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Concept# Projective unitary group

Summary

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 eiθ 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:
The natural map
is an isomorphism, by the second isomorphism theorem, thus
and the special unitary group SU(n) is an n-fold cover of the projective unitary group.
At n = 1, U(1) is abelian and so is equal to its center. Therefore PU(1) = U(1)/U(1) is a trivial group.
At n = 2, , all being representable by unit norm quaternions, and via:
Unitary group#Finite fields
One can also define unitary groups over finite fields: given a field of order q, there is a non-degenerate Hermitian structure on vector spaces over unique up to unitary congruence, and correspondingly a matrix group denoted or and likewise special and projective unitary groups. For convenience, this article uses the convention.
Recall that the group of units of a finite field is cyclic, so the group of units of and thus the group of invertible scalar matrices in is the cyclic group of order The center of has order q + 1 and consists of the scalar matrices which are unitary, that is those matrices with The center of the special unitary group has order gcd(n, q + 1) and consists of those unitary scalars which also have order dividing n.

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In mathematics, the classical groups are defined as the special linear groups over the reals R, the complex numbers C and the quaternions H together with special automorphism groups of symmetric or skew-symmetric bilinear forms and Hermitian or skew-Hermitian sesquilinear forms defined on real, complex and quaternionic finite-dimensional vector spaces. Of these, the complex classical Lie groups are four infinite families of Lie groups that together with the exceptional groups exhaust the classification of simple Lie groups.

Unitary group

In mathematics, the unitary group of degree n, denoted U(n), is the group of n × n unitary matrices, with the group operation of matrix multiplication. The unitary group is a subgroup of the general linear group GL(n, C). Hyperorthogonal group is an archaic name for the unitary group, especially over finite fields. For the group of unitary matrices with determinant 1, see Special unitary group. In the simple case n = 1, the group U(1) corresponds to the circle group, consisting of all complex numbers with absolute value 1, under multiplication.

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In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space V on the associated projective space P(V). Explicitly, the projective linear group is the quotient group PGL(V) = GL(V)/Z(V) where GL(V) is the general linear group of V and Z(V) is the subgroup of all nonzero scalar transformations of V; these are quotiented out because they act trivially on the projective space and they form the kernel of the action, and the notation "Z" reflects that the scalar transformations form the center of the general linear group.

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