Clifford Algebra - Spin and Pin Groups

Spin and Pin Groups

In this section we assume that V is finite dimensional and its bilinear form is non-singular. (If K has characteristic 2 this implies that the dimension of V is even.)

The Pin group Pin_V(K) is the subgroup of the Clifford group Γ of elements of spinor norm 1, and similarly the Spin group Spin_V(K) is the subgroup of elements of Dickson invariant 0 in Pin_V(K). When the characteristic is not 2, these are the elements of determinant 1. The Spin group usually has index 2 in the Pin group.

Recall from the previous section that there is a homomorphism from the Clifford group onto the orthogonal group. We define the special orthogonal group to be the image of Γ0. If K does not have characteristic 2 this is just the group of elements of the orthogonal group of determinant 1. If K does have characteristic 2, then all elements of the orthogonal group have determinant 1, and the special orthogonal group is the set of elements of Dickson invariant 0.

There is a homomorphism from the Pin group to the orthogonal group. The image consists of the elements of spinor norm 1 ∈ K*/K*2. The kernel consists of the elements +1 and −1, and has order 2 unless K has characteristic 2. Similarly there is a homomorphism from the Spin group to the special orthogonal group of V.

In the common case when V is a positive or negative definite space over the reals, the spin group maps onto the special orthogonal group, and is simply connected when V has dimension at least 3. Further the kernel of this homomorphism consists of 1 and −1. So in this case the spin group, Spin(n), is a double cover of SO(n). Please note, however, that the simple connectedness of the spin group is not true in general: if V is Rp,q for p and q both at least 2 then the spin group is not simply connected. In this case the algebraic group Spin_p,q is simply connected as an algebraic group, even though its group of real valued points Spin_p,q(R) is not simply connected. This is a rather subtle point, which completely confused the authors of at least one standard book about spin groups.