Cohomology of Restricted Poisson algebras in characteristic 2
Sofiane Bouarroudj, Quentin Ehret, Jiefeng Liu
TL;DR
The paper addresses the problem of formulating a coherent cohomology and deformation theory for restricted Poisson algebras in characteristic $2$, and ties it to the cohomology of restricted Lie–Rinehart algebras. It develops a restricted Poisson cohomology and shows how, via Kähler differentials, any restricted Poisson algebra induces a restricted Lie–Rinehart structure; when the module of Kähler differentials is free, the two cohomologies coincide. It also introduces a robust cohomology for restricted Lie–Rinehart algebras, proves an abelian-extension classification by $\,H^2_{ m LR}(L;M)$ in suitable cases, and develops a deformation theory in which infinitesimal deformations are governed by $\,H^2_{ m PA}(A)$ with obstructions living in higher cohomology. The work provides explicit examples, including deformation-quantization contexts and several 3-dimensional restricted Poisson algebras, illustrating rigidity and flexibility phenomena. Together, these results extend modular (characteristic $2$) Poisson and Lie–Rinehart theory, offering tools for studying deformations and extensions in the modular setting and advancing understanding of Poisson geometry in positive characteristic.
Abstract
In this paper, we study restricted Poisson algebras in characteristic 2 and their relationship with restricted Lie-Rinehart algebras, for which we develop a cohomology theory and investigate abelian extensions. We also construct a full cohomology complex for restricted Poisson algebras in characteristic 2 that captures formal deformations and prove that it is isomorphic to the cohomology complex of a suitable restricted Lie-Rinehart algebra, under certain assumptions. A number of examples are provided in order to illustrate our constructions.