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I am reading Natale's book [S. Natale, Semisolvability of semisimple Hopf algebras of low dimension, Mem. Amer. Math. Soc. 186 (2007), 123 pp.]. I can not understand a sentence "But this is not possible since ${\rm dim}A^{co q} = 9$....." from the proof of Theorem 12.2.1. In my option, there are severa possible values of ${\rm dim}A^{co q}$ since we do not know the exact order of $\eta$, and the sentence above holds true if and only if $|\eta|=2$. Who can explain it for me? Thanks a lot!

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One has that $dim(A)=dim\;(A^{co\; q})dim\;q(A)$. The claim is that $dim\;q(A)=2$, I don't know if $q$ restricted to $A$ is surjective.

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  • $\begingroup$ It's the transpose of an inclusion, so of course it must be surjective! $\endgroup$
    – darij grinberg
    Commented May 29, 2011 at 10:28

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