Orthochronous Lorentz group from the Cliffordian point of view (Q1362929)

Using the Lorentz quadratic form \(q\) on \(\mathbb{R}^n\) of signature \((n-1,1)\), the orthochronous (future preserving half of) Lorentz group \(O^+(n-1,1)\), forms the full isometry group of the model of hyperbolic space \(\mathbb{H}^{n-1}\), formed by the projective image of the positive cone \(C=\{x\in\mathbb{R}^n\mid q(x,x)<0,\;x_n>0\}\). Alternatively, in the upper half space model of \(\mathbb{H}^{n-1}\) the full isometry group can be described by the action of certain \(2\times 2\) matrices with entries in the Clifford group generated by vectors in a suitably defined Clifford algebra. In this paper, very explicit descriptions are obtained, at the level of elements, between these two representations of this isometry group.