Recall that a unitary group consists of all unitary operators on a complex vector space.
Projective Unitary Group
Projective Unitary Group
Suppose
is a complex vector space, then the projective unitary group is defined as the quotient of by the scalar phases. That is, where two unitary operators if they differ only by a scalar phase factor. Its Lie algebra is denoted as .
Proposition
is the Lie algebra of traceless skew-Hermitian operators. Moreover, .