Cohomologies of Reynolds Lie algebras with derivations and its applications
Basdouri Imed, Sadraoui Mohamed Amin
TL;DR
This work develops a cohesive cohomology theory for Reynolds Lie algebras with derivations (Reynolds LieDer pairs) by integrating Chevalley–Eilenberg cohomology with Reynolds-operator data, enabling systematic treatment of deformations and extensions. The authors construct a total cochain complex for RLieDer pairs, derive foundational deformation theory where infinitesimals are 2-cocycles and rigidity is governed by $\mathcal{H}^2_{RLieDer}(L,L)$, and establish abelian extension theory with 2-cocycles $(\Theta,\xi,\chi)$ classifying such extensions via $\mathcal{H}^2_{RLieDer}(L;V)$. They further develop obstruction theory for extending derivations, showing obstructions live in $\mathcal{H}^2_R(L;V)$ and vanishing of this group ensures extensibility. The results provide a robust framework for cohomology, deformation, and extension problems in Reynolds-operator–based Lie structures with derivations, with potential implications for related algebraic and physical models.
Abstract
The aim of this paper is to study the cohomology theory of Reynolds Lie algebras equipped with derivations and to explore related applications. We begin by introducing the concept of Reynolds LieDer pairs. Subsequently, we construct the associated cohomology. Finally, we investigate formal deformations, abelian extensions, and extensions of a pair of derivations, all interpreted through the lens of cohomology groups.