I have recently encountered a Lie group in a paper called $D_{5}$ as a subgroup of $E_{6}$. I have tried googling with mixed results. Is it just $\operatorname{SO}_{10}(\mathbb C)$?. Is it $\operatorname{Spin}(10)$? I am looking for geometric information and I am approaching this material from the realm of homogeneous spaces and symmetric spaces, if it makes a difference.
What is $D_{5}$
1
$\begingroup$
lie-groups
-
2I know nothing about Lie groups, but have you considered it might just be the dihedral group of order $10$? This might be an inappropriate comment though. – 2012-04-03
-
0IIRC the compact real Lie group $SO_{10}(\mathbf{R})$ has a Lie algebra of the type $D_5$. May be only after complexification? May be also $SO_{10}(\mathbf{C})$? I have ever worked only with algebraic groups or Lie algebras, where we usually assume an algebraically closed ground field, so I'm not fully conversant on the differences between real and complex structures. Alternatively it could refer to the Coxeter group of type $D_5$ (=the Weyl group of the root system of the above mentioned simple Lie algebra). But the latter is not a Lie group, so I'm guessing the first. – 2012-04-03
-
0It may help if you give some context. I don't think there is a standard convention of whether a group called $D_5$ refers to $Spin(10), SO(10)$ or $PSO(10)$. My guess would be $Spin(10)$ since its simply connected (and so singled out). – 2012-04-04