I have a question about the relations of the representations of a compact group $SO(p+q)$ and that of their compact counterparts, especially when there are many non-compact counterparts $SO(p,q)$ available, like the case of $SO(8)$ and $SO(p,q)$ where $p+q=8$.
Most textbooks or notes on group theory will often focus only on compact Lie groups like $SO(n)$, $SU(n)$ and representations. I have not found any accessible (basic) text describing representations of $SO(p,q)$ or $SU(p,q)$. I understand the following facts:
- To obtain the generators of $SO(p,q)$ from those of $SO(p+q)$, choose a maximal subgroup $SO(p)\times SO(q)\subset SO(p+q)$, of which there would be:
$\frac{p(p-1)}{2} + \frac{q(q-1)}{2}$ generators
These will serve as the compact generators of $SO(p,q)$, and the remaining
$\frac{(p+q)(p+q-1)}{2} -\frac{p(p-1)}{2} - \frac{q(q-1)}{2} = pq$ generators
should be multiplied by a factor of $i$ and will serve as the non-compact generators of $SO(p,q)$.
- For the case of $SU(n)$ and $SL(n)$, like $SU(2)$ and its non-compact form, $SL(2,\mathbb R)$, it is straightforward to understand (at least superficially) that their representations are complex conjugates to one another.
However, for the case of $SO(8)$, there are many possible non-compact counterparts, (3 listed above), and so my questions are 1. Are $SO(8)$ representations conjugates to those of $SO(p+q)$, for all combinations of $p+q=8$? 2. What then, are the relations among the representations of $SO(p,q)$? For example, what are the relations among the representations of $SO(4,4)$ and $SO(5,3)$ ?