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It seems that asterisks need no longer be protected
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edit approved Aug 1, 2014 at 15:46
LSpice
edit approved Aug 1, 2014 at 15:46
LSpice
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In question 1, it seems that the subspace $H^{\*}(g(Q))$$H^*(g(Q))$ in $H^{\*}(g(C))$$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).

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).

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).

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

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).

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).

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Pavel Etingof
Pavel Etingof
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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).