Quasi-Twilled Lie Pseudolgebras and Their Deformation Maps

Abstract

In this paper, we present a unified framework for studying cohomology theories of various operators in the context of pseudoalgebras. The central tool in our approach is the notion of a quasi-twilled Lie pseudoalgebra. We introduce two types of deformation maps. Type I unifies modified $r$ matrices, crossed homomorphisms, derivations, and homomorphisms; and Type II provides a uniform treatment of relative Rota-Baxter operators, twisted Rota-Baxter operators, Reynolds operators, and deformation maps of matched pairs of Lie conformal algebras. We construct the corresponding controlling algebras and define cohomology theories for both types of deformation maps. These results recover existing cohomological results for known operators and yield new results, including the cohomology theory for modified $r$-matrices and deformation maps of matched pairs of Lie pseudoalgebras.