Spin(n,C) - Knowlpedia

For every integer $n\ge1$ , $\operatorname{Spin}(n,\mathbb C)$ is the spin group in the even Clifford algebra of $\mathbb C^{n}$ with quadratic form $Q(v)=\sum_jv_j^2$ and convention $v^2=-Q(v)1$ . It is generated by products of an even number of vectors satisfying $Q(v)=1$ .

Dimensions and structure

This group has real dimension $2(n(n-1)/2)$ , complex dimension $n(n-1)/2$ . Conjugation on the generating vector space gives a two-to-one homomorphism to $\mathrm{SO}(n,\mathbb C)$ with kernel $\{\pm1\}$ . It is disconnected at n=1, connected at n>=2, and simply connected at n>=3; the two-dimensional double cover is not universal.

Its complex Lie group structure uses holomorphic multiplication and inversion. Forgetting the complex structure gives the underlying real Lie group; real and holomorphic homomorphisms belong to different categories.

Clifford conjugation gives a two-sheeted covering with kernel {±1}. See $\operatorname{SO}(n,\mathbb C)$ .

References

Pavel Etingof, Lie Groups and Lie Algebras (MIT lecture notes), §31.2, Example 31.3 and Proposition 31.4; §31.3, Example 31.10, pp. 166–168.