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Dual of a Hopf Algebra, sort of commutativity

  1. Jan 27, 2012 #1

    The question is quite easy to write down actually. I was wondering whether it can be proved that for [itex]A[/itex]=([itex]A[/itex], [itex]\mu[/itex], [itex]\eta[/itex], [itex]\Delta[/itex], [itex]\epsilon[/itex], [itex]S[/itex]) a Hopf Algebra with invertible antipode S and [itex]A^*[/itex] the dual of [itex]A[/itex] the following equality holds provided that [itex] \chi(aa')=\chi(a'a)[/itex] for all [itex] a,a' \in A[/itex].

    The equality is:

    [itex] \mu \circ (Id_A \otimes \chi) \circ \Delta = \mu \circ (\chi \otimes Id_A) \circ \Delta[/itex]

    or in sweedler notation [itex]\sum_{(a)} \chi(a_{(1)})a_{(2)} = \sum_{(a)} a_{(1)}\chi(a_{(2)})[/itex]

    Where [itex] a \in A [/itex] [itex] \chi \in A^* [/itex]

    Can anyone see a proof or a counterexample?
    Last edited: Jan 27, 2012
  2. jcsd
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