Deformations and the controlling $L_{\mathrm{\infty }}$-structure of extended Rota-Baxter Lie algebras

In this paper, we introduce the controlling $L_{\mathrm{\infty }}[1]$-algebra of extended Rota-Baxter Lie algebras. As applications, we obtain the cohomology of extended Rota-Baxter Lie algebras. We also study abelian extensions and formal deformations by the lower cohomology group. Lastly, we get the notion of homotopy extended Rota-Baxter Lie algebras. Specifically, we present the controlling $L_{\mathrm{\infty }}[1]$-algebra of modified Rota-Baxter Lie algebras and of Rota-Baxter Lie algebras with weight.