Spin Representation - Set Up

Set Up

Let V be a finite dimensional real or complex vector space with a nondegenerate quadratic form Q. The (real or complex) linear maps preserving Q form the orthogonal group O(V,Q). The identity component of the group is called the special orthogonal group SO(V,Q). (For V real with an indefinite quadratic form, this terminology is not standard: the special orthogonal group is usually defined to be a subgroup with two components in this case.) Up to group isomorphism, SO(V,Q) has a unique connected double cover, the spin group Spin(V,Q). There is thus a group homomorphism Spin(V,Q) → SO(V,Q) whose kernel has two elements denoted {1, −1}, where 1 is the identity element.

O(V,Q), SO(V,Q) and Spin(V,Q) are all Lie groups, and for fixed (V,Q) they have the same Lie algebra, so(V,Q). If V is real, then V is a real vector subspace of its complexification V_C := V ⊗_R C, and the quadratic form Q extends naturally to a quadratic form Q_C on V_C. This embeds SO(V,Q) as a subgroup of SO(V_C, Q_C), and hence we may realise Spin(V,Q) as a subgroup of Spin(V_C, Q_C). Furthermore, so(V_C, Q_C) is the complexification of so(V,Q).

In the complex case, quadratic forms are determined up to isomorphism by the dimension n of V. Concretely, we may assume V=Cn and

The corresponding Lie groups and Lie algebra are denoted O(n,C), SO(n,C), Spin(n,C) and so(n,C).

In the real case, quadratic forms are determined up to isomorphism by a pair of nonnegative integers (p,q) where n:=p+q is the dimension of V, and p-q is the signature. Concretely, we may assume V=Rn and

The corresponding Lie groups and Lie algebra are denoted O(p,q), SO(p,q), Spin(p,q) and so(p,q). We write Rp,q in place of Rn to make the signature explicit.

The spin representations are, in a sense, the simplest representations of Spin(n,C) and Spin(p,q) that do not come from representations of SO(n,C) and SO(p,q). A spin representation is, therefore, a real or complex vector space S together with a group homomorphism ρ from Spin(n,C) or Spin(p,q) to the general linear group GL(S) such that the element −1 is not in the kernel of ρ.

If S is such a representation, then according to the relation between Lie groups and Lie algebras, it induces a Lie algebra representation, i.e., a Lie algebra homomorphism from so(n,C) or so(p,q) to the Lie algebra gl(S) of endomorphisms of S with the commutator bracket.

Spin representations can be analysed according to the following strategy: if S is a real spin representation of Spin(p,q), then its complexification is a complex spin representation of Spin(p,q); as a representation of so(p,q), it therefore extends to a complex representation of so(n,C). Proceeding in reverse, we therefore first construct complex spin representations of Spin(n,C) and so(n,C), then restrict them to complex spin representations of so(p,q) and Spin(p,q), then finally analyse possible reductions to real spin representations.