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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 7 --- by Dror Bar-Natan}

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Online: \url{http://drorbn.net/AcademicPensieve/2013-11/MZV_ex7.pdf}.

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{\bf 1.} Show that the Kontsevich/KZ integral $Z\colon PB_n\to\gr PB_n$ satisfies property U: for $D\in\calG_m\calA_n$, one has $(\gr Z)(\pi(D))=D$. You may take for granted that it is well defined --- namely that it is invariant under deformations of geometrical braids.

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{\bf 2.} Show that the Kontsevich/KZ integral is multiplicative.

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{\bf 3.} Verify Arnold's identity: if $\displaystyle \omega_{ij}=\frac{dz_i-dz_j}{z_i-z_j}$, then $\omega_{12}\wedge\omega_{23}+\omega_{23}\wedge\omega_{31}+\omega_{31}\wedge\omega_{12}=0$. (This will be used within the proof of the invariance of $Z$).

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{\bf 4.} Recall from the last assignment (exercise 3) that there is a naturally-defined ``co-product'' map $\Delta_{A_n}\colon A_n\to A_n\otimes A_n$, where $A_n=\gr\bbQ PB_n$.

(i) Describe $\Delta_{A_n}$ explicitly; in particular, demonstrate that you know how to compute $\Delta_{A_n}$ by computing $\Delta_{A_n}(t^{12}t^{13}t^{23})$.

(ii) Show that the Kontsevich/KZ integral is co-homomorphic in the sense of the last assignment.

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%{\bf 5.} Using the last exercise of the previous assignment and the fact that $PB_n=F_{n-1}\rtimes(F_{n-2}\rtimes(\dots(F_2\rtimes F_1)\dots))$, give a direct algebra-only proof that $\gr PB_n=A_n$ (namely, do not use integration or any other bit of analysis). 

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