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

Quentin Ehret, Abdenacer Makhlouf

TL;DR

This work develops a cohomology- and deformation-theory framework for restricted Lie algebras in positive characteristic, distinguishing the $p\ge3$ and $p=2$ cases. For $p\ge3$, deformations are controlled by Evans–Fuchs restricted cohomology, while deformations of restricted morphisms require a novel cohomology theory; obstructions and equivalences are analyzed through a 2-cocycle perspective. In characteristic $2$, the authors construct a complete cochain complex $H_{*_2}^n(L,M)$ tailored to the 2-map, enabling full computation of cohomology groups and a parallel deformation theory, including morphisms. They also provide explicit computations for restricted Heisenberg algebras, illustrating the classification of restricted structures and the resulting cohomology and deformation phenomena. The results deepen understanding of restricted Lie algebras in positive characteristic, with applications to restricted morphisms and Lie–Rinehart-type structures in low characteristic, and establish groundwork for further explorations of deformations in characteristic two.

Abstract

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 $p\geq3$, 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 $p\geq 2$.

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

TL;DR

This work develops a cohomology- and deformation-theory framework for restricted Lie algebras in positive characteristic, distinguishing the and cases. For , deformations are controlled by Evans–Fuchs restricted cohomology, while deformations of restricted morphisms require a novel cohomology theory; obstructions and equivalences are analyzed through a 2-cocycle perspective. In characteristic , the authors construct a complete cochain complex tailored to the 2-map, enabling full computation of cohomology groups and a parallel deformation theory, including morphisms. They also provide explicit computations for restricted Heisenberg algebras, illustrating the classification of restricted structures and the resulting cohomology and deformation phenomena. The results deepen understanding of restricted Lie algebras in positive characteristic, with applications to restricted morphisms and Lie–Rinehart-type structures in low characteristic, and establish groundwork for further explorations of deformations in characteristic two.

Abstract

The main purpose of this paper is to study cohomology and develop a deformation theory of restricted Lie algebras in positive characteristic . In the case , 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 , 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 . As example, we discuss restricted cohomology with adjoint coefficients of restricted Heisenberg Lie algebras in characteristic .

Paper Structure

This paper contains 33 sections, 46 theorems, 214 equations.

Table of Contents

  1. Introduction
  2. Restricted Lie Algebras
  3. Basics
  4. Cohomology of Restricted Lie Algebras, $p\geq 3$
  5. Ordinary Chevalley-Eilenberg Cohomology
  6. Restricted cohomology of restricted Lie algebras, $p\geq3$
  7. Deformation theory of restricted Lie algebras, $p\geq3$
  8. Restricted formal deformations
  9. Equivalence of restricted formal deformations
  10. Obstructions
  11. Deformation of restricted morphisms
  12. Deformation cohomology of restricted morphisms
  13. Deformations of restricted morphisms
  14. Restricted Nijenhuis operators
  15. Restricted Lie algebras in characteristic $2$
  16. ...and 18 more sections

Key Result

Theorem 2.4

Let $L$ be a $n$-dimensional Lie algebra over a field $\mathop{\mathrm{\mathbb{F}}}\nolimits$ of characteristic $p$. Suppose that $(e_j)_{j\in \{1,\cdots, n\}}$ is a basis of $L$ such that it exists $y_j\in L,~\bigl(\mathop{\mathrm{ad}}\nolimits_{e_j} \bigl)^p=\mathop{\mathrm{ad}}\nolimits_{y_j}$. T

Theorems & Definitions (106)

  • Definition 2.1: Restricted Lie Algebra
  • Remark
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4: Jacobson's Theorem (JN62)
  • Proposition 2.5
  • proof
  • Definition 2.6
  • Definition 2.7
  • Lemma 2.8: EF08
  • ...and 96 more