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Dictionary

Words, grammatical forms and meanings linked to the ontology.

semisimple en · ADJ

Pronunciation

Etymology

From semi- + simple.

Meanings

  1. (not-comparable, of an abelian category, of an algebraic structure, qualifier:most generally) In any of several technical senses, decomposable into sub-objects that have a simple structure. ▸ Containing a collection of simple objects such that all objects in the category are direct sums of these simple objects.
  2. (not-comparable, of a module, of an algebraic structure, qualifier:module theory) In any of several technical senses, decomposable into sub-objects that have a simple structure. ▸ In which each submodule is a direct summand; equivalently, equal to a direct sum of simple submodules.
  3. (not-comparable, of an algebra or ring, of an algebraic structure, qualifier:ring theory) In any of several technical senses, decomposable into sub-objects that have a simple structure. ▸ Semisimple as a module over itself; equivalently, such that all (left) modules are semisimple.
  4. (not-comparable, of a ring, of an algebra or ring, of an algebraic structure, proscribed, qualifier:ring theory) In any of several technical senses, decomposable into sub-objects that have a simple structure. ▸ Semisimple as a module over itself; equivalently, such that all (left) modules are semisimple. ▸ Semiprimitive: having trivial Jacobson radical.
  5. (not-comparable, of an algebraic structure, of an operator or matrix) In any of several technical senses, decomposable into sub-objects that have a simple structure. ▸ For which every invariant subspace has an invariant complement, equivalent to the minimal polynomial being squarefree.
  6. (not-comparable, of a Lie algebra, of an algebraic structure, qualifier:Lie theory) In any of several technical senses, decomposable into sub-objects that have a simple structure. ▸ Being a direct sum of simple Lie algebras.
  7. (not-comparable, of a linear representation of a group or algebra, of an algebraic structure, qualifier:representation theory) In any of several technical senses, decomposable into sub-objects that have a simple structure. ▸ Being a direct sum of simple representations (also known as irreducible representations).
  8. (not-comparable, of an algebraic group, of an algebraic structure) In any of several technical senses, decomposable into sub-objects that have a simple structure. ▸ Being a linear algebraic group whose radical of the identity component is trivial.

Deriveds

nonsemisimple · cosemisimple · semisimply

Synonyms

Jacobson semisimple