Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.)

share|improve this question
    
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 $\sum S(x_{(1)}) x_{(2)} = x_{(1)} S(x_{(2)}) = \varepsilon(x)$. –  mt_ 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! –  mt_ Feb 25 '12 at 13:59
    
Does sbseminar.wordpress.com/2011/07/07/… help? –  David Speyer Nov 13 '12 at 16:02
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.