For a natural number $n \in \mathbb{N}$, the unitary group $U(n)$ is the group of 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 operators on that Hilbert space. For the purposes of studying unitary representations of Lie groups, the topology is chosen to be the strong operator topology, although other topologies on $U(\mathcal{H})$ are frequently considered for other purposes.
The unitary groups are naturally topological groups and Lie groups (infinite dimensional if $\mathcal{H}$ is infinite dimensional).
The unitary group $U(n)$ is compact topological space, hence in particular a compact Lie group.
For $n,k \in \mathbb{N}$, $n \leq k$, then the canonical inclusion of unitary groups
is a 2n-equivalence, hence induces an isomorphism on homotopy groups in degrees $\lt 2n$ and a surjection in degree $2n$.
Consider the coset quotient projection
By prop. 1 and by this corollary, the projection $U(n+1)\to U(n+1)/U(n)$ is a Serre fibration. Furthermore, example 1 identifies the coset with the (2n+1)-sphere
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
Since $\pi_{\leq 2n}(S^{2n+1}) = 0$, this implies that
is an isomorphism and that
is surjective. Hence now the statement follows by induction over $k-n$.
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 groups, have the same homotopy type.
More specifically, $U(\mathcal{H})$ is a maximal compact subgroup of $GL(\mathcal{H})$.
By the Gram-Schmidt process.
(Kuiper’s theorem)
For a separable infinite-dimensional complex Hilbert space $\mathcal{H}$, the unitary group $U(\mathcal{H})$ is contractible.
See also Kuiper's theorem.
This in contrast to the finite dimensional situation. For $n \in \mathbb{N}$ ($n \ge 1$), $U(n)$ is not contractible.
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
The homotopy direct limit over this is written
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.
For all $n \in \mathbb{N}$, the unitary group $U(n)$ is a split group extension of the circle group $U(1)$ by the special unitary group $SU(n)$
Hence it is a semidirect product group
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 first-order integrable case this is precisely a joint orthogonal structure (Riemannian manifold structure), symplectic structure and complex structure.
$U(1)$ is the circle group.
For $n \leq n$, the coset
is called the $n$th real Stiefel manifold of $\mathbb{C}^k$.
The complex Stiefel manifold $V_n(\mathbb{C}^k)$ (example 2) is 2(k-n)-connected.
Consider the coset quotient projection
By prop. 1 and by this corolarry 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. 2 it has the following form in degrees bounded by $n$:
This implies the claim.
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.
The analog of the unitary group for real metric spaces is the orthogonal group.
similarly: quaternionic unitary group
The Lie algebra is the unitary Lie algebra.