Conformal Yang-Baxter equation on $\mathrm{Cur}(\mathrm{sl}_2(\mathbb{C}))$

TL;DR

This paper classifies solutions to the conformal classical Yang-Baxter equation on the current Lie conformal algebra of sl_2(C), advancing the understanding of conformal Lie bialgebra structures.

Contribution

It provides a complete description of all solutions to the conformal CYBE on (sl_2(C)), including the weak version, which was previously unexplored.

Findings

All solutions to the conformal CYBE on (sl_2(C)) are characterized.

Solutions to the weak version of the conformal CYBE are also described.

The results contribute to the classification of conformal Lie bialgebra structures.

Abstract

In 2008, J. Liberati defined what is a conformal Lie bialgebra and introduced the conformal classical Yang-Baxter equation (CCYBE). An $L$-invariant solution to the weak version of CCYBE provides a conformal Lie bialgebra structure. We describe all solutions to the conformal classical Yang-Baxter equation on the current Lie conformal algebra $\mathrm{Cur}(\mathrm{sl}_2(\mathbb{C}))$ and to the weak version of it.