# name of Lie group $\text{SU}(2) \times \text{SU}(2) \times \text{SU}(2)/\mathbb{Z}_2$

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.

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

## 1 Answer

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.

• thank you, Michael! you wouldn't happen to have a reference (any paper at all) for the nomenclature $\text{Spin}^h(4)$, would you? – Jesse Madnick Mar 10 at 23:31
• 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. – Michael Albanese Mar 10 at 23:51