Pre-Lie algebras with divided powers and the Deligne groupoid in positive characteristic

Verstraete, Marvin

doi:10.2140/agt.2025.25.5567

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Pre-Lie algebras with divided powers and the Deligne groupoid in positive characteristic

Marvin Verstraete

Algebraic & Geometric Topology 25 (2025) 5567–5606

DOI: 10.2140/agt.2025.25.5567

Abstract

We develop a deformation theory controlled by pre-Lie algebras with divided powers over a ring of positive characteristic. We show that every differential graded pre-Lie algebra with divided powers comes with operations, called weighted braces, which we use to generalize the classical deformation theory controlled by Lie algebras over a field of characteristic $0$ . Explicitly, we define the Maurer–Cartan set, as well as the gauge group, and prove that there is an action of the gauge group on the Maurer–Cartan set. This new deformation theory moreover admits a Goldman–Millson theorem which remains valid over the integers. As an application, we give the computation of the $π_{0}$ of a mapping space $Map (B^{c} (𝒞), 𝒫)$ with $𝒞$ and $𝒫$ suitable cooperad and operad, respectively.

Keywords

pre-Lie algebras, deformation theory, operads theory

Mathematical Subject Classification

Primary: 17B55

Secondary: 17A30, 18M70, 20L05

References

Publication

Received: 4 December 2023

Revised: 28 October 2024

Accepted: 30 November 2024

Published: 18 December 2025

Authors

Marvin Verstraete
CNRS, UMR 8524 - Laboratoire Paul Painlevé Université de Lille Villeneuve-d’Ascq France

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Volume 25, issue 9 (2025)