On the Goldman-Millson theorem for \(A_\infty\)-algebras in arbitrary characteristic

Complete filtered \(A_\infty\)-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 \(A_\infty\) analogs of several key theorems from the Maurer-Cartan...

Full description

Saved in:
Bibliographic Details
Published in:arXiv.org 2023-06
Main Authors: Milham, Alex, Rogers, Christopher L
Format: Article
Language:English
Subjects:
Algebra
Commutators
Deformation
Equivalence
Fields (mathematics)
Group theory
Homology
Isomorphism
Nerves
Theorems
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:Complete filtered \(A_\infty\)-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 \(A_\infty\) analogs of several key theorems from the Maurer-Cartan theory for \(L_\infty\)-algebras. In contrast with the \(L_\infty\) case, our results hold over a field of arbitrary characteristic. We first leverage some abstract homotopical algebra to give a concise proof of the \(A_\infty\)-Goldman-Millson theorem: The nerve functor, which assigns a simplicial set \(\mathcal{N}_{\bullet}(A)\) to an \(A_\infty\)-algebra \(A\), sends filtered quasi-isomorphisms to homotopy equivalences. We then characterize the homotopy groups of \(\mathcal{N}_\bullet(A)\) 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 \(L_\infty\)-algebra. This answers a question posed by N. de Kleijn and F. Wierstra in arXiv:1809.07743.
ISSN:2331-8422