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unitary group

Definition

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 ${ℂ}^n$. This is canonically identified with the group of $n\times n$ unitary matrices.

More generally, for a Hilbert space $\mathscr{H}$, $U(\mathscr{H})$ 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 $U(\mathscr{H})$ are frequently considered for other purposes.

Properties

The unitary groups are naturally topological groups and Lie groups (infinite dimensional if $\mathscr{H}$ is infinite dimensional).

Proposition

For $\mathscr{H}$ a Hilbert space, which can be either finite or infinite dimensional, the unitary group $U(\mathscr{H})$ and the general linear group $\mathrm{GL}(\mathscr{H})$, regarded as topological groups, have the same homotopy type.

More specifically, $U(\mathscr{H})$ is a maximal compact subgroup of $\mathrm{GL}(\mathscr{H})$.

Proof

By the Gram-Schmidt process.

Theorem

(Kuiper’s theorem)

For a separable infinite-dimensional complex Hilbert space $\mathscr{H}$, the unitary group $U(\mathscr{H})$ is contractible.

See also Kuiper's theorem.

Remark

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

Write $BU(n)$ for the classifying space of the topological group $U(n)$. Inclusion of matrices into larger matrices gives a canonical sequence of inclusions

$$\cdots \to BU(n)\hookrightarrow BU(n+1)\hookrightarrow BU(n+2)\to \cdots .$$\cdots \to B U(n) \hookrightarrow B U(n+1) \hookrightarrow B U(n+2) \to \cdots \,.

The homotopy direct limit over this is written

$$BU:={\underrightarrow{\lim }}_{n}BU(n)$$B U := {\lim_\to}_n B U(n)

or sometimes $BU(\mathrm{\infty })$. Notice that this is very different from $BU(\mathscr{H})$ for $\mathscr{H}$ an infinite-dimensional Hilbert space. See topological K-theory for more on this.

Proposition

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 $\mathrm{SU}(n)$

$$\mathrm{SU}(n)\to U(n)\to U(1).$$SU(n) \to U(n) \to U(1) \,.

Hence it is a semidirect product group

$$U(n)\simeq \mathrm{SU}(n)⋊U(1).$$U(n) \simeq SU(n) \rtimes U(1) \,.

Examples

$U(1)$ is the circle group.

Related concepts

• 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.

• The Lie algebra is the unitary Lie algebra.

• unitary representation