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[FORTHCOMING] Cohomology and deformations of restricted Lie algebras and their morphisms in positive characteristic

[FORTHCOMING] Cohomologie et déformations des algèbres de Lie restreintes et leurs morphismes en caractéristique positive


Quentin Ehret
Université de Haute-Alsace
France

Abdenacer Makhlouf
Université de Haute-Alsace
France



Validated on 28 March 2025   DOI : TBA

Abstract

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The main purpose of this paper is to study cohomology and develop a deformation theory of restricted Lie algebras in positive characteristic p > 0. In the case p3, it is shown that the deformations of restricted Lie algebras are controlled by the restricted cohomology introduced by Evans and Fuchs. Moreover, we introduce a new cohomology that controls the deformations of restricted morphisms of restricted Lie algebras. In the case p=2, we provide a full restricted cohomology complex with values in a restricted module and investigate its connections with formal deformations. Furthermore, we introduce a full deformation cohomology that controls deformations of restricted morphisms of restricted Lie algebras in characteristic 2. As example, we discuss restricted cohomology with adjoint coefficients of restricted Heisenberg Lie algebras in characteristic p2.

The main purpose of this paper is to study cohomology and develop a deformation theory of restricted Lie algebras in positive characteristic p > 0. In the case p3, it is shown that the deformations of restricted Lie algebras are controlled by the restricted cohomology introduced by Evans and Fuchs. Moreover, we introduce a new cohomology that controls the deformations of restricted morphisms of restricted Lie algebras. In the case p=2, we provide a full restricted cohomology complex with values in a restricted module and investigate its connections with formal deformations. Furthermore, we introduce a full deformation cohomology that controls deformations of restricted morphisms of restricted Lie algebras in characteristic 2. As example, we discuss restricted cohomology with adjoint coefficients of restricted Heisenberg Lie algebras in characteristic p2.

Modular Lie algebra restricted cohomology deformation

Modular Lie algebra restricted cohomology deformation

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