Mathematics – Quantum Algebra
Scientific paper
2011-03-05
Mathematics
Quantum Algebra
23 pages, 2 figures. New version: change of title, other minor changes. Version 3: 31 pages, added material on the Deligne 2-g
Scientific paper
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.
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