M6bius transformations and Clifford algebras
- P. Lounesto, A. Springer
The space of 2-by-2 Hermitian matrices is isometric to Minkowski space. This is commonly used to exhibit the group SL(2, C) as a twofold cover of the identity component of the Lorentz group. That these Hermitian matrices also represent equations of circles in the Euclidean plane leads to the group PSL(2, C) as the Mibbius group of the Euclidean plane. Clifford algebras naturally arise in the construction of covers of the orthogonal group by spin groups. By considering in addition the Clifford algebra of the space of equations of spheres, we are able to extend these ideas to the M6bius group of finite-dimensional vector spaces over general fields.