Commutative post-Lie algebra structures on Kac--Moody algebras

TL;DR

This paper classifies commutative post-Lie algebra structures on infinite-dimensional Lie algebras, showing they are trivial or almost trivial on loop and affine Kac--Moody algebras, extending finite-dimensional results.

Contribution

It provides a comprehensive classification of commutative post-Lie algebra structures on certain infinite-dimensional Lie algebras, extending known finite-dimensional results.

Findings

All such structures on loop algebras are trivial.

All such structures on affine Kac--Moody algebras are almost trivial.

Extension of finite-dimensional results to infinite-dimensional cases.

Abstract

We determine commutative post-Lie algebra structures on some infinite-dimensional Lie algebras. We show that all commutative post-Lie algebra structures on loop algebras are trivial. This extends the results for finite-dimensional perfect Lie algebras. Furthermore we show that all commutative post-Lie algebra structures on affine Kac--Moody Lie algebras are "almost trivial".