### Thursday at the Streetfest II

#### Posted by Guest

David here

I thought I’d talk a bit about Ezra Getzler’s talk (which was only loosely connected to his abstract) on some Lie theory for ${L}_{\mathrm{\infty}}$ algebras. He started off by saying that is is just as worthwhile working with the nerve of a groupoid and so we defined something to work with:

Given a simplicial set ${X}_{\u2022}$, we can consider the horn ${\Lambda}_{k}^{i}$ of the $k$-simplex $\Delta $. If the maps ${\xi}_{k}^{i}:{X}_{k}\to \mathrm{Hom}({\Lambda}_{k}^{i},{X}_{k})$ are surjective and for $k>n$ bijective, then ${X}_{k}$ is the nerve of an $n$-groupoid. We are far more interested in Lie $n$-groupoids and so need ${X}_{\u2022}$ to be a simplicial manifold and the various $\xi $ maps to be surjective submersions. This agrees with the usual defintion of Lie groupoid when $n=1$ and we call this beast a Li $n$-group if ${X}_{0}=*$ (i.e. has space of objects a point).

I believe ${L}_{\mathrm{\infty}}$ algebras have been discussed before on this blog (see here for links to various papers), but I’ll give a lightning definition:

An ${L}_{\mathrm{\infty}}$ algebra is a $Z$-graded vector space ${g}^{\u2022}=\{\dots ,{V}^{-2},{V}^{-1},{V}^{0},{V}^{1},{V}^{2},\dots \}$ with $k$-brackets (functions on $k$-fold tensor product) satisfying a generalised Jacobi property, which encodes the Leibniz rule for $k=2$, makes the 1-bracket a differential, among others. If the $k$-brackets are all zero for $k>2$ then we are only talking about a dg-Lie algebra.

We then found out that if the graded vector space is concentrated in degrees -1 and 0, i.e. $$0\to {V}^{-1}\to {V}^{0}\to 0,$$ then ${V}^{0}$ is a Lie algebra. More generally a Lie n-algebra is an ${L}_{\mathrm{\infty}}$ algebra concentratd in degrees $-n,\dots ,0$.

He then went to define differential forms on an $n$-simplex such that the complex of such things is a simplicial dg-algebra. After some nested defintions, we got to the Maurer-Cartan simplicial set of ${g}^{\u2022}$ which is the same as the nerve of a dg-Lie algebra.

The next talk is starting. More…

D

PS - Fri 1.42PM local - Many apologies for not getting this one up, or even finishing it off - I just couldn’t get the XHTML to work -actually it was a LaTeX problem $>$:(. I’ll just give some references. Sorry Urs - this was one you would have liked.

Hinich, V and Schechtman, V, Homotopy Lie algebras

Hinich, V, DG coalgebras as formal stacks

There was also reference to the work of J. Duskin and T. Beke, but a search just now on MathSciNet didn’t turn up anything I could conceptually link to the talk.

Must dash

David