MC elements in pronilpotent DG Lie algebras

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• 作者：Amnon Yekutieli ^{amyekut@math.bgu.ac.il}
• 关键词：Primary ; 53D55 ; Secondary ; 17B40 ; 14B10 ; 18D05
• 刊名：Journal of Pure and Applied Algebra
• 出版年：2012
• 期刊代码：35_00224049
• 类别：mt
• 出版时间：November, 2012
• 卷：216
• 期：11
• 页码：2338-2360
• 文件大小：488 K

摘要

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.