Differential Graded Lie Research Articles

Formality conjecture for minimal surfaces of Kodaira dimension 0

Let $\mathcal {F}$ be a polystable sheaf on a smooth minimal projective surface of Kodaira dimension 0. Then the differential graded (DG) Lie algebra $R\operatorname {Hom}(\mathcal {F},\mathcal {F})$ of derived endomorphisms of $\mathcal {F}$ is formal. The proof is based on the study of equivariant $L_{\infty }$ minimal models of DG Lie algebras equipped with a cyclic structure of degree 2 which is non-degenerate in cohomology, and does not rely (even for K3 surfaces) on previous results on the same subject.

Differential Graded Lie Algebras and Leibniz Algebra Cohomology

Abstract In this note, we interpret Leibniz algebras as differential graded (DG) Lie algebras. Namely, we consider two fully faithful functors from the category of Leibniz algebras to that of DG Lie algebras and show that they naturally give rise to the Leibniz cohomology and the Chevalley–Eilenberg cohomology. As an application, we prove a conjecture stated by Pirashvili in [ 9].

MC elements in pronilpotent DG Lie algebras

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.