Lie theory for nilpotent L∞-algebras

By E. Getzler. 2004

397 citations recorded by the source corpus.

The Deligne groupoid is a functor from nilpotent differential graded Lie algebras concentrated in positive degrees to groupoids; in the special case of Lie algebras over a field of characteristic zero, it gives the associated simply connected Lie group. We generalize the Deligne groupoid to a functor γ from L ∞ -algebras concentrated in degree > -n to n-groupoids. (We actually construct the nerve of the n-groupoid, which is an enriched Kan complex.) The construction of gamma is quite explicit (it is based on Dupont's proof of the de Rham theorem) and yields higher dimensional analogues of holonomy and of the Campbell-Hausdorff formula. In the case of abelian L ∞ algebras (i.e., chain complexes), the functor γ is the Dold-Kan simplicial set.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/140476ec1a0a0878f5a10c9bd06e713f123a32f9

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Homological mirror symmetry for curves of higher genus (2009, neutral): “rove the statement of Lemma 2.6. It is in fact standard, but we could not ﬁnd a reference. Fix some basic ﬁeld k of characteristic zero. 36 ALEXANDER I. EFIMOV Infact one can deﬁne, following Getzler [Ge] asimplicial set MC•(g) for any pro-nilpotent L∞-algebra, such that π0(|MC•(g)|) is the set of e”
Deformation quantization of gerbes (2005, methodology): “gether with the composition, provide the deﬁnition of the Deligne two-groupoid of L. Remark 3.1.1. Recently Getzlergave adeﬁnition ofaDelignen-groupoid of a DGLA concentrated in degrees above −n, cf. [G]. Now let L be a sheaf of DGLAs on M. An L-stack on M is the following data: 1) A Maurer-Cartan e”
On weak Lie 2‐algebras (2007, background): “ticular, the 2-group of automorphisms of γ is obtained by integrating the corresponding Lie algebra crossed module under the derived bracket (i.e. a strict Lie 2-algebra in our sense). It was Getzler [2] who generalized this construction to L∞-algebras and higher values of n. The result is a weak n-”
Homotopy theory of homotopy algebras (2014, methodology): “During the preparation of this paper, Dolgushev–Hoffnung–Rogers [5] used the integration theory of L∞-algebras [12, 18] to endow the category of P∞-algebras with another model of ∞-category, that we present now using the langage of the present paper.”
Deformation theory and rational homotopy type (2012, methodology): “Getzler obtains a groupoid by1137 replacing MC• by a much smaller sub-simplicial set (scheme) γ•(L) given by the vanishing1138 of the “Dupont gauge ” s : MC(L)• → MC•−1(L).”