Spin(3,C) - Knowlpedia

$\operatorname{Spin}(3,\mathbb C)$ is the spin group in the even Clifford algebra of $\mathbb C^{3}$ 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 $6$ , complex dimension $3$ . Conjugation on the generating vector space gives a two-to-one homomorphism to $\mathrm{SO}(3,\mathbb C)$ with kernel $\{\pm1\}$ . For dimension at least three it is connected and simply connected, so the covering is 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 $\operatorname{SO}(3,\mathbb C)$ .

The complex spin group in dimension three is SL(2,C); its vector action gives the two-to-one covering of SO(3,C). See $\operatorname{SL}(2,\mathbb C)$ .

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.