Knowlpedia

Definition

U(3)

U(3) as a separately identified real Lie group, with its defining operations and dimension.

U⁡(3)\operatorname{U}(3) is the compact

U⁡(3)={A∈M3(C):A∗A=I},\operatorname{U}(3)=\{A\in M_{3}(\mathbb C):A^*A=I\},

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

Dimensions and structure

This group has real dimension 99. The tangent algebra is formed by skew-Hermitian complex matrices.

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

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