The group algebra $k(G)$ of any group $G$ satisfies as a Hopf algebra the following identities: $$ S\otimes S\circ \varDelta=\sigma\circ\Delta\circ S $$ $$ \nabla\circ S\otimes S=S\circ\nabla\circ\sigma $$ where $S$ is the antipode, $\Delta$, the comultiplication, $\nabla$, the multiplication, and $\sigma:x\otimes y\mapsto y\otimes x$.

Is this valid for all Hopf algebras or only for some special class?

EDIT. I asked this question later at MathOverflow.


Your first identity says that $S$ is a coalgebra anti-homomorphism, and your second says $S$ is an algebra anti-homomorphism. Both of these are true for all Hopf algebras, see for example http://library.msri.org/books/Book43/files/nik.pdf or http://ncatlab.org/nlab/show/Hopf+algebra, or one of the books by Sweedler or Abe.

  • $\begingroup$ Do these Hopf algebras lie in the category of vector spaces, or in an arbitrary braided monoidal category? $\endgroup$ – Sergei Akbarov Aug 12 '14 at 17:51
  • $\begingroup$ These are Hopf algebra objects in the category of vector spaces, I don't know a reference for more general categories, possibly the ncatlab has something. $\endgroup$ – Matthew Towers Aug 12 '14 at 18:18
  • $\begingroup$ My category is not the category of vector spaces. :( $\endgroup$ – Sergei Akbarov Aug 13 '14 at 7:16
  • $\begingroup$ Try expressing the proof diagrammatically, you probably find it works anywhere. $\endgroup$ – Matthew Towers Aug 13 '14 at 18:27

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