Cohomology and Crossed Modules of Modified Rota-Baxter Pre-Lie Algebras

The goal of the present paper is to provide a cohomology theory and crossed modules of modified Rota-Baxter pre-Lie algebras. We introduce the notion of a modified Rota-Baxter pre-Lie algebra and its bimodule. We define a cohomology of modified Rota-Baxter pre-Lie algebras with coefficients in a suitable bimodule. Furthermore, we study the infinitesimal deformations and abelian extensions of modified Rota-Baxter pre-Lie algebras and relate them with the second cohomology groups. Finally, we investigate skeletal and strict modified Rota-Baxter pre-Lie 2-algebras. We show that skeletal modified Rota-Baxter pre-Lie 2-algebras can be classified into the third cohomology group, and strict modified Rota-Baxter pre-Lie 2-algebras are equivalent to the crossed modules of modified Rota-Baxter pre-Lie algebras.