Contents

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).

###### Proposition

The unitary group $U(n)$ is compact topological space, hence in particular a compact Lie group.

### Homotopy groups

###### Proposition

For $n,N \in \mathbb{N}$, $n \leq N$, then the canonical inclusion of unitary groups

$U(n) \hookrightarrow U(N)$

is a 2n-equivalence, hence induces an isomorphism on homotopy groups in degrees $\lt 2n$ and a surjection in degree $2n$.

###### Proof

Consider the coset quotient projection

$U(n) \longrightarrow U(n+1) \longrightarrow U(n+1)/U(n) \,.$

By prop. and by this corollary, the projection $U(n+1)\to U(n+1)/U(n)$ is a Serre fibration. Furthermore, example identifies the coset with the (2n+1)-sphere

$S^{2n+1}\simeq U(n+1)/U(n) \,.$

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

$\cdots \to \pi_{\bullet+1}(S^{2n+1}) \longrightarrow \pi_\bullet(U(n)) \longrightarrow \pi_\bullet(U(n+1)) \longrightarrow \pi_\bullet(S^{2n+1}) \to \cdots$

Since $\pi_{\leq 2n}(S^{2n+1}) = 0$, this implies that

$\pi_{\lt 2n}(U(n)) \overset{\simeq}{\longrightarrow} \pi_{\lt 2n}(U(n+1))$

is an isomorphism and that

$\pi_{2n}(U(n)) \longrightarrow \pi_{2n}(U(n+1))$

is surjective. Hence now the statement follows by induction over $N-n$.

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

$\array{ \pi_{2k+0}(U) &= 0\\ \pi_{2k+1}(U) &= \mathbb{Z}. }$

In the unstable range for low $n$ they instead start out as follows

$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$$\pi_15$
$U(1)$$\mathbb{Z}$00000000000000
$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}$

Due to the 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 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.

### 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 group is in (Espinoza-Uribe).

###### 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.

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

###### 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 in the norm topology.

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

###### 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.

### Coset spaces

###### Example

The (2n+1)-spheres are coset spaces of unitary groups

$S^{2n+1} \simeq U(n+1)/U(n) \,.$
###### Example

For $n \leq k$, the coset

$V_n(\mathbb{C}^k) \coloneqq U(k)/U(k-n)$

is called the $n$th real Stiefel manifold of $\mathbb{C}^k$.

###### Proposition

The complex Stiefel manifold $V_n(\mathbb{C}^k)$ (example ) is 2(k-n)-connected.

###### Proof

Consider the coset quotient projection

$U(k-n) \longrightarrow U(k) \longrightarrow U(k)/U(k-n) = V_n(\mathbb{C}^k) \,.$

By prop. and by 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. it has the following form in degrees bounded by $n$:

$\cdots \to \pi_{\bullet \leq 2(k-n)}(U(k-n)) \overset{epi}{\longrightarrow} \pi_{\bullet \leq 2(k-n)}(U(k)) \overset{0}{\longrightarrow} \pi_{\bullet \leq 2(k-n)}(V_n(\mathbb{C}^k)) \overset{0}{\longrightarrow} \pi_{\bullet-1 \lt 2(k-n)}(U(k)) \overset{\simeq}{\longrightarrow} \pi_{\bullet-1 \lt 2(k-n)}(U(k-n)) \to \cdots \,.$

This implies the claim.

• M. Mimura and H. Toda, Homotopy Groups of $SU(3)$, $SU(4)$ and $Sp(2)$, J. Math. Kyoto Univ. Volume 3, Number 2 (1963), 217-250. (Euclid)

• Jesus Espinoza, Bernardo Uribe, Topological properties of the unitary group, JP Journal of Geometry and Topology 16 (2014) Issue 1, pp 45-55. journal, arXiv:1407.1869