Knot at Lunch on January 30 2008

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Invitation

Dear Knot at Lunch People,

We will have our next lunch on Wednesday January 30, at the usual place, Bahen 6180, at 12 noon.

As always, please bring brown-bag lunch and fresh ideas. The agenda: we may talk a bit about Artin's theorem on the braid group and automorphisms of the free group, or we may talk about something else.

Further information about this meeting will/may appear at https://drorbn.net/drorbn/index.php?title=Knot_at_Lunch_on_January_30_2008.

As always, if you know anyone I should add to this mailing list or if you wish to be removed from this mailing list please let me know. To prevent junk accumulation in mailboxes, I will actively remove inactive people unless they request otherwise.

Best,

Dror.

Another Invitation

Madre, at Aluna Theatre, January 29 - February 17, featuring Hernando's daughter Juliette Burgos.

Possible Agenda

• The Artin representation of the pure braid group on $n$ strand $PB_{n}$ into the automorphism group $\operatorname {Aut} (F_{n})$ of the free group on $n$ generators $F_{n}$ (hey, I'm not even sure that's what it's called!).
  • Abstract definition.
  • Practical computation from the Wirtinger presentation.
  • Practical computation by stirring the soup.
  • Properties.
  • Artin's theorem (is that what it's called?).
  • Is there an Artin's theorem for pure virtual braids?
  • What is the finite type completion of $\operatorname {Aut} (F_{n})$? How about some of its obvious subgroups?

• Infinitesimal Artin - derivations on free Lie algebras.
  • Derivations, tangential derivations, special derivations (see BBS/Alekseev-080116-162451.jpg).
  • Drinfel'd's theorem on special derivations (see Drinfel'd's On quasitriangular Quasi-Hopf algebras and a group closely connected with $\operatorname {Gal} ({\bar {\mathbb {Q}}}/{\mathbb {Q}})$, Leningrad Math. J. 2 (1991) 829-860, though I wouldn't be surprised if the theorem is actually older).
  • Can you prove that ${\mathcal {A}}^{\mbox{hor}}$ injects into ${\mathcal {A}}^{\mbox{trees}}$?
  • The VS structure on several kinds of algebras of derivations. As always, can you find associators?
  • The Alekseev-Torossian "divergence" (BBS/Alekseev-080121-093327.jpg).