Homogeneous conformal averaging operators on semisimple Lie algebras

TL;DR

This paper explores the connection between the classical Yang-Baxter equation, conformal algebras, and averaging operators on Lie algebras, specifically classifying homogeneous averaging operators on current conformal algebras of semisimple Lie algebras.

Contribution

It provides a complete description of all homogeneous averaging operators on current conformal algebras of semisimple Lie algebras, linking them to CYBE solutions.

Findings

All homogeneous averaging operators on  g correspond to CYBE solutions with a pole at zero.

The singular part of CYBE solutions determines averaging operators on conformal algebras.

Classification of these operators is achieved for semisimple Lie algebras.

Abstract

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 $\mathrm{Cur} \mathfrak g$. For a finite-dimensional semisimple Lie algebra $\mathfrak g$, we describe all homogeneous averaging operators on $\mathrm{Cur}\mathfrak g$. It turns out that all these operators actually define solutions of CYBE with a pole at the origin.