In this chapter, we will take a closer look at the groups SO(N) and their double covers, Spin(N). We assume that N ≥ 3 and that N = 2n + 1 or 2n. The group Spin(N) was constructed at the end of Chapter 13 as the universal cover of SO(N). Since we proved that πl (SO(N)) ≅ ℤ/2ℤ, it is a double cover. In this chapter, we will construct and study the interesting and important spin representations of the group Spin(N). We will also show how to compute the center of Spin(N).
KeywordsWeyl Group Short Exact Sequence Double Cover Semidirect Product Clifford Algebra
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