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 (or
linear transformation of
finite-dimensional vector spaces) is one for which every
invariant subspace has an invariant complement. This is equivalent to the
minimal polynomial having only
irreducible factors with multiplicity one. Over a
perfect field, this amounts to saying that the matrix has simple roots in the algebraic closure (or any larger algebraically closed field), i.e., it becomes
diagonalizable over the algebraic closure. Thus, over an algebraically closed field, “semisimple” and “diagonalizable” are synonymous for matrices.A
semisimple Lie algebra is a
Lie algebra which is a direct sum of
simple Lie algebras. A Lie algebra is simple if its dimension is larger than one and if it does not contain any nontrivial ideals. This means that if is such that for any if , then is either zero or the whole Lie algebra.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 that 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.An
abelian category A is said to be semisimple if every
short exact sequence splits in A. For example, the
representation categories of
compact groups are semisimple abelian.
See more at Wikipedia.org...
Free Download Now!
