Spin(4,C) - Knowlpedia

$\operatorname{Spin}(4,\mathbb C)$ is the spin group generated by even products of vectors of squared length 1 in the Clifford algebra of $\mathbb C^{4}$ with relation $v^2=-\sum_jv_j^2$ . Its conjugation action on vectors is the standard double covering of $\mathrm{SO}(4,\mathbb C)$ .

Dimensions and structure

This group has real dimension $12$ , complex dimension $6$ . Its tangent algebra is the skew-symmetric $4\times4$ matrices. The group is simply connected.

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.

The simply connected complex spin group in dimension four is SL(2,C) × SL(2,C). See $\mathrm{SL}(2,\mathbb C)\times\mathrm{SL}(2,\mathbb C)$ .

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.