On restricted Rota-Baxter Lie algebras of arbitrary weight

Recently, Ehret and Gilliers introduced the notion of a (trivially) restricted post-Lie algebra, recovering the concepts of a restricted Lie algebra and a restricted pre-Lie algebra. In this paper, we specifically introduce restricted Rota-Baxter Lie algebras of arbitrary weight with an intrinsic graph subalgebra characterization. We show that, via the splitting property, they give rise to restricted post-Lie algebras, and furthermore possess a novel replication property. We then present two natural constructions of such restricted Rota-Baxter structures in prime characteristic: one arising from Rota-Baxter associative algebras of arbitrary weight, and the other from Rota-Baxter Lie algebras of weight $1$. The Rota-Baxter $p$-envelopes of a Rota-Baxter Lie algebra are also examined.