Is the projective special unitary group a lie group?

Question

Is the projective special unitary group a lie group? If so then what is its Lie algebra and where can I find out more about these objects?

Answer 1

Yes, $PSU(n)$ is a Lie group. Indeed, the quotient of a Lie group by a closed normal subgroup is itself a Lie group in a natural way. Furthermore, $PSU(n)$ is the quotient of $SU(n)$ by the center $\mathbb{Z}/n\mathbb{Z}$. Since $\mathbb{Z}/n\mathbb{Z}$ is a finite group, its Lie algebra is $\{0\}$. Therefore, the Lie algebra of $PSU(n)$ is the quotient of $\frak{su}(n)$ by $\{0\}$, and so is just $\frak{su}(n)$.