Reductive - Definition and Overview

In mathematics, the term semisimple is used in a number of related ways, within different subjects. The common theme is the idea of a decomposition into 'simple' parts, that fit together in the cleanest way (by direct sum).

• A semisimple module is one in which each submodule is a direct summand. In particular, a semisimple representation is completely reducible, i.e., is a direct sum of irreducible representations (under a descending chain condition). One speaks of an abelian category as being semisimple when every object has the corresponding property.

• A semisimple ring or semisimple algebra is one that is semisimple as a module over itself.

• A semisimple matrix is diagonalizable over any algebraically closed field containing its entries. In practice this means that it has a diagonal matrix as its Jordan normal form.

• A semisimple Lie algebra is a Lie algebra which is a direct sum of simple Lie algebras.

• A connected Lie group is called semisimple when its Lie algebra is; and the same for algebraic groups. Every finite dimensional representation of a semisimple Lie algebra, Lie group, or algebraic group in characteristic 0 is semisimple, i.e., completely reducible, but the converse is not true. (See reductive group.) Moreover, in characteristic p>0, semisimple Lie groups and Lie algebras have finite dimensional representations which are not semisimple. An element of a semisimple Lie group or Lie algebra is itself semisimple if its image in every finite-dimensional representation is semisimple in the sense of matrices.

• A linear algebraic group G is called semisimple if the radical of the identity component G⁰ of G is trivial. G is semisimple if and only if G has no nontrivial connected abelian normal subgroup.