Indefinite Signature
In indefinite signature, the spin group Spin(p, q) is constructed through Clifford algebras in a similar way to standard spin groups. It is a connected double cover of SO0(p, q), the connected component of the identity of the indefinite orthogonal group SO(p, q) (there are a variety of conventions on the connectedness of Spin(p, q); in this article, it is taken to be connected for p + q > 2 ). As in definite signature, there are some accidental isomorphisms in low dimensions:
- Spin(1, 1) = GL(1, R)
- Spin(2, 1) = SL(2, R)
- Spin(3, 1) = SL(2,C)
- Spin(2, 2) = SL(2, R) × SL(2, R)
- Spin(4, 1) = Sp(1, 1)
- Spin(3, 2) = Sp(4, R)
- Spin(5, 1) = SL(2, H)
- Spin(4, 2) = SU(2, 2)
- Spin(3, 3) = SL(4, R)
Note that Spin(p, q) = Spin(q, p).
Read more about this topic: Spin Group
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