The Superization of Hochschild's Lemma and Restricted Lie-Rinehart Superalgebras
Sofiane Bouarroudj, Quentin Ehret, Abdenacer Makhlouf, Nurtas Shyntas
TL;DR
The paper introduces restricted Lie-Rinehart superalgebras over fields of characteristic $p>2$, motivated by a superized version of Hochschild's lemma. It extends Hochschild’s and Schauenburg’s results to the super setting, establishing a Hochschild-type compatibility between the $p$-map and the anchor action, and defines the restricted structure for Lie-Rinehart superalgebras. Representations and a semi-direct product construction are developed, leading to the formulation of restricted and universal enveloping algebras with the appropriate universal properties. The work combines Hopf superalgebra techniques with parity-sensitive computations to provide a coherent theory and concrete examples. The resulting framework lays groundwork for further study of cohomology, representations, and enveloping algebras in the modular, superalgebra context.
Abstract
The main goal of this paper is to introduce the notion of restricted Lie-Rinehart superalgebra over a field of characteristic $p>2$, motivated by a generalization of Hochschild's lemma to the super setting. We extend Schauenburg's proof of Hochschild's lemma to Lie-Rinehart superalgebras and we prove a superized version that serves as a foundation for our construction. Building upon this, we define restricted Lie-Rinehart superalgebras, investigate their representations, construct the semi-direct product with a restricted module, and provide several examples. Finally, we construct the corresponding universal enveloping algebra and show that this algebra satisfies the expected universal property.