Deformation maps of quasi-twilled Lie algebras

TL;DR

This paper introduces a unified framework using quasi-twilled Lie algebras to study cohomology and deformation theories of various operators in Lie algebras, leading to new insights and results.

Contribution

It develops two types of deformation maps that unify multiple operator theories and provides the associated controlling algebras and cohomologies, including new results for modified r-matrices.

Findings

Unified deformation maps for multiple Lie algebra operators.

Identification of controlling algebras and cohomologies for these operators.

New results on the cohomology of modified r-matrices and matched pairs.

Abstract

In this paper, we provide a unified approach to study the cohomology theories and deformation theories of various types of operators in the category of Lie algebras, including modified $r$-matrices, crossed homomorphisms, derivations, homomorphisms, relative Rota-Baxter operators, twisted Rota-Baxter operators, Reynolds operators and deformation maps of matched pairs of Lie algebras. The main ingredients are quasi-twilled Lie algebras. We introduce two types of deformation maps of a quasi-twilled Lie algebra. Deformation maps of type I unify modified $r$-matrices, crossed homomorphisms, derivations and homomorphisms between Lie algebras, while deformation maps of type II unify relative Rota-Baxter operators, twisted Rota-Baxter operators, Reynolds operators and deformation maps of matched pairs of Lie algebras. We further give the controlling algebras and cohomologies of these two types of deformation maps, which not only recover the existing results for crossed homomorphisms, derivations, homomorphisms, relative Rota-Baxter operators, twisted Rota-Baxter operators and Reynolds operators, but also leads to some new results which are unable to solve before, e.g. the controlling algebras and cohomologies of modified $r$-matrices and deformation maps of matched pairs of Lie algebras.