Reality conditions on representations of classical groups

[hideelo]

I'm reading "Division Algebras and Quantum Theory" by John Baez

https://arxiv.org/abs/1101.5690

In the last paragraph of section 5 (Applications) he says the following

"SU(2) is not the only compact Lie group with the property that all its irreducible continuous unitary representations on complex Hilbert spaces are real or quaternionic. ...All compact simple Lie groups have this property except those of type An for n > 1, Dn with n odd, and E6. For the symmetric groups Sn, the orthogonal groups O(n), and the special orthogonal groups SO(n) for n ≥ 3, all representations are in fact real"On the one hand he says that for Dn with n odd we have irreducible continuous unitary representations that are neither real nor quaternionic (so they're complex). But then he says that the representations of SO(n) are real for n ≥ 3.

But Dn is SO(2n) so which of these is true?

 

[lpetrich]

One has to distinguish between SO(n) the group and SO(n) the algebra. I've seen the group and the algebra given different typography, like SO(n) for the group and so(n) for the algebra. Thus, D(n) = so(2n), and SO(n) is the vector representation of so(n). The spinor representation of so(n) is sometimes called Spin(n).

 

[hideelo]

Yeah, but surely SO(2n) the group counts as a representation of so(2n) the algebra, doesn't it?

 

[fresh_42]

Yeah, but surely SO(2n) the group counts as a representation of so(2n) the algebra, doesn't it?

The algebra $\mathfrak{g}$ counts as a representation of the group $G$.