Homogeneous conformal averaging operators on semisimple Lie algebras

In this note we show a close relation between the following objects: Classical Yang – Baxter equation (CYBE), conformal algebras (also known as vertex Lie algebras), and averaging operators on Lie algebras. It turns out that the singular part of a solution of CYBE (in the operator form) on a Lie algebra $\mathfrak{g}$ determines an averaging operator on the corresponding current conformal algebra $\mathrm{C}\mathrm{u}\mathrm{r}\mathfrak{g}$. For a finite-dimensional semisimple Lie algebra $\mathfrak{g}$, we describe all homogeneous averaging operators on $\mathrm{C}\mathrm{u}\mathrm{r}\mathfrak{g}$. It turns out that all these operators actually define solutions of CYBE with a pole at the origin.