Editing Linear algebraic group (section)

==Semisimple and reductive groups==
{{main|Reductive group}}
A connected linear algebraic group ''G'' over an algebraically closed field is called '''semisimple''' if every smooth connected solvable normal subgroup of ''G'' is trivial. More generally, a connected linear algebraic group ''G'' over an algebraically closed field is called '''[[reductive group|reductive]]''' if every smooth connected unipotent normal subgroup of ''G'' is trivial.<ref>Milne (2017), Definition 6.46.</ref> (Some authors do not require reductive groups to be connected.) A semisimple group is reductive. A group ''G'' over an arbitrary field ''k'' is called semisimple or reductive if <math>G_{\overline k}</math> is semisimple or reductive. For example, the group ''SL''(''n'') of ''n'' × ''n'' matrices with determinant 1 over any field ''k'' is semisimple, whereas a nontrivial torus is reductive but not semisimple. Likewise, ''GL''(''n'') is reductive but not semisimple (because its center ''G''<sub>''m''</sub> is a nontrivial smooth connected solvable normal subgroup).

Every compact connected Lie group has a '''[[complexification (Lie group)|complexification]]''', which is a complex reductive algebraic group. In fact, this construction gives a one-to-one correspondence between compact connected Lie groups and complex reductive groups, up to isomorphism.<ref>Bröcker & tom Dieck (1985), section III.8; Conrad (2014), section D.3.</ref>

A linear algebraic group ''G'' over a field ''k'' is called '''simple''' (or ''k''-'''simple''') if it is semisimple, nontrivial, and every smooth connected normal subgroup of ''G'' over ''k'' is trivial or equal to ''G''.<ref>Conrad (2014), after Proposition 5.1.17.</ref> (Some authors call this property "almost simple".) This differs slightly from the terminology for abstract groups, in that a simple algebraic group may have nontrivial center (although the center must be finite). For example, for any integer ''n'' at least 2 and any field ''k'', the group ''SL''(''n'') over ''k'' is simple, and its center is the group scheme μ<sub>''n''</sub> of ''n''th roots of unity.

Every connected linear algebraic group ''G'' over a perfect field ''k'' is (in a unique way) an extension of a reductive group ''R'' by a smooth connected unipotent group ''U'', called the '''unipotent radical''' of ''G'':
:<math>1\to U\to G\to R\to 1.</math>
If ''k'' has characteristic zero, then one has the more precise '''[[Levi decomposition]]''': every connected linear algebraic group ''G'' over ''k'' is a semidirect product <math>R\ltimes U</math> of a reductive group by a unipotent group.<ref>Conrad (2014), Proposition 5.4.1.</ref>