This post is a report on a talk that Mikhail Kapranov gave in Berkeley a few weeks ago on the “fundamental Lie algebra” — it is a Lie algebra associated to a manifold and a point on the manifold much like the fundamental group. Before telling about what Kapranov said, I’d like to start with some background.
Let be a manifold and . Suppose we consider pairs where is a vector bundle and is a flat connection on . Two such pairs may be tensored together to produce a new vector bundle with flat connection .
Define a group by the following procedure. An element is a collection of linear isomorphisms for each which are natural with respect to maps and which obey the rule .
In other words, an element of is a linear automorphism of the fibre at of any vector bundle with flat connection. So, what is ?
As you might have guessed, is the fundamental group of with respect to the basepoint . To see why this is reasonable, note that any loop in based at gives rise to a monodromy map by integrating the connection. However, homotopic loops give rise to the same map since the connection is flat.
Now, what happens when we change our setup to include all connections (not necessarily flat). Then we end up with a much bigger group which is roughly the group of loops in based at modulo reparmetrization and going back and forth. This is a kind of infinite dimensional Lie group. Since this group is a bit hard to study, we will consider its Lie algebra.
To define this Lie algebra, consider the vector space whose elements are collections of linear maps where now ranges over all vector bundles with connection. We will impose the naturality condition and also that . This vector space is naturally a Lie algebra. Kapranov calls this Lie algebra the fundamental Lie algebra.
To see some elements of , recall that if is a vector bundle with connection, then we have a curvature which is an valued 2-form on . Hence if we have any element of then we can pair it against the value of at to get a endomorphism of . This construction gives rise to a map .
Kapranov has proven the following theorem which describes . Before stating it, we need to recall the notion of the free Lie algebra associated to a vector space , . As you might expect it is just the free Lie algebra generated by elements of . It is graded with graded pieces . For example and .
is isomorphic (not canonically) to .
In particular it is a free Lie algebra, since every Lie subalgebra of a free Lie algebra is free.
I should mention in closing is that what we have been doing is Tannakian constructions of groups (and Lie algebras). Namely we started with a tensor category with fibre functor and then we considered automorphisms of this fibre functor. Those who know me well will understand that I like this procedure since it is the key idea in the proof of the geometric Satake correspondence.