Homogeneous conformal averaging operators on 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.

9 pages

References in corpus (3)

• Averaging algebras, Schröder numbers, rooted trees and operads
• Rota-Baxter operators on $sl(2,C)$ and solutions of the classical Yang-Baxter equation
• Operads of decorated trees and their duals

Cited by in corpus (8)

• Simple finite-dimensional double algebras
• Rota-Baxter Operators on Quadratic Algebras
• Conformal classical Yang-Baxter equation, $S$-equation and $\mathcal{O}$-operators
• Rota-Baxter operators on a sum of fields
• Spectrum of Rota-Baxter operators
• Rota-Baxter operators on $\mathrm{Cur}({\mathrm{sl}}_2(\mathbb{C}))$
• Rota-Baxter operators of nonzero weight on the matrix algebra of order three
• Double Lie algebras of a nonzero weight