On some formality criteria for DG-Lie algebras

Abstract

Abstract. We give some formality criteria for a differential graded Lie algebra to be formal. For instance, we show that a DG-Lie algebra L is formal if and only if the natural spectral sequence computing the Chevalley-Eilenberg cohomology H * CE (L, L) degenerates at E 2 .

Formality criteria in terms of higher Whitehead brackets

“…There is literature relating the collapse of a certain spectral sequence with (co)formality. For instance, a characterization of the formality of a DGL (in fact, of an L ∞ algebra) has been given in terms of the Chevalley-Eilenberg spectral sequence in [21]. Similar claims, extending these results for algebras over operads, have been recently given in [23].…”

On the adjoint map of homotopy abelian DG-Lie algebras

“…is a surjective morphism of DG-vector spaces. of L [6,Section 2]. By standard décalage isomorphisms, CE(L, L) can be described as the product total complex of the complex of DG-vector spaces…”

Noncommutative formality implies commutative and Lie formality

“…However, the author could not find in the literature an exposition that was optimized for the context of this paper. Obstructions to formality in CDGA k is treated in [HS79] and obstructions to formality in DGL k is treated in [Man15]. We start by recalling an easy consequence of the homotopy transfer theorem for P ∞ -algebras, where P is a Koszul operad.…”