Properties
Since the determinant of a unitary matrix is a complex number with norm 1, the determinant gives a group homomorphism
The kernel of this homomorphism is the set of unitary matrices with determinant 1. This subgroup is called the special unitary group, denoted SU(n). We then have a short exact sequence of Lie groups:
This short exact sequence splits so that U(n) may be written as a semidirect product of SU(n) by U(1). Here the U(1) subgroup of U(n) consists of matrices of the form diag(eiθ, 1, 1, ..., 1).
The unitary group U(n) is nonabelian for n > 1. The center of U(n) is the set of scalar matrices λI with λ ∈ U(1). This follows from Schur's lemma. The center is then isomorphic to U(1). Since the center of U(n) is a 1-dimensional abelian normal subgroup of U(n), the unitary group is not semisimple.
Read more about this topic: Unitary Group
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—Ralph Waldo Emerson (1803–1882)
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—John Locke (1632–1704)