# YBL

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 Apr7 awarded Guru Apr7 revised What is special about polylogarithms that leads to so many interesting identities and applications? small improvements/corrections Jan25 awarded Revival Nov20 awarded Yearling Nov19 awarded Good Question Nov15 awarded Notable Question Oct5 asked Pull back of D-modules and Koszul resolution Sep29 awarded Necromancer Sep29 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). Sep10 revised Can I define the polynomial ring A[x] with an isomorphism f: A ---> A[x]? added 5 characters in body Aug31 revised Hopf algebra structure on the ring of quasisymmetric functions added 213 characters in body Aug31 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. Aug31 comment Hopf algebra structure on the ring of quasisymmetric functions I edited to answer your questions. Aug31 revised Hopf algebra structure on the ring of quasisymmetric functions added 201 characters in body Aug31 answered Hopf algebra structure on the ring of quasisymmetric functions Aug29 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 $f:X\to \mathbb C$. Aug28 comment Poisson structure on the cotangent bundle The problem is that you are forgetting the last axiom in B-B's definition of a TDO, namely how $gr^1 A$ is identified with $T_X$ through $\sigma(d)(f) = fd - df$. The Poisson bracket on the cotengant bundle is the equivalent of a first ordre non commutative deformation. A simple isomorphism of algebras $gr A \simeq Sym T_X$ forgets about this information. The Poisson bracket is just a Lie bracket that is also a derivation. To get the desired result you just have to check that $\sigma$ given in degree 1 extends uniquely to the whole symetric algebra. Aug27 revised Microlocal geometry - A theorem of Verdier edited tags Aug27 comment Poisson structure on the cotangent bundle Which definition of a sheaf of twisted differential operators are you working with? If you take the one from Beilinson and Bernstein, the isomorphism of Poisson structures is straightforward. Aug27 comment Microlocal geometry - A theorem of Verdier I posted the question quite a while ago, tried the bounty but still got no answer or comment. Is it because the problem is actually difficult or because the question is poorly formulated. I guess what seemed elementary to Verdier really isn't for most of us. Anyways, once again any comment, advice or reference would be truly appreciated. Thanks.