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July 14, 2005

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 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 , we can consider the horn Λ k i of the k-simplex Δ. If the maps ξ k i:X kHom(Λ 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 to be a simplicial manifold and the various ξ 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 algebras have been discussed before on this blog (see here for links to various papers), but I’ll give a lightning definition:

An L algebra is a Z-graded vector space g ={,V 2,V 1,V 0,V 1,V 2,} 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. 0V 1V 00, then V 0 is a Lie algebra. More generally a Lie n-algebra is an L algebra concentratd in degrees n,,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 which is the same as the nerve of a dg-Lie algebra.

The next talk is starting. More…


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


Posted at July 14, 2005 4:21 AM UTC

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