SU(1) - Knowlpedia

$\operatorname{SU}(1)$ is the compact special unitary group

\operatorname{SU}(1)=\{A\in M_{1}(\mathbb C):A^*A=I,\ \det A=1\},

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

Dimensions and structure

This group has real dimension $0$ . The tangent algebra is formed by skew-Hermitian complex matrices of trace zero. The determinant-one condition makes this the trivial group.

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.

Both groups consist of the identity alone. See $\operatorname{SL}(1,\mathbb R)$ .

References

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