Return to Revisions

In question 1, it seems that the subspace $H^*(g(Q))$ in $H^*(g(C))$ indeed depends on the rational form of g. Consider the example of $sl(3)$ and two rational forms, $su(3,Q)$ and $sl(3,Q)$, and look at the fundamental classes in $H^8$ (i.e. the generators of the top exterior power). A basis of $su(3,Q)$ is $i(E_{11}-E_{22}), i(E_{22}-E_{33}), E_{12}-E_{21}, E_{13}-E_{31}, E_{23}-E_{32}, i(E_{12}+E_{21}), i(E_{13}+E_{31}), i(E_{23}+E_{32})$. Since we have 5 factors of i, the two fundamental classes differ by a factor of i (up to rational factors).