Lie theory for nilpotent L-algebras

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108 Scopus citations

Abstract

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.

Original languageEnglish (US)
Pages (from-to)271-301
Number of pages31
JournalAnnals of Mathematics
Volume170
Issue number1
DOIs
StatePublished - 2009

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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