Editing Unitary group (section)

== Related groups ==

=== 2-out-of-3 property ===
The unitary group is the 3-fold intersection of the [[orthogonal group|orthogonal]], [[linear complex structure|complex]], and [[symplectic group|symplectic]] groups:
: <math>\operatorname{U}(n) = \operatorname{O}(2n) \cap \operatorname{GL}(n, \mathbf{C}) \cap \operatorname{Sp}(2n, \mathbf{R}) .</math>

Thus a unitary structure can be seen as an orthogonal structure, a complex structure, and a symplectic structure, which are required to be ''compatible'' (meaning that one uses the same ''J'' in the complex structure and the symplectic form, and that this ''J'' is orthogonal; writing all the groups as matrix groups fixes a ''J'' (which is orthogonal) and ensures compatibility).

In fact, it is the intersection of any ''two'' of these three; thus a compatible orthogonal and complex structure induce a symplectic structure, and so forth.<ref>{{cite book |last=Arnold |first=V.I. |title=Mathematical Methods of Classical Mechanics |url=https://archive.org/details/mathematicalmeth00arno_565 |url-access=limited |publisher=Springer |edition=Second |year=1989 |page=[https://archive.org/details/mathematicalmeth00arno_565/page/n238 225]}}</ref><ref>{{cite web |url=http://math.ucr.edu/home/baez/symplectic.html |title=Symplectic, Quaternionic, Fermionic |last=Baez |first=John |access-date=1 February 2012}}</ref>

At the level of equations, this can be seen as follows:
: <math>\begin{array}{r|r}
  \text{Symplectic} &
  A^\mathsf{T}JA = J \\
  \hline
  \text{Complex} &
  A^{-1}JA = J \\
  \hline
  \text{Orthogonal} &
  A^\mathsf{T} = A^{-1}
\end{array}</math>

Any two of these equations implies the third.

At the level of forms, this can be seen by decomposing a Hermitian form into its real and imaginary parts: the real part is symmetric (orthogonal), and the imaginary part is skew-symmetric (symplectic)—and these are related by the complex structure (which is the compatibility). On an [[almost Kähler manifold]], one can write this decomposition as {{nowrap|1=''h'' = ''g'' + ''iω''}}, where ''h'' is the Hermitian form, ''g'' is the [[Riemannian metric]], ''i'' is the [[almost complex manifold|almost complex structure]], and ''ω'' is the [[almost symplectic manifold|almost symplectic structure]].

From the point of view of [[Lie group]]s, this can partly be explained as follows: O(2''n'') is the [[maximal compact subgroup]] of {{nowrap|GL(2''n'', '''R''')}}, and U(''n'') is the maximal compact subgroup of both {{nowrap|GL(''n'', '''C''')}} and Sp(2''n''). Thus the intersection {{nowrap|O(2''n'') ∩ GL(''n'', '''C''')}} or {{nowrap|O(2''n'') ∩ Sp(2''n'')}} is the maximal compact subgroup of both of these, so U(''n''). From this perspective, what is unexpected is the intersection {{nowrap|1=GL(''n'', '''C''') ∩ Sp(2''n'') = U(''n'')}}.

=== Special unitary and projective unitary groups ===
[[File:PSU-PU.svg|right]]
{{see also|Special unitary group|Projective unitary group}}

Just as the orthogonal group O(''n'') has the [[special orthogonal group]] SO(''n'') as subgroup and the [[projective orthogonal group]] PO(''n'') as quotient, and the [[projective special orthogonal group]] PSO(''n'') as [[subquotient]], the unitary group U(''n'') has associated to it the [[special unitary group]] SU(''n''), the [[projective unitary group]] PU(''n''), and the [[projective special unitary group]] PSU(''n''). These are related as by the [[commutative diagram]] at right; notably, both projective groups are equal: {{nowrap|1=PSU(''n'') = PU(''n'')}}.

The above is for the classical unitary group (over the complex numbers) – for [[#Finite_fields|unitary groups over finite fields]], one similarly obtains special unitary and projective unitary groups, but in general {{nowrap|PSU(''n'', ''q''<sup>2</sup>) ≠ PU(''n'', ''q''<sup>2</sup>)}}.