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unitary group

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Contents

Definition

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

More generally, for a Hilbert space \mathcal{H}, U()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()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)U(n) is compact topological space, hence in particular a compact Lie group.

Homotopy groups

Proposition

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

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

is a 2n-equivalence, hence induces an isomorphism on homotopy groups in degrees <2n\lt 2n and a surjection in degree 2n2n.

Proof

Consider the coset quotient projection

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

By prop. and by this corollary, the projection U(n+1)U(n+1)/U(n)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+1U(n+1)/U(n). S^{2n+1}\simeq U(n+1)/U(n) \,.

Therefore the long exact sequence of homotopy groups of the fiber sequence U(n)U(n+1)S 2n+1U(n)\to U(n+1) \to S^{2n+1} is of the form

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

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

is an isomorphism and that

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

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

It follows that the homotopy groups π k(U(n))\pi_k(U(n)) are independent of nn for n>k2n \gt \frac{k}{2} (the “stable range”). So if U=lim nU(n)U = \underset{\longrightarrow}{\lim}_n U(n), then π k(U(n))=π k(U)\pi_k(U(n)) = \pi_k(U). By Bott periodicity we have

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

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

GGπ 1\pi_1π 2\pi_2π 3\pi_3π 4\pi_4π 5\pi_5π 6\pi_6π 7\pi_7π 8\pi_8π 9\pi_9π 10\pi_10π 11\pi_11π 12\pi_12π 13\pi_13π 14\pi_14π 15\pi_15
U(1)U(1)\mathbb{Z}00000000000000
U(2)U(2)0\mathbb{Z} 12\mathbb{Z}_{12} 2\mathbb{Z}_2 2\mathbb{Z}_2 3\mathbb{Z}_3 15\mathbb{Z}_{15} 2\mathbb{Z}_2 2 2\mathbb{Z}_2^{\oplus 2} 3 12\mathbb{Z}_3\oplus\mathbb{Z}_{12} 2 2 84\mathbb{Z}_2^{\oplus 2}\oplus\mathbb{Z}_{84} 2 2\mathbb{Z}_2\oplus\mathbb{Z}_2
U(3)U(3) 6\mathbb{Z}_60 12\mathbb{Z}_{12} 3\mathbb{Z}_3 30\mathbb{Z}_{30} 4\mathbb{Z}_4 60\mathbb{Z}_{60} 6\mathbb{Z}_6 2 84\mathbb{Z}_2\oplus\mathbb{Z}_{84} 36\mathbb{Z}_{36}
U(4)U(4)0\mathbb{Z} 24\mathbb{Z}_{24} 2\mathbb{Z}_2 2 120\mathbb{Z}_2\oplus\mathbb{Z}_{120} 4\mathbb{Z}_4 60\mathbb{Z}_{60} 4\mathbb{Z}_4 2 1680\mathbb{Z}_2\oplus\mathbb{Z}_{1680} 2 72\mathbb{Z}_2\oplus\mathbb{Z}_{72}
U(5)U(5)0\mathbb{Z} 120\mathbb{Z}_{120}0 360\mathbb{Z}_{360} 4\mathbb{Z}_4 1680\mathbb{Z}_{1680} 6\mathbb{Z}_6
U(6)U(6)0\mathbb{Z} 720\mathbb{Z}_{720} 2\mathbb{Z}_2 2 5040\mathbb{Z}_2\oplus\mathbb{Z}_{5040} 6\mathbb{Z}_6
U(7)U(7)0\mathbb{Z} 5040\mathbb{Z}_{5040}0
U(8)U(8)0\mathbb{Z}

Due to the relation to the special unitary group, the higher homotopy groups of U(n)U(n) and SU(n)SU(n) agree. The U(2)U(2) row can be found using the fact that SU(2)SU(2) is diffeomorphic to S 3S^3. The U(3)U(3) row can be found using Mimura-Toda 63. Otherwise the table is given in columns π k\pi_k, k=6,,15k= 6,\ldots, 15, and in rows U(n)U(n), n=4,,8n=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()U(\mathcal{H}) and how they all agree and make U()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()U(\mathcal{H}) and the general linear group GL()GL(\mathcal{H}), regarded as topological groups, have the same homotopy type.

Additionally, U()U(\mathcal{H}) is a maximal compact subgroup of GL()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()U(\mathcal{H}) is contractible in the norm topology.

See also Kuiper's theorem. Note that U()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 nn \in \mathbb{N} (n1n \ge 1), U(n)U(n) is not contractible.

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

BU(n)BU(n+1)BU(n+2). \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:=lim nBU(n) B U := {\lim_\to}_n B U(n)

or sometimes BU()B U(\infty). Notice that this is very different from BU()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 nn \in \mathbb{N}, the unitary group U(n)U(n) is a split group extension of the circle group U(1)U(1) by the special unitary group SU(n)SU(n)

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

Hence it is a semidirect product group

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

Relation to orthogonal, symplectic and general linear group

The unitary group U(n)U(n) is equivalently the intersection of the orthogonal group O(2n)O(2n), the symplectic group Sp(2n,)Sp(2n,\mathbb{R}) and the complex general linear group GL(n,)GL(n,\mathbb{C}) inside the real general linear group GL(2n,)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)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)U(1) is the circle group.

Coset spaces

Example

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

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

For nkn \leq k, the coset

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

is called the nnth real Stiefel manifold of k\mathbb{C}^k.

Proposition

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

Proof

Consider the coset quotient projection

U(kn)U(k)U(k)/U(kn)=V n( k). 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)U(k)/U(kn)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 nn:

π 2(kn)(U(kn))epiπ 2(kn)(U(k))0π 2(kn)(V n( k))0π 1<2(kn)(U(k))π 1<2(kn)(U(kn)). \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.

References

  • M. Mimura and H. Toda, Homotopy Groups of SU(3)SU(3), SU(4)SU(4) and Sp(2)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

Last revised on March 29, 2018 at 09:23:49. See the history of this page for a list of all contributions to it.