Gauge equivalence for complete L∞-algebras

Guan, Ai (2021) Gauge equivalence for complete L∞-algebras. Homology, Homotopy and Applications, 23 (2). pp. 283-297. ISSN 1532-0073

Text (Gauge equivalence for complete L-infinity algebras)
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Abstract

We introduce a notion of left homotopy for Maurer–Cartan elements in L∞‑algebras and A∞‑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∞‑algebra.