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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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References:
[1] N. Ashley, ”Simplicial \(T\)-complexes and crossed complexes: a nonabelian version of a theorem of Dold and Kan,” Dissertationes Math. \((\)Rozprawy Mat.\()\), vol. 265, p. 61, 1988. · Zbl 1003.55500
[2] T. Beke, ”Higher \vCech theory,” \(K\)-Theory, vol. 32, iss. 4, pp. 293-322, 2004. · Zbl 1070.18008 · doi:10.1007/s10977-004-0840-0
[3] A. K. Bousfield and V. K. A. M. Gugenheim, On \({ PL}\) de Rham Theory and Rational Homotopy Type, , 1976. · Zbl 0338.55008
[4] M. K. Dakin, Kan complexes and multiple groupoid structures, Univ. Amiens, 1983. · Zbl 0566.55010
[5] P. Deligne, Letter to L. Breen, Feb. 28, 1994. · math.northwestern.edu
[6] A. Dold, ”Homology of symmetric products and other functors of complexes,” Ann. of Math., vol. 68, pp. 54-80, 1958. · Zbl 0082.37701 · doi:10.2307/1970043
[7] J. L. Dupont, ”Simplicial de Rham cohomology and characteristic classes of flat bundles,” Topology, vol. 15, iss. 3, pp. 233-245, 1976. · Zbl 0331.55012 · doi:10.1016/0040-9383(76)90038-0
[8] J. L. Dupont, Curvature and Characteristic Classes, New York: Springer-Verlag, 1978. · Zbl 0373.57009
[9] J. Duskin, ”Higher-dimensional torsors and the cohomology of topoi: the abelian theory,” in Applications of Sheaves, Fourman, M. P., Mulvey, C. J., and Scott, D. S., Eds., New York: Springer-Verlag, 1979, pp. 255-279. · Zbl 0444.18014
[10] J. W. Duskin, ”Simplicial matrices and the nerves of weak \(n\)-categories. I: Nerves of bicategories,” Theory Appl. Categ., vol. 9, pp. 198-308, 2001. · Zbl 1046.18009 · emis:journals/TAC/#vol9 · eudml:123775
[11] S. Eilenberg and S. Mac Lane, ”On the groups of \(H(\Pi,n)\). I,” Ann. of Math., vol. 58, pp. 55-106, 1953. · Zbl 0050.39304 · doi:10.2307/1969820
[12] Z. Fiedorowicz and J. Loday, ”Crossed simplicial groups and their associated homology,” Transactions Amer. Math. Soc., vol. 326, iss. 1, pp. 57-87, 1991. · Zbl 0755.18005 · doi:10.2307/2001855
[13] E. Getzler, ”A Darboux theorem for Hamiltonian operators in the formal calculus of variations,” Duke Math. J., vol. 111, iss. 3, pp. 535-560, 2002. · Zbl 1100.32008 · doi:10.1215/S0012-7094-02-11136-3
[14] P. G. Glenn, ”Realization of cohomology classes in arbitrary exact categories,” J. Pure Appl. Algebra, vol. 25, iss. 1, pp. 33-105, 1982. · Zbl 0487.18015 · doi:10.1016/0022-4049(82)90094-9
[15] W. M. Goldman and J. J. Millson, ”The deformation theory of representations of fundamental groups of compact Kähler manifolds,” Inst. Hautes Études Sci. Publ. Math., vol. 67, pp. 43-96, 1988. · Zbl 0678.53059 · doi:10.1007/BF02699127 · numdam:PMIHES_1988__67__43_0 · eudml:104031
[16] V. Hinich, ”Descent of Deligne groupoids,” Internat. Math. Res. Notices, vol. 5, pp. 223-239, 1997. · Zbl 0948.22016 · doi:10.1155/S1073792897000160 · arxiv:alg-geom/9606010
[17] D. M. Kan, ”Functors involving c.s.s. complexes,” Trans. Amer. Math. Soc., vol. 87, pp. 330-346, 1958. · Zbl 0090.39001 · doi:10.2307/1993103
[18] T. Lada and M. Markl, ”Strongly homotopy Lie algebras,” Comm. Algebra, vol. 23, iss. 6, pp. 2147-2161, 1995. · Zbl 0999.17019 · doi:10.1080/00927879508825335 · arxiv:hep-th/9406095
[19] L. Lambe and J. Stasheff, ”Applications of perturbation theory to iterated fibrations,” Manuscripta Math., vol. 58, iss. 3, pp. 363-376, 1987. · Zbl 0632.55011 · doi:10.1007/BF01165893 · eudml:155226
[20] P. J. May, Simplicial Objects in Algebraic Topology, Chicago, IL: University of Chicago Press, 1992. · Zbl 0769.55001
[21] A. Nijenhuis and R. W. Richardson Jr., ”Cohomology and deformations in graded Lie algebras,” Bull. Amer. Math. Soc., vol. 72, pp. 1-29, 1966. · Zbl 0136.30502 · doi:10.1090/S0002-9904-1966-11401-5
[22] S. Paoli, ”Semistrict Tamsamani \(n\)-groupoids and connected \(n\)-types,” , preprint , 2007.
[23] D. Quillen, ”Rational homotopy theory,” Ann. of Math., vol. 90, pp. 205-295, 1969. · Zbl 0191.53702 · doi:10.2307/1970725
[24] D. Sullivan, ”Infinitesimal computations in topology,” Inst. Hautes Études Sci. Publ. Math., vol. 47, pp. 269-331, 1977. · Zbl 0374.57002 · doi:10.1007/BF02684341 · numdam:PMIHES_1977__47__269_0 · eudml:103948
[25] H. Whitney, Geometric Integration Theory, Princeton, NJ: Princeton Univ. Press, 1957. · Zbl 0083.28204
[26] M. Kuranishi, ”On the locally complete families of complex analytic structures,” Ann. of Math. (2), vol. 75, pp. 536-577, 1962. · Zbl 0106.15303 · doi:10.2307/1970211
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