Biquaternions Lie Algebra and Complex-Projective Spaces

In this paper, Lie group and Lie algebra structures of unit complex 3-sphere S 3 C are studied. In order to do this, adjoint representation of unit biquaternions (complexified quaternions) is obtained. Also, a correspondence between the elements of S 3 C and the special bicomplex unitary matrices SU C2 (2) is given by expressing biquaternions as 2-dimensional bicomplex numbers C 2 2. The relation SO(R 3 ) ∼= S 3 /{±1} = RP 3 among the special orthogonal group SO(R 3 ), the quotient group of unit real quaternions S 3 /{±1} and the projective space RP 3 is known as the Euclidean-projective space [1]. This relation is generalized to the Complex-projective space and is obtained as SO(C 3 ) ∼= S 3 C/{±1} = CP 3 .