Do we need to show these are bialgebras first before we show they are Hopf algebras?
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.)
I am not sure why, but I cannot add a post. Can someone please add this post to the forum for me.
(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)?”
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.
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.
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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