Controlling $L_{\infty }$-algebra and deformation complex of deformation maps on quasi-twilled pre-Lie algebras

Abstract

In this paper, we introduce quasi-twilled pre-Lie algebras to provide a unified framework for investigating the cohomology theories and deformation theories of various types of operators in the category of pre-Lie algebras. We introduce two types of deformation maps of quasi-twilled pre-Lie algebras, which unify various operators on pre-Lie algebras. Right deformation maps unify modified Rota–Baxter operators of weight $\lambda$, derivations, homomorphisms and crossed homomorphisms. Left deformation maps unify relative Rota–Baxter operators of weight $0$, twisted Rota–Baxter operators, Reynolds operators and deformation maps of matched pairs of pre-Lie algebras. Moreover, we give the controlling algebras and the cohomologies of these two types of deformation maps, which not only recover some previous results for derivations, homomorphisms, crossed homomorphisms, relative Rota–Baxter operators of weight $0$, twisted Rota–Baxter operators and Reynolds operators but also lead to some new results, such as the controlling algebra and the cohomology of a modified Rota–Baxter operator of weight $\lambda$ on a pre-Lie algebra, the controlling algebra and the cohomology of a deformation map of a matched pair of pre-Lie algebras.