Lie algebras and coalgebras via the configuration pairing

I will outline recent advances generalizing, simplifying, and applying the configuration pairing of graphs and trees originally developed to compute the homology of configuration spaces. Earlier work applies this framework to investigate knot invariants as well as rational homotopy theory. I will present an new application directly to Lie algebras and Lie coalgebras, where this framework provides an interesting tool for making computations and an alternate view of some basic, classical Lie algebra constructions.

In the theory of finite type knot invariants, we filter knots and their invariants into towers of Lie algebras; reducing the study of knots to the study of Lie algebras. This work is morally dual -- filtering Lie algebras to reduce them to towers of "simplified knots.