Spin(5,C) - Knowlpedia

$\operatorname{Spin}(5,\mathbb C)$ is the spin group generated by even products of vectors of squared length 1 in the Clifford algebra of $\mathbb C^{5}$ with relation $v^2=-\sum_jv_j^2$ . Its conjugation action on vectors is the standard double covering of $\mathrm{SO}(5,\mathbb C)$ .

Dimensions and structure

This group has real dimension $20$ , complex dimension $10$ . Its tangent algebra is the skew-symmetric $5\times5$ 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 complex spin group in dimension five is Sp(4,C), where 4 is the complex matrix size. See $\mathrm{Sp}(4,\mathbb C)$ .

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.