If $S_0, S_1$ are two solutions of Maurer-Cartan equation $dS+\frac{1}{2}{S,S}=0$ for a dg-Lie algebra $g$, do we have a suitable concept of homotopy between $S_0$ and $S_1$?
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1$\begingroup$ There is a standard notion of "gauge equivalence" of Maurer-Cartan solutions, at least when $g^0$ (the degree zero part of $g$) is nilpotent. It comes from the "gauge action", which is just the exponentiated $\operatorname{ad}$ action of $g^0$. See for example these notes of Manetti: arxiv.org/abs/math/0507286 $\endgroup$– Kevin H. LinCommented Dec 7, 2010 at 10:15
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$\begingroup$ If you take the infinitesimal neighbourhood of a Maurer Cartan solution (ie the twisted dg-Lie algebra) then the differential certainly gives a notion of homotopy. I look forward to an expert's take on this nice question. $\endgroup$– James GriffinCommented Dec 7, 2010 at 11:03
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2 Answers
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$S_0$ and $S_1$ are said to be homotopy equivalent if there is a Maurer-Cartan element $S(t,dt)$ in the dgla $g[t,dt]$ such that $S(0)=S_0$ and $S(1)=S_1$. It is not completely clear at first that this is an equivalence relation, but actually it is. Indeed much more is true and the homotopy equivalence just described is just the tip of the iceeberg. To see this, rewrite $g[t,dt]$ as $g\otimes\Omega^1$, where $\Omega^1$ is the differential graded commutative algebra of polynomial differential forms on the (algebraic) 1-simplex. Then one sees this is the beginning of a simplicial dgla $g\otimes\Omega^\bullet$, and taking Maurer-Cartan elements produces a simplicial set $MC(g\otimes \Omega^\bullet)$. This simplicial set turns out to be a Kan complex and the fact that the homotopy relation between solution of the Maurer-Cartan equation on $g$ is an equivalence relation is precisely the 'horn filling' property of this Kan complex.
A good reference is Getzler's [math/0404003] Lie theory for nilpotent L-infinity algebras
(in formal deformation theory one produces a nilpotent dgla out of an arbitrary one by tensoring it with the maximal ideal $m_A$ of a local Artin algebra $A$)
answered Dec 7, 2010 at 16:28
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$\begingroup$ You mean maximal ideal of a local Artin algebra? $\endgroup$ Commented Dec 7, 2010 at 17:11
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To continue what Kevin Lin mentioned, a discussion on the relation between homotopy and gauge equivalence for solutions of the Maurer-Cartan equation is in the last section of Manetti's "Deformation theory via differential graded Lie algebras". To sum up, you can define two solutions $S_0,S_1\in MC_{\mathfrak{g}}(A)$ to be homotopic iff there exists an element $S\in MC_{\mathfrak{g}[t,dt]}(A)$ with $S(0)=S_0$ and $S(1)=S_1$. Now it turns out that in a certain sense this homotopy equivalence and the gauge equivalence are the same thing, however the homotopy definition is preferable since it extends to $L_\infty$ algebras.
answered Dec 7, 2010 at 11:29