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In several papers I've seen on Combinatorial Hopf Algebras, the algebra and coalgebra structures are described, but no antipode is defined. CHAs generally have a natural grading, and are of finite dimension in each degree, and I have seen one comment which suggested that this leads to an obvious or unique antipode. If so, please can someone spell it out.

(Are there any good introductory papers on CHAs, as the ones I've found so far all go straight in at the deep end.)

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If an antipode S exists then it is unique. If you know the comultiplication you should be able to recover the antipode from the relation $\sum S(x_{(1)}) x_{(2)} = x_{(1)} S(x_{(2)}) = \varepsilon(x)$. – m_t_ Feb 25 '12 at 9:52
@mt_ - Ok, so the antipode (if it exists) is determined by the bialgebra structure. Can it always be derived from the bialgebra structure, or are restrictions necessary, such as requiring the bialgebra to be graded, or more? (I can imagine that a grading enables an inductive definition, whereas without a grading it might not be clear where to start - however, this is all a bit fuzzy to me.) – DavidA Feb 25 '12 at 13:48
You may perhaps be able to extract a construction of S from uniqueness proofs for the antipode - I don't know. There is a link in the wiki article, or maybe try Sweedler's book or any other Hopf algebra book. I read a proof not so long ago where the antipode was recovered from the coalgebra structure but I've forgotten where - sorry! – m_t_ Feb 25 '12 at 13:59
Does… help? – David Speyer Nov 13 '12 at 16:02

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