Clifford Algebra - The Clifford Group Γ

The Clifford Group Γ

In this section we assume that V is finite dimensional and the quadratic form Q is nondegenerate.

The invertible elements of the Clifford algebra act on it by twisted conjugation: conjugation by x maps y ↦ xy α(x)−1.

The Clifford group Γ is defined to be the set of invertible elements x that stabilize vectors, meaning that for all v in V we have:

This formula also defines an action of the Clifford group on the vector space V that preserves the norm Q, and so gives a homomorphism from the Clifford group to the orthogonal group. The Clifford group contains all elements r of V of nonzero norm, and these act on V by the corresponding reflections that take v to v−2⟨v,r⟩r/Q(r) (In characteristic 2 these are called orthogonal transvections rather than reflections.)

The Clifford group Γ is the disjoint union of two subsets Γ0 and Γ1, where Γi is the subset of elements of degree i. The subset Γ0 is a subgroup of index 2 in Γ.

If V is a finite dimensional real vector space with positive definite (or negative definite) quadratic form then the Clifford group maps onto the orthogonal group of V with respect to the form (by the Cartan-Dieudonné theorem) and the kernel consists of the nonzero elements of the field K. This leads to exact sequences

Over other fields or with indefinite forms, the map is not in general onto, and the failure is captured by the spinor norm.