On the Goldman-Millson theorem for A∞-algebras in arbitrary characteristic

Abstract

Complete filtered A∞-algebras model certain deformation problems in the noncommutative setting. The formal deformation theory of a group representation is a classical example. With such applications in mind, we provide the A∞ analogs of several key theorems from the Maurer-Cartan theory for L∞-algebras. In contrast with the L∞ case, our results hold over a field of arbitrary characteristic. We first leverage some abstract homotopical algebra to give a concise proof of the A∞-Goldman-Millson theorem: The nerve functor, which assigns a simplicial set N•(A) to an A∞-algebra A, sends filtered quasi-isomorphisms to homotopy equivalences. We then characterize the homotopy groups of N•(A) in terms of the cohomology algebra H(A), and its group of quasi-invertible elements. Finally, we return to the characteristic zero case and show that the nerve of A is homotopy equivalent to the simplicial Maurer-Cartan set of its commutator L∞-algebra. This answers a question posed by N. de Kleijn and F. Wierstra in [26].