Rota-Baxter operators on compact simple Lie groups and algebras

TL;DR

This paper classifies Rota-Baxter operators on compact simple Lie groups and algebras, showing they are limited to trivial and inverse maps, thus advancing understanding of these operators in Lie theory.

Contribution

It provides a complete classification of Rota-Baxter operators on compact simple Lie groups and algebras, identifying only trivial and inverse maps as solutions.

Findings

Only trivial and inverse maps are Rota-Baxter operators on compact simple Lie groups.

Similar classification achieved for Rota-Baxter operators on compact simple Lie algebras.

Results clarify the structure of Rota-Baxter operators in the context of compact simple Lie groups and algebras.

Abstract

A Rota-Baxter operator on a Lie group $ G $ is a smooth map $ B : G \to G $ such that $ B(g)B(h) = B(gB(g)hB(g)^{-1}) $ for all $ g, h \in G $. This concept was introduced in 2021 by Guo, Lang and Sheng as a Lie group analogue of Rota-Baxter operators of weight 1 on Lie algebras. We show that the only Rota-Baxter operators on compact simple Lie groups are the trivial map and the inverse map. A similar description for Rota-Baxter operators of weight 1 on compact simple Lie algebras is provided.