group theory

# Contents

## 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 $\mathbb{C}^n$. 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.

## Properties

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

### In infinite dimension

###### Proposition

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 $GL(\mathcal{H})$.

###### Proof

By the Gram-Schmidt process.

###### Theorem

(Kuiper’s theorem)

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

###### Remark

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

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

The homotopy direct limit over this is written

$B U := {\lim_\to}_n B U(n)$

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.

### Relation to special unitary group

###### 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 $SU(n)$

$SU(n) \to U(n) \to U(1) \,.$

Hence it is a semidirect product group

$U(n) \simeq SU(n) \rtimes U(1) \,.$

### Relation to orthogonal, symplectic and general linear 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.

## Examples

$U(1)$ is the circle group.

Revised on January 22, 2015 01:01:45 by Urs Schreiber (88.100.66.95)