Abstract
The chapter starts with a discussion of the Clifford group Γ(V), for any vector space V with a non-degenerate symmetric bilinear form B. Using the norm homomorphism on the Clifford group, we define the spin group Spin(V), and discuss its topology for the cases V=ℝn,m and V=ℂn. Following a discussion of Clifford modules in general, we focus on the case that the bilinear form is split. We show that in this case, there is an essentially unique irreducible Clifford module, the so-called spinor module. We give a discussion of pure spinors and their relation with Lagrangian subspaces, followed by a proof of Cartan’s triality principle. The classification of spinor modules for the case V=ℂn is used to derive interesting properties of the spin groups, with applications to compact Lie groups.
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Notes
- 1.
In more detail, recall that the bilinear form on \(\varDelta _{7}\cong{\mathsf{S}}_{8}^{\bar{0}}\) is ℂl(7)≅ℂl 0(8)-invariant. Restricting to ℂl(6)⊆ℂl(7), we obtain a ℂl(6)-invariant bilinear form on \(\varDelta _{6}^{+}\oplus \varDelta _{6}^{-}\cong {\mathsf{S}}_{6}\), which must agree with the canonical bilinear form up to scalar multiple. But the latter vanishes on the even and odd part of S 6.
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Meinrenken, E. (2013). The spin representation. In: Clifford Algebras and Lie Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, vol 58. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36216-3_3
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DOI: https://doi.org/10.1007/978-3-642-36216-3_3
Publisher Name: Springer, Berlin, Heidelberg
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