Projective special 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), embedded as scalars. Abstractly, it is the holomorphic isometry group of complex projective space, just as the projective orthogonal group is the isometry group of real projective space.

In terms of matrices, elements of U(n) are complex n×n unitary matrices, and elements of the center are diagonal matrices equal to e^iθ multiplied by the identity matrix. Thus, elements of PU(n) correspond to equivalence classes of unitary matrices under multiplication by a constant phase θ.

Abstractly, given a Hermitian space V, the group PU(V) is the image of the unitary group U(V) in the automorphism group of the projective space P(V).

The projective special unitary group PSU(n) is equal to the projective unitary group, in contrast to the orthogonal case.

The connections between the U(n), SU(n), their centers, and the projective unitary groups is shown at right.

The center of the special unitary group is the scalar matrices of the nth roots of unity: ${\displaystyle Z({\mbox{SU}}(n))={\mbox{SU}}(n)\cap Z({\mbox{U}}(n))\cong \mathbf {Z} /n}$

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