Gauge equivalence for complete $L_{\infty }$-algebras

Abstract

We introduce a notion of left homotopy for Maurer--Cartan elements in $L_{\infty }$-algebras and $A_{\infty }$-algebras, and show that it corresponds to gauge equivalence in the differential graded case. From this we deduce a short formula for gauge equivalence, and provide an entirely homotopical proof to Schlessinger--Stasheff's theorem. As an application, we answer a question of T. Voronov, proving a non-abelian Poincaré lemma for differential forms taking values in an $L_{\infty }$-algebra.

References in corpus (6)

• Lie theory for nilpotent L-infinity algebras
• Deformation theory via differential graded Lie algebras
• Cohomology theories for homotopy algebras and noncommutative geometry
• A tale of three homotopies
• Homotopy theory of complete Lie algebras and Lie models of simplicial sets
• A model structure for the Goldman-Millson theorem

Cited by in corpus (2)

• Review of deformation theory II: a homotopical approach
• Koszul duality for compactly generated derived categories of second kind