Editing Symplectic group (section)

==Relationship between the symplectic groups==
Every complex, [[semisimple Lie algebra]] has a [[Real form (Lie theory)#Split real form|split real form]] and a [[Real form (Lie theory)#Compact real form|compact real form]]; the former is called a [[complexification]] of the latter two.

The Lie algebra of {{math|Sp(2''n'', '''C''')}} is [[Semisimple Lie algebra|semisimple]] and is denoted {{math|'''sp'''(2''n'', '''C''')}}. Its [[Real form (Lie theory)#Split real form|split real form]] is {{math|'''sp'''(2''n'', '''R''')}} and its [[Real form (Lie theory)#Compact real form|compact real form]] is {{math|'''sp'''(''n'')}}. These correspond to the Lie groups {{math|Sp(2''n'', '''R''')}} and {{math|Sp(''n'')}} respectively.

The algebras, {{math|'''sp'''(''p'', ''n'' − ''p'')}}, which are the Lie algebras of {{math|Sp(''p'', ''n'' − ''p'')}}, are the [[Metric signature|indefinite signature]] equivalent to the compact form.