I am trying to understand some things stated in these notes on Kontsevich's lectures of 1994 on deformation theory. I am especially interested on the part on SHLA (strong homotopy Lie algebras, or $L_\infty$-algebras). It is around page 27.

By definition, a SHLA is a cofree cocommutative coalgebra $C = S^c(V)$ endowed with a square-zero differential $d_C$ (the operations of the SHLA are then given by the projection of the action of $d_C$ onto $V=C^{(1)}$, the weight $1$ part). In the notes Kontsevich works on the tangent space of $C$.

Question 1: What is the tangent space of $C$? I find the definition given in the notes very vague (also, I'm not familiar with supermanifolds).

If I understand correctly, he then defines the Maurer-Cartan locus as some leaf of a foliation in $C$ (maybe as $\ker[d_C,-]$, but he might also be working in the tangent space), and gauge equivalences as induced by $\mathrm{im}[d_C,-]\subseteq\ker[d_C,-]$, or something like that.

Question 2: What is the correct picture? What is the correct definition for the Deligne groupoid in the context of SHLAs (seen as coalgebras)?

I would be grateful for any help or references.


1 Answer 1


The following is a partial answer.

I found the answer to my question 1 reading Kontsevich's Deformation Quantization of Poisson Manifolds. There (section 4), he says that we look at $C$ as a formal pointed manifold (whatever that means; I think that the idea behind this is that, at least in the finite dimensional case, the dual of $C$ is a graded commutative algebra, and thus a higher analogue of an affine variety), and he explains that $V$ is the tangent space of $C$ at the basepoint (i.e. the point $0$). But $C$ has only one geometric point, so this is all the data we need.

A vector field on $C$ is then given by a map $Q:C\to V$, which automatically determines a morphism $C\to C$ by universal property. If $Q$ is odd and $[Q,Q]=0$, by noticing that $[Q,Q] = 2Q^2$ we have that $Q$ is the same thing as a differential on $C$ (and in particular its first Taylor coefficient makes $V$ into a chain complex).

I am still unclear on the good settings for Maurer-Cartan elements. Kontsevich says some things about formal neighborhoods and such, but I don't know enough about that yet. Any help or comments are more than welcome.

Edit: In section 4.5.2, he explains how to think of Maurer-Cartan elements in the $L_\infty$ algebra tensored by a finite dimensional, nilpotent commutative algebra (i.e. the maximal ideal of a local aritinian algebra). I think I more or less understand what is going on, but I would like to understand what the Deligne groupoid is for the original $L_\infty$-algebra, without tensoring by something nice.

A possible idea would be to consider another cocommutative algebra $$M:=\prod_{n\ge1}k\cdot a^{\otimes n}$$ with coproduct defined in a way similar to the cofree cocommutative coalgebra, and look at the morphisms $M\to C$ landing in the kernel of $Q$. But what is a gauge equivalence in this setting? And in particular, why should gauge equivalences be preserved by $L_\infty$ morphisms?


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