Rota-Baxter operators on compact simple Lie groups and algebras

A Rota-Baxter operator on a Lie group $G$ is a smooth map $B:G\to G$ such that $B(g)B(h)=B(gB(g)hB(g{)}^{-1})$ for all $g,h\in G$. This concept was introduced in 2021 by Guo, Lang and Sheng as a Lie group analogue of Rota-Baxter operators of weight 1 on Lie algebras. We show that the only Rota-Baxter operators on compact simple Lie groups are the trivial map and the inverse map. A similar description for Rota-Baxter operators of weight 1 on compact simple Lie algebras is provided.