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# Unitary group

(Redirected from U(1))

In mathematics, the unitary group of degree n over the field F (which is either the field $\mathbb{R}$ of real numbers or the field $\mathbb{C}$ of complex numbers) is the group of n by n unitary matrices with entries from F, with the group operation that of matrix multiplication. This is a subgroup of the general linear group GL(n,F).

In the simple case n = 1, the group U(1) is the unit circle in the complex plane, under multiplication. All the complex unitary groups contain copies of this group.

If the field F is the field of real numbers then the unitary group coincides with the orthogonal group $\mathrm{O}(n,\mathbb{R})$. If F is the field of complex numbers one usually writes U(n) for the unitary group of degree n.

The unitary group U(n) is a real Lie group of dimension n2. The Lie algebra of U(n) consists of complex n-by-n Skew-hermitian matrices, with the Lie bracket given by the commutator.