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.
- There is an invariant complex antilinear map r: S → S with r2 = idS. The fixed point set of r is then a real vector subspace SR of S with SR ⊗ C = S. This is called a real structure.
- There is an invariant complex antilinear map j: S → S with j2 = −idS. It follows that the triple i, j and k:=ij make S into a quaternionic vector space SH. This is called a quaternionic structure.
- 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.
| p−q mod 8 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|---|
| Structure | R + R | R | C | H | H + H | H | C | R |
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.
Read more about this topic: Spin Representation
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