# Connected compact Lie groups with Lie algebra so(4n, R)

I am trying to write a complete list of connected compact simple Lie groups (or of connected complex simple Lie groups, both tasks are equivalent). I am missing just one case.

Consider the Lie algebra $\mathfrak{so}(n, \mathbb{R})$ where $n$ is a multiple of 4. The simply connected Lie group with this Lie algebra is $\operatorname{Spin}(n, \mathbb{R})$, and has center $(\mathbb{Z}/2\mathbb{Z})^2$. The center has five distinct subgroups: itself, the trivial group, and three subgroups isomorphic to $(\mathbb{Z}/2\mathbb{Z})$; hence there are five Lie groups with Lie algebra $\mathfrak{so}(n, \mathbb{R})$.

Three of them are $\operatorname{PSO}(n, \mathbb{R})$, $\operatorname{SO}(n, \mathbb{R})$ and $\operatorname{Spin}(n, \mathbb{R})$. The other two are obtained by taking the quotient

$$\operatorname{Spin}(n, \mathbb{R})/\{\operatorname{Id}, x\},$$

where $x$ is one of the two preimages of $-\operatorname{Id}$ by the double cover $\operatorname{Spin}(n, \mathbb{R}) \to \operatorname{SO}(n, \mathbb{R})$. These two groups are isomorphic, by the Dynkin diagram automorphism. Each of them has $\operatorname{Spin}(n, \mathbb{R})$ as a double cover and $\operatorname{PSO}(n, \mathbb{R})$ as a quotient of order 2. (So does $\operatorname{SO}(n, \mathbb{R})$, but the two groups I am talking about is not in general isomorphic to the latter.)

I have found so far no mention of these extra groups in the literature. Thus my question is:

Are these groups cited anywhere? Do they have a name?

As a bonus, if someone knows a place where a complete list of connected compact (or complex) simple Lie groups is published, it would be great. All the sources I have found so far say "such a list can be easily established by knowing the centers of the simply connected simple compact Lie groups". It is indeed an easy exercise, but doing the calculation on one's own does not give names of the groups!

Note also than when $n=4$, this group is isomorphic to $\operatorname{SO}(3, \mathbb{R}) \times \operatorname{Spin}(3, \mathbb{R})$, and when $n=8$, it is isomorphic to $\operatorname{SO}(8, \mathbb{R})$ by triality. So the simplest nondegenerate case is $n=12$.

• One can also wonder what's its smallest faithful representation. – YCor Jan 10 '15 at 12:46
• These groups are usually called "semi-spinor" groups and denoted $Ss^\pm(4n)$. – Christian Nassau Jan 10 '15 at 12:57
• I think that these two groups (which are isomorphic, of course) are generally denoted $\mathrm{SO}'(4m)$, where $n=4m$. At least, this is the notation I have seen in Wolf and Helgason. – Robert Bryant Jan 10 '15 at 12:57
• Here's a physics paper abut their role in string theory: Brett McInnis, The semispin groups in string theory, arxiv.org/pdf/hep-th/9906059v1.pdf – Christian Nassau Jan 10 '15 at 12:58
• And here's a computation of their homology: Baum, Brodwer: The cohomology of quotients of classical groups, Topology 3: 305-336, 1965. – Christian Nassau Jan 10 '15 at 12:59