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

2023/10/31 by Verstraete, Marvin
#17B55 18M70 17A30 20L05 #Algebraic Topology (math.AT) #FOS: Mathematics

paper · doi:10.48550/arxiv.2310.20300

Abstract

The purpose of this paper is to 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 on the integers. As an application, we give the computation of the π₀ of a mapping space Map(B^c(C),P) with C and P suitable cooperad and operad respectively.

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