Dual of a Hopf Algebra, sort of commutativity

1. Jan 27, 2012

conquest

Hi,

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

The equality is:

$\mu \circ (Id_A \otimes \chi) \circ \Delta = \mu \circ (\chi \otimes Id_A) \circ \Delta$

or in sweedler notation $\sum_{(a)} \chi(a_{(1)})a_{(2)} = \sum_{(a)} a_{(1)}\chi(a_{(2)})$

Where $a \in A$ $\chi \in A^*$

Can anyone see a proof or a counterexample?

Last edited: Jan 27, 2012