Spin(8,C) - Knowlpedia

$\operatorname{Spin}(8,\mathbb C)$ is the spin group generated by even products of vectors of squared length 1 in the Clifford algebra of $\mathbb C^{8}$ with relation $v^2=-\sum_jv_j^2$ . Its conjugation action on vectors is the standard double covering of $\mathrm{SO}(8,\mathbb C)$ .

Dimensions and structure

This group has real dimension $56$ , complex dimension $28$ . Its tangent algebra is the skew-symmetric $8\times8$ matrices. The group is simply connected. Dimension eight is the setting for triality; the vector and two half-spin representations live naturally on the spin group.

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.

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.