Once I read a paper in which the author(s) gave a sensible (possibly well-established) name to the $9$-dimensional compact Lie group $$\frac{\text{SU}(2) \times \text{SU}(2) \times \text{SU}(2)}{\mathbb{Z}_2} \cong \frac{\text{Sp}(1) \times \text{Sp}(1) \times \text{Sp}(1)}{\mathbb{Z}_2}.$$ I can't seem to recall what the name was (or which paper I read it in), and it's driving me mad.

This is a silly question, granted, but I would appreciate it if anyone who knows a standard (or just a sensible) name for this group could kindly remind me what it is. If I recall right, the name it was supposed to call to mind the isomorphism $\text{SO}(4) \cong (\text{SU}(2) \times \text{SU}(2))/\mathbb{Z}_2$. Thanks.

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    $\begingroup$ Presumably you have in mind the diagonal action of $\mathbb{Z}_2$? $\endgroup$ – Qiaochu Yuan Mar 10 at 20:50
  • $\begingroup$ @QiaochuYuan: yes, the diagonal action $\endgroup$ – Jesse Madnick Mar 10 at 23:25

Note that $SU(2)\times SU(2) = \operatorname{Spin}(4)$ so

$$(SU(2)\times SU(2)\times SU(2))/\mathbb{Z}_2 \cong (\operatorname{Spin}(4)\times SU(2))/\mathbb{Z}_2$$

which is sometimes called $\operatorname{Spin}^h(4)$. It is also referred to as $\operatorname{Spin}_{SU(2)}(4)$, see for example $(2.15)$ of A New $SU(2)$ Anomaly by Wang, Wen, and Witten.

  • $\begingroup$ thank you, Michael! you wouldn't happen to have a reference (any paper at all) for the nomenclature $\text{Spin}^h(4)$, would you? $\endgroup$ – Jesse Madnick Mar 10 at 23:31
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    $\begingroup$ There's definition 5 (page 24) of Xuan Chen's thesis. In this paper, they denote the group by $G_0(4)$, see the bottom of page 65. You might also be interested in this MathOverflow question. $\endgroup$ – Michael Albanese Mar 10 at 23:51

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