Unitary Groups

Unitary Group

A unitary group consists of all unitary operators on a complex vector space: ${\mathrm{U}}_n:=\{A\in {\text{GL}}_n(\text{C})\mid A^{\ast }A=I_n\},$where ${}^{\ast }$ denotes the conjugate transpose. The special unitary group is ${\mathrm{SU}}_n={\mathrm{SL}}_n(\text{C})\cap {\text{U}}_n$.

Projective Unitary Group

Projective Unitary Group

Suppose $V$ is a complex vector space, then the projective unitary group $\mathrm{PU}(V)$ is defined as the quotient of $U(V)$ by the scalar phases. That is, $\text{PU}(V):=U(V)/\sim ,$ where two unitary operators $T_1\sim T_2$ if they differ only by a scalar phase factor. Its Lie algebra is denoted as $\mathfrak{pu}(V)$.

Proposition

$\text{pu}(V)$ is the Lie algebra of traceless skew-Hermitian operators. Moreover, $\breve{(}V)\cong \text{pu}(V)\oplus \mathbb{R}$.

Other Useful Resources

Handwiki, Projective Unitary Groups