Number 5

Do we need to show these are bialgebras first before we show they are Hopf algebras?

 

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5 comments

  1. If what you mean is that in order to prove that the examples given are Hopf algebras we need to prove that they are bialgebras, then I think that the answer is “yes.” It is one of the steps involved –at least for me this is the case.

    (A Hopf algebra is a bialgebra with an additional map called the antipode.)

  2. karenewalters

    I am not sure why, but I cannot add a post. Can someone please add this post to the forum for me.

    6b
    (b) is written “Prove that F(x, y) is an addition formula if and only if F(x, y) = x+y+ (higher order terms) and F(x, F(y, z)) = F(F(x,y),z).”

    Should it be, “Prove that F(x, y) is an addition formula if and only if f(x, y) = x+y+ (higher order terms) and F(x, F(y, z)) = F(F(x,y),z)?”

  3. Karen, the answer is ‘no’. Here f is a function of one variable and F is a function of two variables.
    I don’t know why you weren’t registered as an author. I just sent you an invitation.

  4. diegcif

    Fabian and I think there is some Hopf algebra inside problem 6. However we are not sure how to continue.
    We can see f as a map in C[[x]] (or in C[[x,y]]) and F as a map in C[[x]]*C[[x]]. The “addition formula” may be written as a commutative diagram and we think it should be the antipode, by approriately defining the product and coproduct.

  5. karenewalters

    I think you guys may be on to something. I couldn’t make the connection myself but when I found f(x) for each F(x, y) I can see how f can be viewed as a map in C[[x]] and F as a map in C[[x]]*C[[x]]. I am curious what you find. I hope you are successful before midnight.

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