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{\bf Universit\'e de Gen\`eve

Section de math\'ematiques
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A.~Alekseev and P.~Severa

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{\bf Braids and Associators, problem set 6 --- by Dror Bar-Natan}

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\noindent Online: \url{http://drorbn.net/AcademicPensieve/2013-10/MZV_ex6.pdf}.

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{\bf 1.} With $R=C^\infty(\bbR^n)$ and $I=\{f\in R\colon f(0)=0\}$, find the set $\calZ$ of all expansions $Z\colon R\to A:=\gr R=\hat\bigoplus I^m/I^{m+1}$.

Bonus (hard). Can you find an algebraic condition that characterises the Taylor expansion $Z_T$ within $\calZ$? (You may want to read question 3).

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{\bf 2.} Find a homomorphic expansion for $\bbZ F_n$, the group ring (over the integers) of the free group on $n$ generators? (The simplest one is known as ``the Magnus expansion''.

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{\bf 3.} Let $G$ be a group and $R$ be a ring, let $RG=\{\sum a_ig_i\colon a_i\in R\}$ be the group ring of G with coefficients in $R$, and let $\Delta\colon RG\to RG\otimes_RRG$ be the $R$-linear extension of the map $\Delta(g)=g\otimes g$. Let $I:=\{\sum a_ig_i\colon \sum a_i=0\}$ be the augmentation ideal of $RG$, and let $A:=\gr RG$.

(i) Explain how $\Delta$ induces a map $\Delta_A\colon A\to A\otimes_R A$.

(ii) Describe $\Delta_A$ in the case where $RG=\bbZ F_n$.

(iii) We say that an expansion $Z\colon RG\to A$ is co-homomorphic if $(Z\otimes Z)\circ\Delta=\Delta_A\circ Z$. Is there a co-homomorphic expansion for $\bbZ F_n$? For $\bbQ F_n$?

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{\bf 3.} Recall that $A_n:=\gr PB_n=\langle t^{ij}=t^{ji}\colon 1\leq i\neq j\leq n\rangle/\calR$, where $\calR$ consists of the relations $[t^{ij},t^{kl}]=0$ when $|\{i,j,k,l\}|=4$ and $[t^{jk},t^{ij}+t^{ik}]=0$ when $|\{i,j,k\}|=3$. Show that every degree $m$ element of $A_n$ can be written as a linear combination of sorted elements; namely, of elements of the form $t^{i_1j_1}t^{i_2,j_2}\cdots t^{i_mj_m}$, where $i_\alpha<j_\alpha$ for every $1\leq\alpha\leq m$ and where $j_1\leq j_2\leq\dots\leq j_m$.

(This should remind you of $PB_n=F_{n-1}\rtimes(F_{n-2}\rtimes(\dots(F_2\rtimes F_1)\dots))$. Does it?)

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