U(2) - Knowlpedia

$\operatorname{U}(2)$ is the compact unitary group

\operatorname{U}(2)=\{A\in M_{2}(\mathbb C):A^*A=I\},

where $A^*=\overline A^{\mathsf T}$ . The defining Hermitian form is positive definite.

Dimensions and structure

This group has real dimension $4$ . The tangent algebra is formed by skew-Hermitian complex matrices.

The scalar field of its matrix entries does not assign a complex Lie group structure to this real Lie group. In particular, a group homomorphism is not called linear merely because the group has a matrix representation.

References

Pavel Etingof, Lie Groups and Lie Algebras (MIT lecture notes), §6.1, pp. 38–39; Proposition 6.3 and Exercise 6.5.