Degenerate Spin Groups as Semi-Direct Products

Abstract

Let Q be a symmetric bilinear form on \({\mathbb{R}}^n={\mathbb{R}}^{p+q+r}\) with corank r, rank p + q and signature type (p, q), p resp. q denoting positive resp. negative dimensions. We consider the degenerate spin group Spin(Q) = Spin(p, q, r) in the sense of Crumeyrolle and prove that this group is isomorphic to the semi-direct product of the nondegenerate and indefinite spin group Spin(p, q) with the additive matrix group \(M at\left(p + q, r\right)\).