Editing Semisimple Lie algebra (section)

==Semisimple and reductive groups==
{{Main|Semisimple Lie group|Reductive group}}
A connected Lie group is called [[Semisimple Lie group|semisimple]] if its Lie algebra is a semisimple Lie algebra, i.e. a direct sum of simple Lie algebras. It is called [[Reductive group|reductive]] if its Lie algebra is a direct sum of simple and trivial (one-dimensional) Lie algebras. Reductive groups occur naturally as symmetries of a number of mathematical objects in algebra, geometry, and physics. For example, the group <math>GL_n(\mathbb{R})</math> of symmetries of an ''n''-dimensional real [[vector space]] (equivalently, the group of invertible matrices) is reductive.