Abstract. There is compelling evidence including some long-route
indirect proofs, that the following are the same, or at least closely and
directly related to each other:
- The Kashiwara-Vergne conjecture, which says, more or less, that
convolutions of Ad-invariant functions on a Lie group are the same as
convolutions of Ad-invariant functions on its Lie algebra.
- Drinfel'd associators, at least "restricted to one loop", and the work of
Alekseev and Torossian.
- Knot invariants, and in particular, the multi-variable Alexander
polynomial.
- Virtual knots in which overcrossings (but not undercrossings) are made to
commute.
- The Etingof-Kazhdan formalism in the case of co-commutative Lie
bi-algebras.
There is also some weaker evidence that all these topics remain related even
when the various commutativity, co-commutativity and other restrictions are
dropped.
If all goes well, in 5-10 years I will be able to give a coherent
explanation of all that (perhaps sooner, if others will get interested
and do the work). For now, all I have are some disorganized tidbits.
Partially I will follow the topics and handouts of several talks that I
gave recently. In particular, with base:=http://www.math.toronto.edu/~drorbn/Talks,
see
base/Sandbjerg-0810,
base/Istanbul-0606,
base/Zurich-080513,
base/MSRI-0808,
and
base/Copenhagen-081009.
