Degenerate Spin Groups as Semi-Direct Products

Abstract

Let Q be a symmetric bilinear form on R n =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 Mat p + q, r . Mathematics Subject Classification (2000). 15A66.Let , be a symmetric bilinear form on R n and let us consider the subspace W of R n W = {w ∈ R n | w, x = 0 for all x ∈ R n }.W is called the radical and the dimension of W is called the co-rank of , . If corank is zero/non-zero, then , is called non-degenerate/degenerate. If W ⊂ R n is any complementary subspace to W , then the restriction of , to W is nondegenerate. If the co-rank of , is r and the the restriction of , to W has signature (p, q) (in the sense that W has an orthogonal decomposition W ⊕ W , where the dimensions of W respectively W are p resp. q and the restriction of , to W resp. W is positive resp. negative definite), then the pair (R n , , ) is said to be of type R p,q,r (see [1], [2]).

Kinematic mappings for Cayley–Klein geometries via Clifford algebras

“…Spin group elements are generated by pairs of reflections, so they are rotations. The Spin group for degenerated Clifford algebras as in the case of Euclidean geometry are semi-direct products of Spin groups for the non-degenerated part and an additive matrix group, see [4].…”

“…Every point on Study's quadric stands for an element of SO(4) since Spin (4,0,0) is a double cover of SO (4). In this case it is not necessary to slice the quadric, since Q 6 2 has no real point.…”

Canonical forms for families of anti-commuting diagonalizable linear operators

“…But as opposed to the previous case, if A 2 a is zero on some subspace, A a need not be zero, hence we may have a family of anti-commuting matrices whose squares are positive, negative or zero. This is just the representation of some degenerate Clifford algebra [2] for which the construction of canonical forms is not known.…”