   4. (linear algebra, of an operator or matrix) For which every invariant subspace has an invariant complement, equivalent to the minimal polynomial being squarefree.
   5. (Lie theory, of a Lie algebra) Being a direct sum of simple Lie algebras.
   6. (representation theory, of a linear representation of a group or algebra) Being a direct sum of simple representations (also known as irreducible representations).
   7. (group theory, of an algebraic group) Being a linear algebraic group whose radical of the identity component is trivial.

French

Alternative forms

• semi-simple

Etymology

From semi- +‎ simple.

Adjective

semisimple (plural semisimples)

1. (mathematics) semisimple