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\begin{document}

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\section*{unitary group}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{group_theory}{}\paragraph*{{Group Theory}}\label{group_theory}

[[!include group theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{HomotopyGroups}{Homotopy groups}\dotfill \pageref*{HomotopyGroups} \linebreak
\noindent\hyperlink{in_infinite_dimension}{In infinite dimension}\dotfill \pageref*{in_infinite_dimension} \linebreak
\noindent\hyperlink{SUn}{Relation to special unitary group}\dotfill \pageref*{SUn} \linebreak
\noindent\hyperlink{RelationToOrthogonalSymplecticAndGeneralLinearGroup}{Relation to orthogonal, symplectic and general linear group}\dotfill \pageref*{RelationToOrthogonalSymplecticAndGeneralLinearGroup} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{coset_spaces}{Coset spaces}\dotfill \pageref*{coset_spaces} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

For a [[natural number]] $n \in \mathbb{N}$, the \textbf{unitary group} $U(n)$ is the [[group]] of [[isometry|isometries]] of the $n$-dimensional complex [[Hilbert space]] $\mathbb{C}^n$. This is canonically identified with the group of $n \times n$ [[unitary matrices]].

More generally, for a Hilbert space $\mathcal{H}$, $U(\mathcal{H})$ is the group of [[unitary operator]]s on that Hilbert space. For the purposes of studying unitary representations of Lie groups, the topology is chosen to be the [[operator topology|strong operator topology]], although other topologies on $U(\mathcal{H})$ are frequently considered for other purposes.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

The unitary groups are naturally [[topological group]]s and [[Lie groups]] (infinite dimensional if $\mathcal{H}$ is infinite dimensional).

\begin{prop}
\label{UnitaryGroupIsCompact}\hypertarget{UnitaryGroupIsCompact}{}
The unitary group $U(n)$ is [[compact topological space]], hence in particular a [[compact Lie group]].

\end{prop}
\hypertarget{HomotopyGroups}{}\subsubsection*{{Homotopy groups}}\label{HomotopyGroups}

\begin{prop}
\label{InclusionOfUnitaryGroupnIntoUnitaryGroupnPlusIneIsnMinus1Equivalence}\hypertarget{InclusionOfUnitaryGroupnIntoUnitaryGroupnPlusIneIsnMinus1Equivalence}{}
For $n,N \in \mathbb{N}$, $n \leq N$, then the canonical inclusion of unitary groups

\begin{displaymath}
U(n) \hookrightarrow U(N)
\end{displaymath}
is a [[n-equivalence|2n-equivalence]], hence induces an [[isomorphism]] on [[homotopy groups]] in degrees $\lt 2n$ and a [[surjection]] in degree $2n$.

\end{prop}
\begin{proof}
Consider the [[coset]] [[quotient]] [[projection]]

\begin{displaymath}
U(n)
  \longrightarrow
  U(n+1)
  \longrightarrow
  U(n+1)/U(n)
  \,.
\end{displaymath}
By prop. \ref{UnitaryGroupIsCompact} and by \href{coset#QuotientProjectionForCompactLieSubgroupIsPrincipal}{this corollary}, the projection $U(n+1)\to U(n+1)/U(n)$ is a [[Serre fibration]]. Furthermore, example \ref{nSphereAsUnitaryCosetSpace} identifies the [[coset]] with the [[n-sphere|(2n+1)-sphere]]

\begin{displaymath}
S^{2n+1}\simeq U(n+1)/U(n)
  \,.
\end{displaymath}
Therefore the [[long exact sequence of homotopy groups]] of the [[fiber sequence]] $U(n)\to U(n+1) \to S^{2n+1}$ is of the form

\begin{displaymath}
\cdots
    \to
  \pi_{\bullet+1}(S^{2n+1})
    \longrightarrow
  \pi_\bullet(U(n))
    \longrightarrow
  \pi_\bullet(U(n+1))
    \longrightarrow
  \pi_\bullet(S^{2n+1})
   \to
  \cdots
\end{displaymath}
Since $\pi_{\leq 2n}(S^{2n+1}) = 0$, this implies that

\begin{displaymath}
\pi_{\lt 2n}(U(n))
    \overset{\simeq}{\longrightarrow}
  \pi_{\lt 2n}(U(n+1))
\end{displaymath}
is an isomorphism and that

\begin{displaymath}
\pi_{2n}(U(n))
    \longrightarrow
  \pi_{2n}(U(n+1))
\end{displaymath}
is surjective. Hence now the statement follows by induction over $N-n$.

\end{proof}
It follows that the [[homotopy groups]] $\pi_k(U(n))$ are independent of $n$ for $n \gt \frac{k}{2}$ (the ``stable range''). So if $U = \underset{\longrightarrow}{\lim}_n U(n)$, then $\pi_k(U(n)) = \pi_k(U)$. By [[Bott periodicity]] we have

\begin{displaymath}
\itexarray{
\pi_{2k+0}(U) &= 0\\
\pi_{2k+1}(U) &= \mathbb{Z}.
}
\end{displaymath}
In the unstable range for low $n$ they instead start out as follows

\begin{tabular}{l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l}
$G$&$\pi_1$&$\pi_2$&$\pi_3$&$\pi_4$&$\pi_5$&$\pi_6$&$\pi_7$&$\pi_8$&$\pi_9$&$\pi_10$&$\pi_11$&$\pi_12$&$\pi_13$&$\pi_14$&$\pi_15$\\
\hline 
$U(1)$&$\mathbb{Z}$&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\
$U(2)$&``&0&$\mathbb{Z}$&$\mathbb{Z}_2$&$\mathbb{Z}_2$&$\mathbb{Z}_{12}$&$\mathbb{Z}_2$&$\mathbb{Z}_2$&$\mathbb{Z}_3$&$\mathbb{Z}_{15}$&$\mathbb{Z}_2$&$\mathbb{Z}_2^{\oplus 2}$&$\mathbb{Z}_3\oplus\mathbb{Z}_{12}$&$\mathbb{Z}_2^{\oplus 2}\oplus\mathbb{Z}_{84}$&$\mathbb{Z}_2\oplus\mathbb{Z}_2$\\
$U(3)$&``&``&``&0&$\mathbb{Z}$&$\mathbb{Z}_6$&0&$\mathbb{Z}_{12}$&$\mathbb{Z}_3$&$\mathbb{Z}_{30}$&$\mathbb{Z}_4$&$\mathbb{Z}_{60}$&$\mathbb{Z}_6$&$\mathbb{Z}_2\oplus\mathbb{Z}_{84}$&$\mathbb{Z}_{36}$\\
$U(4)$&``&``&``&``&``&0&$\mathbb{Z}$&$\mathbb{Z}_{24}$&$\mathbb{Z}_2$&$\mathbb{Z}_2\oplus\mathbb{Z}_{120}$&$\mathbb{Z}_4$&$\mathbb{Z}_{60}$&$\mathbb{Z}_4$&$\mathbb{Z}_2\oplus\mathbb{Z}_{1680}$&$\mathbb{Z}_2\oplus\mathbb{Z}_{72}$\\
$U(5)$&``&``&``&``&``&``&``&0&$\mathbb{Z}$&$\mathbb{Z}_{120}$&0&$\mathbb{Z}_{360}$&$\mathbb{Z}_4$&$\mathbb{Z}_{1680}$&$\mathbb{Z}_6$\\
$U(6)$&``&``&``&``&``&``&``&``&``&0&$\mathbb{Z}$&$\mathbb{Z}_{720}$&$\mathbb{Z}_2$&$\mathbb{Z}_2\oplus\mathbb{Z}_{5040}$&$\mathbb{Z}_6$\\
$U(7)$&``&``&``&``&``&``&``&``&``&``&``&0&$\mathbb{Z}$&$\mathbb{Z}_{5040}$&0\\
$U(8)$&``&``&``&``&``&``&``&``&``&``&``&``&``&0&$\mathbb{Z}$\\
\end{tabular}

Due to the [[unitary group\#SUn | relation to the special unitary group]], the higher homotopy groups of $U(n)$ and $SU(n)$ agree. The $U(2)$ row can be found using the fact that $SU(2)$ is diffeomorphic to $S^3$. The $U(3)$ row can be found using \hyperlink{MimuraToda63}{Mimura-Toda 63}. Otherwise the table is given in columns $\pi_k$, $k= 6,\ldots, 15$, and in rows $U(n)$, $n=4,\ldots,8$, by the [[Encyclopedic Dictionary of Mathematics]], Table 6.VII in Appendix A.

\hypertarget{in_infinite_dimension}{}\subsubsection*{{In infinite dimension}}\label{in_infinite_dimension}

A good discussion about the various topologies one might place on $U(\mathcal{H})$ and how they all agree and make $U(\mathcal{H})$ a [[Polish space|Polish]] group is in (\hyperlink{EspinozaUribe}{Espinoza-Uribe}).

\begin{prop}
\label{}\hypertarget{}{}
For $\mathcal{H}$ a [[Hilbert space]], which can be either finite or infinite dimensional, the unitary group $U(\mathcal{H})$ and the [[general linear group]] $GL(\mathcal{H})$, regarded as [[topological group]]s, have the same [[homotopy type]].

Additionally, $U(\mathcal{H})$ is a [[maximal compact subgroup]] of $GL(\mathcal{H})$ for finite-dimensional $\mathcal{H}$.

\end{prop}
\begin{proof}
By the [[Gram-Schmidt process]].

\end{proof}
\begin{theorem}
\label{}\hypertarget{}{}
\textbf{(Kuiper's theorem)}

For a separable infinite-dimensional complex [[Hilbert space]] $\mathcal{H}$, the unitary group $U(\mathcal{H})$ is [[contractible]] in the [[norm topology]].

\end{theorem}
See also [[Kuiper's theorem]]. Note that $U(\mathcal{H})$ is also contractible in the [[strong operator topology]] (due to Dixmier and Douady).

\begin{remark}
\label{}\hypertarget{}{}
This in contrast to the finite dimensional situation. For $n \in \mathbb{N}$ ($n \ge 1$), $U(n)$ is not contractible.

\end{remark}
Write $B U(n)$ for the [[classifying space]] of the [[topological group]] $U(n)$. Inclusion of matrices into larger matrices gives a canonical sequence of inclusions

\begin{displaymath}
\cdots \to B U(n) \hookrightarrow B U(n+1) \hookrightarrow B U(n+2) \to \cdots
  \,.
\end{displaymath}
The [[homotopy limit|homotopy]] [[direct limit]] over this is written

\begin{displaymath}
B U := {\lim_\to}_n B U(n)
\end{displaymath}
or sometimes $B U(\infty)$. Notice that this is very different from $B U(\mathcal{H})$ for $\mathcal{H}$ an infinite-dimensional Hilbert space. See [[topological K-theory]] for more on this.

\hypertarget{SUn}{}\subsubsection*{{Relation to special unitary group}}\label{SUn}

\begin{prop}
\label{}\hypertarget{}{}
For all $n \in \mathbb{N}$, the [[unitary group]] $U(n)$ is a [[split exact sequence|split]] [[group extension]] of the [[circle group]] $U(1)$ by the [[special unitary group]] $SU(n)$

\begin{displaymath}
SU(n) \to U(n) \to U(1)
  \,.
\end{displaymath}
Hence it is a [[semidirect product group]]

\begin{displaymath}
U(n) \simeq SU(n) \rtimes U(1)
  \,.
\end{displaymath}
\end{prop}
\hypertarget{RelationToOrthogonalSymplecticAndGeneralLinearGroup}{}\subsubsection*{{Relation to orthogonal, symplectic and general linear group}}\label{RelationToOrthogonalSymplecticAndGeneralLinearGroup}

The unitary group $U(n)$ is equivalently the [[intersection]] of the [[orthogonal group]] $O(2n)$, the [[symplectic group]] $Sp(2n,\mathbb{R})$ and the complex [[general linear group]] $GL(n,\mathbb{C})$ inside the real [[general linear group]] $GL(2n,\mathbb{R})$.

Actually it is already the intersection of any two of these three, a fact also known as the ``2 out of 3-property'' of the unitary group.

This intersection property makes a [[G-structure]] for $G = U(n)$ (an [[almost Hermitian structure]]) precisely a joint [[orthogonal structure]], [[almost symplectic structure]] and [[almost complex structure]]. In the [[integrability of G-structure|first-order integrable case]] this is precisely a joint [[orthogonal structure]] ([[Riemannian manifold]] structure), [[symplectic structure]] and [[complex structure]].

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

$U(1)$ is the [[circle group]].

\hypertarget{coset_spaces}{}\subsubsection*{{Coset spaces}}\label{coset_spaces}

\begin{example}
\label{nSphereAsUnitaryCosetSpace}\hypertarget{nSphereAsUnitaryCosetSpace}{}
The [[n-spheres|(2n+1)-spheres]] are [[coset spaces]] of unitary groups

\begin{displaymath}
S^{2n+1} \simeq U(n+1)/U(n)
  \,.
\end{displaymath}
\end{example}
\begin{example}
\label{ComplexStiefelManifold}\hypertarget{ComplexStiefelManifold}{}
For $n \leq k$, the [[coset]]

\begin{displaymath}
V_n(\mathbb{C}^k) \coloneqq U(k)/U(k-n)
\end{displaymath}
is called the $n$th \emph{real [[Stiefel manifold]]} of $\mathbb{C}^k$.

\end{example}
\begin{prop}
\label{}\hypertarget{}{}
The complex [[Stiefel manifold]] $V_n(\mathbb{C}^k)$ (example \ref{ComplexStiefelManifold}) is [[n-connected topological space|2(k-n)-connected]].

\end{prop}
\begin{proof}
Consider the [[coset]] [[quotient]] [[projection]]

\begin{displaymath}
U(k-n)
    \longrightarrow
  U(k)
    \longrightarrow
  U(k)/U(k-n) 
    = 
  V_n(\mathbb{C}^k)
  \,.
\end{displaymath}
By prop. \ref{UnitaryGroupIsCompact} and by \href{QuotientProjectionForCompactLieSubgroupIsPrincipal}{this corollary} the projection $U(k)\to U(k)/U(k-n)$ is a [[Serre fibration]]. Therefore there is induced the [[long exact sequence of homotopy groups]] of this [[fiber sequence]], and by prop. \ref{InclusionOfUnitaryGroupnIntoUnitaryGroupnPlusIneIsnMinus1Equivalence} it has the following form in degrees bounded by $n$:

\begin{displaymath}
\cdots
    \to
  \pi_{\bullet \leq 2(k-n)}(U(k-n))
    \overset{epi}{\longrightarrow}
  \pi_{\bullet \leq 2(k-n)}(U(k))
    \overset{0}{\longrightarrow}
  \pi_{\bullet \leq 2(k-n)}(V_n(\mathbb{C}^k))
    \overset{0}{\longrightarrow}
  \pi_{\bullet-1 \lt 2(k-n)}(U(k))
    \overset{\simeq}{\longrightarrow}
  \pi_{\bullet-1 \lt 2(k-n)}(U(k-n))
    \to
  \cdots
  \,.
\end{displaymath}
This implies the claim.

\end{proof}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[stable unitary group]]


\item The subgroup of unitary matrices with [[determinant]] equal to 1 is the [[special unitary group]]. The [[quotient]] by the [[center]] is the [[projective unitary group]]. The space of equivalence classes of unitary matrices under conjugation is the [[symmetric product of circles]].


\item The analog of the unitary group for real metric spaces is the [[orthogonal group]].


\item similarly: [[quaternionic unitary group]]


\item The [[Lie algebra]] is the [[unitary Lie algebra]].


\item [[unitary representation]]


\item [[unitary operator]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item M. Mimura and H. Toda, \emph{Homotopy Groups of $SU(3)$, $SU(4)$ and $Sp(2)$}, J. Math. Kyoto Univ. Volume 3, Number 2 (1963), 217-250. (\href{http://projecteuclid.org/euclid.kjm/1250524818}{Euclid})


\item Jesus Espinoza, Bernardo Uribe, \emph{Topological properties of the unitary group}, JP Journal of Geometry and Topology \textbf{16} (2014) Issue 1, pp 45-55. \href{http://www.pphmj.com/abstract/8730.htm}{journal}, arXiv:\href{https://arxiv.org/abs/1407.1869v1}{1407.1869}



\end{itemize}
[[!redirects unitary group]] [[!redirects unitary groups]]

[[!redirects U(n)]]



\end{document}