Projective unitary group

In mathematics, the projective unitary group PU(N) is the quotient of the unitary group U(N) by the right multiplication of its center, U(1). In terms of matrices, elements of U(N) are complex $N\times N$ unitary matrices, and elements of the center are diagonal matrices equal to $e^{i\theta }$ multiplied by the the identity matrix. Thus elements of PU(N) correspond to equivalence classes of unitary matrices under multiplication by a constant phase $\theta$.

Examples

At N=1, U(1) is abelian and so is equal to its center. Therefore PU(1)=U(1)/U(1)=0.

At N=2, the center of U(2) consists of two by two matrices proportional to the identity, so

PU(2)=U(2)/U(1)=SO(3).

Notice that PU(2)=SO(3)=SU(2)/Z₂. This is a general feature, the special unitary groups SU(N) are quotients of a degree N map to U(1), and so they are N-fold covers of PU(N). That is:

PU(N)=SU(N)/Z_N.

PU(H) Is a classifying space for circle bundles

The same construction may be applied to matrices acting on an infinite-dimensional Hilbert space ${\mathcal {H}}$.

The unitary operators U(${\mathcal {H}}$) acting on such a space are those operators that may be written as the identity plus a compact operator. As the space of compact operators is contractible, the space of unitary operators on an infinite-dimensional Hilbert space is also therefore contractible, in contrast with their finite-dimensional cousins and their topologically nontrivial limit U($\infty$).

The center of the infinite-dimensional unitary group U(${\mathcal {H}}$) is, as in the finite dimensional case, U(1), which again acts on the unitary group via multiplication by a phase. As the unitary group does not contain the zero matrix, this action is free. Thus U(${\mathcal {H}}$) is a contractible space with a U(1) action, which identifies it as EU(1) and the space of U(1) orbits as BU(1), the classifying space for U(1).

PU(${\mathcal {H}}$) is defined precisely to be the space of orbits of the U(1) action on U(${\mathcal {H}}$), thus PU(${\mathcal {H}}$) is a realization of the classifying space BU(1). In particular, using the isomorphism

$\pi _{n}(X)=\pi _{n+1}(BX)$

between the homotopy groups of a space X and the homotopy groups of its classifying space BX, combined with the homotopy type of the circle U(1)

$\pi _{1}(U(1))=\mathbf {Z} ,$ $\pi _{k\neq 1}(U(1))=0$

we find the homotopy groups of PU(${\mathcal {H}}$)

$\pi _{2}(PU({\mathcal {H}}))=\mathbf {Z} ,$ $\pi _{k\neq 2}(PU({\mathcal {H}}))=0$

thus identifying PU(${\mathcal {H}}$) as a representative of the Eilenberg-MacLane space K(Z,2).

As a consequence, PU(${\mathcal {H}}$) must be of the same homotopy type as the infinite-dimensional complex projective space, which also represents K(Z,2). This means in particular that they have isomorphic homology and cohomology groups

H²ⁿ(PU(${\mathcal {H}}$))=H_2n(PU(${\mathcal {H}}$))=Z

and

H²ⁿ⁺¹(PU(${\mathcal {H}}$))=H_2n+1(PU(${\mathcal {H}}$))=0.

Representations

PU(N) in general has no N-dimensional representations, just as SO(3) has no two-dimensional representations (here it is crucial that I am referring to the group and not the algebra, as the algebra so(3) is isomorphic to su(2)).

PU(N) has an adjoint action on SU(N), thus it has an (N²-1)-dimensional representation. When N=2 this corresponds to the three dimensional representation of SO(3). The adjoint action is defined by thinking of an element of PU(N) as an equivalence class of elements of U(N) that differ by phases. One can then take the adjoint action with respect to any of these U(N) representatives, and the phases commute with everything and so cancel. Thus the action is independent of the choice of representative and so it is well-defined.

This construction is useful in the definition of twisted K-theory.

Projective representations

However it does have projective representions, which are representations up to a phase which is independent of the vector on which one acts. These are useful in quantum mechanics, as physical states are only defined up to an over all phase. For example, massive fermionic states transform under a projective representation but not under a representation of the little group PU(2)=SO(3).

The projective representations of a group are classified by its second integral cohomology, which in this case is

H²(PU(N))=Z_N or H²(PU(${\mathcal {H}}$))=Z.

The cohomology groups in the finite case can be derived from the long exact sequence for bundles and the above fact that SU(N) is a Z_N bundle over PU(N). The cohomology in the infinite case was argued above from the isomorphism with the cohomology of the infinite complex projective space.

Thus PU(N) enjoys N projective representations while PU(${\mathcal {H}}$) has a countably infinite number. As usual, the projective representations of a group are ordinary representations of a central extension of the group. In this case the central extended group corresponding to the first projective representation of each projective unitary group is just the original unitary group that we quotiented by U(1) in the definition of PU.

See also

• unitary group
• special unitary group
• orthogonal group
• symplectic group
• unitary operators