Spin Representation - Real Representations

Real Representations

The complex spin representations of so(n,C) yield real representations S of so(p,q) by restricting the action to the real subalgebras. However, there are additional "reality" structures that are invariant under the action of the real Lie algebras. These come in three types.

1. There is an invariant complex antilinear map r: S → S with r2 = id_S. The fixed point set of r is then a real vector subspace S_R of S with S_R ⊗ C = S. This is called a real structure.
2. There is an invariant complex antilinear map j: S → S with j2 = −id_S. It follows that the triple i, j and k:=ij make S into a quaternionic vector space S_H. This is called a quaternionic structure.
3. There is an invariant complex antilinear map b: S → S∗ that is invertible. This defines a hermitian bilinear form on S and is called a hermitian structure.

The type of structure invariant under so(p,q) depends only on the signature p−q modulo 8, and is given by the following table.

Here R, C and H denote real, hermitian and quaternionic structures respectively, and R+R and H+H indicate that the half-spin representations both admit real or quaternionic structures respectively.