Spin(3,R) - Knowlpedia

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

Clifford conjugation gives a two-sheeted covering with kernel {±1}. See $\operatorname{SO}(3,\mathbb R)$ .

The compact spin group in dimension three is isomorphic to SU(2), with the spin map corresponding to the adjoint double cover of SO(3). See $\operatorname{SU}(2)$ .

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.