© | << < ? > >> | Dror Bar-Natan: Talks:

The Pure Virtual Braid Group is Quadratic

With Peter Lee at the University of Oregon

August 5, 2011, Columbia Hall 45, 10:00-1:00
Handout: Quadraticity.html, Quadraticity.pdf, Quadraticity.png. (Also, B4Syzygy.pdf).

Sources: Quadraticity.zip.    Pensieve: Projects: Quadraticity.

Not a traditional talk, this will be a visit to explore a certain subject. It's defining email was sent on May 2, 2011:

Dear Friends and Potential Future Friends in Oregon,

I'd like to make use of a nice alignment of circumstances that is to occur this summer, and I wonder if you'd want to help (or, if you wish, also make use of that alignment).

[I will be in Oregon for a personal reason]. My student Peter Lee has a math result that may interest some people in Oregon (you) and he and I need a stage to motivate ourselves to sharpen it to perfection. So it may be an excellent idea for Peter and for me to visit Oregon and spend some time telling our story, right before [my personal reason]. Here are the details:

[Timing details]

The math result is about groups and quadratic algebras. There is a simple way to define functor Q that takes a group to "the quadratic approximation to its unipotent completion". A famed example is the pure braid group, which is taken by the functor Q to the Drinfel'd-Kohno algebra (aka "horizontal chord diagrams").

We do not know how good is the functor Q. We do not know which quadratic algebras occur in its image, and when a quadratic algebra does occur, we don't know how well it approximates the unipotent completion of the group it came from.

Peter's work is about a general methodology that allows one to show that in certain cases the said quadratic approximation is "good" (in some natural sense). Furthermore, in the case of the pure virtual braid group the quadratic approximation is a well known algebra, the one with generators aij (1≤i,j≤n) and with relations [aij,akl] = 0 = [aij,aik] + [aij,ajk] + [aik,ajk], (the classical Yang-Baxter relations). Peter shows that this quadratic approximation is "good". In the case of the classical braid group, the parallel of Peter's result is most famously deduced from the existence of the Kontsevich integral (though it has other proofs as well). In the case of the virtual braid group, his result was conjectured but not proven in a paper by Bartholdi, Enriquez, Etingof, and Rains (arXiv:math.RA/0509661).

Would any of you be around for the said dates (right before August 7th)? Would you be interested in hearing 2-4 lectures on the subject, or perhaps holding some sessions on the subject in the "Russian seminar" style?

My grant could cover all relevant expenses. We're really mostly looking for an audience to motivate us.