Title:
MC
Elements in Pronilpotent DG
Lie Algebras
Authors:
Amnon Yekutieli
Publication status:
J.
Pure Appl. Algebra
216
(2012),
2338–2360
Abstract:
Consider a pronilpotent DG (differential graded) Lie algebra
over a field of characteristic 0. In the first part of the paper we
introduce the reduced Deligne groupoid associated to this DG Lie
algebra. We prove that a DG Lie quasi-isomorphism between two such
algebras induces an equivalence between the corresponding reduced
Deligne groupoids. This extends the famous result of Goldman- Millson
(attributed to Deligne) to the unbounded pronilpotent case.
In
the second part of the paper we consider the Deligne 2-groupoid. We
show it exists under more relaxed assumptions than known before (the
DG Lie algebra is either nilpotent or of quasi quantum type). We
prove that a DG Lie quasi-isomorphism between such DG Lie algebras
induces a weak equivalence between the corresponding Deligne
2-groupoids.
In the third part of the paper we prove that an
L-infinity quasi-isomorphism between pronilpotent DG Lie algebras
induces a bijection between the sets of gauge equivalence classes of
Maurer-Cartan elements. This extends a result of Kontsevich and
others to the pronilpotent case.
Electronic
Preprint:
paper
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updated 15 Sep 2012