Higher Structures of Rota--Baxter Lie $H$-Pseudoalgebras

This paper investigates Rota--Baxter Lie $H$-pseudoalgebras. We develop a cohomology theory for $\lambda$-weighted relative Rota--Baxter operators via a Maurer--Cartan approach, constructing the underlying differential graded Lie algebra. We classify non-abelian extensions using second cohomology and derive the Wells exact sequence to address the inducibility of automorphisms. Furthermore, we explore the homotopy theory of these structures by introducing $2$-term skeletal and strict Rota--Baxter $L_\infty$-$H$-pseudoalgebras. In particular, we establish a one-to-one correspondence between strict $2$-term structures and crossed modules of Rota--Baxter Lie $H$-pseudoalgebras. These results establish a foundational framework for future advancements in the higher categorical theory of pseudoalgebras with algebraic operators. Ultimately, this work provides a robust foundation for the higher categorical study of pseudoalgebras equipped with algebraic operators.