Rota--Baxter operators and post-Lie algebra structures on semisimple Lie algebras

TL;DR

This paper explores the relationship between Rota--Baxter operators and post-Lie algebra structures on semisimple Lie algebras, establishing conditions for their existence and classifying structures in specific cases.

Contribution

It provides a classification of post-Lie algebra structures on pairs of semisimple Lie algebras, especially when one is a direct sum of two  algebras, using Rota--Baxter operators.

Findings

Existence of post-Lie structures implies isomorphism of simple Lie algebras.

Classification difficulty increases when  is involved in semisimple pairs.

Complete classification achieved for  case with  as the second algebra.

Abstract

Rota--Baxter operators $R$ of weight $1$ on $\mathfrak{n}$ are in bijective correspondence to post-Lie algebra structures on pairs $(\mathfrak{g},\mathfrak{n})$, where $\mathfrak{n}$ is complete. We use such Rota--Baxter operators to study the existence and classification of post-Lie algebra structures on pairs of Lie algebras $(\mathfrak{g},\mathfrak{n})$, where $\mathfrak{n}$ is semisimple. We show that for semisimple $\mathfrak{g}$ and $\mathfrak{n}$, with $\mathfrak{g}$ or $\mathfrak{n}$ simple, the existence of a post-Lie algebra structure on such a pair $(\mathfrak{g},\mathfrak{n})$ implies that $\mathfrak{g}$ and $\mathfrak{n}$ are isomorphic, and hence both simple. If $\mathfrak{n}$ is semisimple, but $\mathfrak{g}$ is not, it becomes much harder to classify post-Lie algebra structures on $(\mathfrak{g},\mathfrak{n})$, or even to determine the Lie algebras $\mathfrak{g}$ which can arise. Here only the case $\mathfrak{n}=\mathfrak{sl}_2(\Bbb{C})$ was studied. In this paper we determine all Lie algebras $\mathfrak{g}$ such that there exists a post-Lie algebra structure on $(\mathfrak{g},\mathfrak{n})$ with $\mathfrak{n}=\mathfrak{sl}_2(\Bbb{C})\oplus \mathfrak{sl}_2(\Bbb{C})$.