A connected Lie Group is called ''semisimple'' when its Lie algebra is; and the same for Algebraic Group s. 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.
