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Let $H$ be a Hopf algebra with antipode $S$. For $h \in H$, we have $(S \otimes S) \circ \Delta(h) = \tau \circ \Delta \circ S(h)$, where $\tau(a \otimes b) = b \otimes a$ and $\Delta$ is the commultiplication. How to prove this identity? Thank you very much.

LJR
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    See proposition 4.0.1 4) of Hopf algebras by Sweedler and use that $\tau\circ \tau=Id$. Or see proposition 4.2.6 iii) of Hopf algebras by Dascalescu, Nastasescu and Raianu if you prefer modern writing. The proof of the property is a nice trick though. – Mathematician 42 Aug 02 '16 at 13:05
  • @Mathematician42, thank you very much. – LJR Aug 02 '16 at 13:11

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