Some examples of DG-Lie formality transfer

TL;DR

The paper establishes a two-direction formality transfer theorem for morphisms of DG-Lie algebras: if $M$ is formal and the induced map on second Chevalley–Eilenberg cohomology is injective, then $L$ is formal, and dually if $L$ is formal with an injective pullback on CE cohomology, then $M$ is formal. The proof hinges on the Chevalley–Eilenberg double complex, its spectral sequence, and the Euler derivation $e_f$, together with degeneration criteria. It then derives concrete applications: equivalence of formality with the formality of universal enveloping algebras, persistence of formality under invariant subalgebras and under free finite-group actions on KS-formal spaces, and a non-example clarifying the limitations of transfer. Collectively, these results provide a versatile, cohomology-driven criterion to deduce formality in deformation-theoretic and geometric contexts, with practical impact on algebraic and complex-analytic problems.

Abstract

We give a convenient reformulation, a slight generalization and some applications of the formality transfer theorem for DG-Lie algebras.

Theorems & Definitions (19)

• Theorem 1.1
• Theorem 1.2: =Theorem \ref{['thm.mainlater']}
• Corollary 1.3
• Corollary 1.4
• Proposition 2.1
• proof
• Definition 2.2
• Lemma 2.3: Ma15
• Lemma 2.4
• proof