Spin(5,R) - Knowlpedia

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

Dimensions and structure

This group has real dimension $10$ . Its tangent algebra is the skew-symmetric $5\times5$ matrices. The group is simply connected.

The compact spin group in dimension five is the compact quaternionic group Sp(2). See $\operatorname{Sp}(2)$ .

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.