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unitary group

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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

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.

More specifically, U()U(\mathcal{H}) is a maximal compact subgroup of GL()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()U(\mathcal{H}) is contractible.

See also Kuiper's theorem.

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.

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) \,.

Examples

U(1)U(1) is the circle group.

Revised on November 4, 2013 01:48:30 by Urs Schreiber (89.204.154.47)