Biquaternions Lie Algebra and Complex-Projective Spaces

In this paper, Lie group and Lie algebra structures of unit complex 3sphere mathbb{S}^3_{ mathbb{C} are studied. In order to do this, adjoint representation of unit biquaternions (complexified quaternions) is obtained. Also, a correspondence between the elements of mathbb{S}^3_{ mathbb{C} and the special bicomplex unitary matrices SU_{ mathbb{}^2}(2) is given by expressing biquaternions as 2dimensional bicomplex numbers mathbb{C}^2_2 . The relation SO( mathbb{R}^3)= mathbb{S}^3/{ { pm 1 }}= mathbb{R}P^3 among the special orthogonal group SO( mathbb{R}^3) , the quotient group of unit real quaternions mathbb{S}^3/{ { pm 1 }} and the projective space = mathbb{R}P^3 is known as the Euclideanprojective space [1]. This relation is generalized to the Complexprojective space and is obtained as SO( mathbb{C}^3) cong mathbb{S}^3_{ mathbb{C}}/{ { pm 1 }}= mathbb{C}P^3 .