The [[Matrix Lie group]], $\mbox{U}(1)$ is a subgroup of $\mbox{GL}(1, \mathbb{C})$ ([[GL(n;F)]]) containing $1 \times 1$ [unitary matrices.](Unitary%20operators.md#Unitary%20matrices) This means that $\mbox{U}(1)$ describes rotations of a plane. Therefore it contains [complex numbers](Complex%20analysis%20(index).md#Numbers%20on%20a%20complex%20plane) along a unit circle on the [complex plane](Complex%20analysis%20(index).md#The%20Complex%20plane). The set of complex numbers in $U(1)$ is the set of complex numbers, $e^{i\phi}$, that may be parameterized by, $\phi$ which defines an angle relative to the real axis.  # Corresponding [[Lie Algebra]] to $\mbox{U}(1)$ %%Here the graphic shows theta instead of phi. This is a simple graphic you should remake yourself. In addition you should define elsewhere the concept of the complex plane%% #MathematicalFoundations/Algebra/AbstractAlgebra/GroupTheory/Lie/LieGroups/Algebras/LieAlgebras #MathematicalFoundations/Algebra/AbstractAlgebra/GroupTheory/Lie/LieGroups #MathematicalFoundations/Analysis/ComplexAnalysis