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revised What is special about polylogarithms that leads to so many interesting identities and applications?
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comment Hopf algebra structure on the ring of quasisymmetric functions
Indeed he is. Btw if you are looking for references on the subject, you should know that the product is often called the "stuffle product" (as it is somewhow opposite to the usual "shuffle product" of the Tensor Hopf algebra).
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revised Can I define the polynomial ring A[x] with an isomorphism f: A ---> A[x]?
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revised Hopf algebra structure on the ring of quasisymmetric functions
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comment Hopf algebra structure on the ring of quasisymmetric functions
This is basic duality. Basically if you have a $k$-Hopf algebra $H$ with $k$-basis $h_s$ then $Hom_{k-mod}(H,A) = H^\vee \otimes_k A$. And, by duality, an element $\sum a_s h_s^\vee$ in this last $k$-module corresponds to a $k$-algebra morphism iff it is diagonal. Rather than paraphrasing what I remember about this stuff I added a link to a short simple article on the subject.
Aug
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comment Hopf algebra structure on the ring of quasisymmetric functions
I edited to answer your questions.
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revised Hopf algebra structure on the ring of quasisymmetric functions
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answered Hopf algebra structure on the ring of quasisymmetric functions
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comment Iterated Milnor fibrations and Thom's a_f condition
Thanks for the link, I had already glanced at it but I don't think it is relevant since they are only considering a dimension 1 base, i.e. a single function $f:X\to \mathbb C$.