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Lie theory for nilpotent $$L_{\infty}$$-algebras. (English) Zbl 1246.17025
Roughly speaking, in the present article Getzler constructs a procedure to integrate $$L_{\infty}$$-algebras. In another very nice, closely related article, A. Henriques [“Integrating $$L_\infty$$-algebras”, Compos. Math. 144, No. 4, 1017–1045 (2008; Zbl 1152.17010)] integrates $$L_{\infty}$$-algebras from a slightly different perspective.
More precisely, in the article under review, Getzler associates to a nilpotent $$L_{\infty}$$-algebra $${\mathfrak g}$$ a simplicial set $$\gamma_{\bullet}({\mathfrak g})$$ which is isomorphic to the nerve $$N_{\bullet}G$$ of the simply connected Lie group $$G$$ associated to $${\mathfrak g}$$ in case the latter is a nilpotent Lie algebra.
The subject has at least two origins different from the one which the title suggests. In rational homotopy theory, Sullivan’s spatial realization functor [D. Sullivan, “Infinitesimal computations in topology”, Publ. Math., Inst. Hautes Étud. Sci. 47, 269–331 (1977; Zbl 0374.57002)] associates to a commutative differential graded algebra (dga) $$A$$ the simplicial set $\text{Spec}_{\bullet}(A):=\text{Hom}_{\text{dga}}(A,\Omega_{\bullet}),$ where $$\Omega_n$$ is the simplicial dga of polynomial differential forms on the standard $$n$$-simplex. This functor permits to associate to a commutative dga $$A$$ a topological space (namely, the geometric realization of the above simplicial set) which represents the rational homotopy type of $$A$$.
On the other hand, a well-known folklore result states that all (formal) algebraic deformation problems in characteristic zero are governed by an $$L_{\infty}$$-algebra $${\mathfrak g}$$ such that the space of (moduli of inequivalent) deformations is given by the quotient of the Maurer-Cartan set $$\text{MC}({\mathfrak g})$$ by the action of the group $$\exp({\mathfrak g}^0)$$ (in case one can make sense of these notions).
Getzler points out that (for a dg Lie algebra $${\mathfrak g}$$ whose underlying cochain complex is bounded below and finite dimensional in each degree) $$\text{Spec}_n(C^*({\mathfrak g}))$$ identifies with $$\text{MC}_n({\mathfrak g}):= \text{MC}({\mathfrak g}\otimes \Omega_n)$$, where $$C^*({\mathfrak g})$$ is the Chevalley-Eilenberg complex associated to $${\mathfrak g}$$. In the remainder of the article, Getzler concentrates on the Maurer-Cartan point of view.
Talking about integration, you may ask “integration into what ?”. The integration procedure will give a simplicial set $$\gamma_{\bullet}({\mathfrak g})$$ with nice properties. The main theorem of the present paper is that $$\gamma_{\bullet}({\mathfrak g})$$ is a Kan complex, i.e., all horns (boundaries of faces missing exactly one component) may be filled. In fact, Getzler shows that for a nilpotent $$L_{\infty}$$-algebra $${\mathfrak g}$$, $$\gamma_{\bullet}({\mathfrak g})$$ is an $${\infty}$$-groupoid.
The definition of $$\gamma_{\bullet}({\mathfrak g})$$ is simply $\gamma_{\bullet}({\mathfrak g}):=\{\alpha\in MC_{\bullet}({\mathfrak g})\,|\, s_{\bullet}\alpha=0\},$ where $$s_{\bullet}:\Omega_{\bullet}^*\to\Omega_{\bullet}^{*-1}$$ is some chain homotopy coming from Dupont’s explicit proof of the de Rham theorem in [J. L. Dupont, “Simplicial de Rham cohomology and characteristic classes of flat bundles”, Topology 15, 233–245 (1976; Zbl 0331.55012)]. The main achievement of Getzler’s article is to use the calculus of differential forms and Dupont’s operators to construct explicitly (and in an elementary way) the differential forms filling in the horns.

##### MSC:
 17B55 Homological methods in Lie (super)algebras 17B30 Solvable, nilpotent (super)algebras 20L05 Groupoids (i.e. small categories in which all morphisms are isomorphisms) 22E60 Lie algebras of Lie groups 18G55 Nonabelian homotopical algebra (MSC2010) 13D10 Deformations and infinitesimal methods in commutative ring theory 55U10 Simplicial sets and complexes in algebraic topology 55P62 Rational homotopy theory
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