TY - JOUR

T1 - MC elements in pronilpotent DG Lie algebras

AU - Yekutieli, Amnon

N1 - Funding Information:
I wish to thank James Stasheff, Vladimir Hinich, Michel Van den Bergh, William Goldman, Oren Ben Bassat, Marco Manetti and Ronald Brown for useful conversations. Thanks also to the referee for reading the paper carefully and providing several constructive remarks. This research was supported by the Israel Science Foundation.

PY - 2012/11/1

Y1 - 2012/11/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84861806938&partnerID=8YFLogxK

U2 - 10.1016/j.jpaa.2012.03.002

DO - 10.1016/j.jpaa.2012.03.002

M3 - Article

AN - SCOPUS:84861806938

SN - 0022-4049

VL - 216

SP - 2338

EP - 2360

JO - Journal of Pure and Applied Algebra

JF - Journal of Pure and Applied Algebra

IS - 11

ER -