Unitary Group
A unitary group consists of all unitary operators on a complex vector space:
where denotes the conjugate transpose.
The special unitary group is.
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 operatorsif 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, .