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 $n 2$ . The Lie algebra of $U(n)$ consists of complex $n$ -by- $n$ Skew-hermitian matrices, with the Lie bracket given by the commutator.

See also: Special unitary group

Categories: Lie groups

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01-04-2007 01:21:04

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