Editing Symplectic group (section)

=={{math|Sp(''n'')}}==

The '''compact symplectic group'''<ref>{{harvnb|Hall|2015}} Section 1.2.8</ref> {{math|Sp(''n'')}} is the intersection of {{math|Sp(2''n'', '''C''')}} with the <math>2n\times 2n</math> unitary group:

:<math>\operatorname{Sp}(n):=\operatorname{Sp}(2n;\mathbf C)\cap\operatorname{U}(2n)=\operatorname{Sp}(2n;\mathbf C)\cap\operatorname {SU} (2n).</math>

It is sometimes written as {{math|USp(2''n'')}}. Alternatively, {{math|Sp(''n'')}} can be described as the subgroup of {{math|GL(''n'', '''H''')}} (invertible [[quaternion]]ic matrices) that preserves the standard [[hermitian form]] on {{math|'''H'''<sup>''n''</sup>}}:

:<math>\langle x, y\rangle = \bar x_1 y_1 + \cdots + \bar x_n y_n.</math>

That is, {{math|Sp(''n'')}} is just the [[Classical group#Sp(p, q) – the quaternionic unitary group|quaternionic unitary group]], {{math|U(''n'', '''H''')}}.<ref>{{harvnb|Hall|2015}} p. 14</ref> Indeed, it is sometimes called the '''hyperunitary group'''. Also Sp(1) is the group of quaternions of norm {{math|1}}, equivalent to {{math|[[SU(2)]]}} and topologically a [[3-sphere|{{math|3}}-sphere]] {{math|S<sup>3</sup>}}.

Note that {{math|Sp(''n'')}} is ''not'' a symplectic group in the sense of the previous section&mdash;it does not preserve a non-degenerate skew-symmetric {{math|'''H'''}}-bilinear form on {{math|'''H'''<sup>''n''</sup>}}: there is no such form except the zero form. Rather, it is isomorphic to a subgroup of {{math|Sp(2''n'', '''C''')}}, and so does preserve a complex [[symplectic manifold | symplectic form]] in a vector space of twice the dimension. As explained below, the Lie algebra of {{math|Sp(''n'')}} is the compact [[real form]] of the complex symplectic Lie algebra {{math|'''sp'''(2''n'', '''C''')}}.

{{math|Sp(''n'')}} is a real Lie group with (real) dimension {{math|''n''(2''n'' + 1)}}. It is [[Compact space|compact]] and [[simply connected]].<ref>{{harvnb|Hall|2015}} Prop. 13.12</ref>

The Lie algebra of {{math|Sp(''n'')}} is given by the quaternionic [[skew-Hermitian]] matrices, the set of {{math|''n''-by-''n''}} quaternionic matrices that satisfy

:<math>A+A^{\dagger} = 0</math>

where {{math|A<sup>†</sup>}} is the [[conjugate transpose]] of {{math|A}} (here one takes the quaternionic conjugate). The Lie bracket is given by the commutator.

===Important subgroups===
Some main subgroups are:

: <math>\operatorname{Sp}(n) \supset \operatorname{Sp}(n-1)</math>
: <math>\operatorname{Sp}(n) \supset \operatorname{U}(n) </math>
: <math>\operatorname{Sp}(2) \supset \operatorname{O}(4)</math>

Conversely it is itself a subgroup of some other groups:

: <math>\operatorname{SU}(2n) \supset \operatorname{Sp}(n)</math>
: <math>\operatorname{F}_4 \supset \operatorname{Sp}(4)</math>
: <math>\operatorname{G}_2 \supset \operatorname{Sp}(1)</math>

There are also the [[isomorphism]]s of the [[Lie algebras]] {{math|1='''sp'''(2) = '''so'''(5)}} and {{math|1='''sp'''(1) = '''so'''(3) = '''su'''(2)}}.