Inversion of the Covering Map for Indefinite Spin Groups

The double covering of Orthogonal groups by the Spin groups is vital for many applications. In this dissertation, we address the important question of inverting the covering map φ_{p,q} : Spin⁺(p,q) → SO⁺(p,q) for (p,q)∈ (2,1),(2,2),(3,1),(3,2),(3,3) in the indefinite case. We also develop explicit formulae for the forward map φ_{p,q} : Spin⁺(p,q) → SO⁺(p,q) for (p,q)∈ (1,2),(1,3),(2,3). We do not work with the even subalgebra of Cl(p,q) and thus our formulae are quite explicit. Our method relies significantly on standard Givens and hyperbolic Givens.