# Commutative Post-Lie algebra structures on nilpotent Lie algebras and Poisson algebras

## TL;DR

This paper classifies commutative post-Lie algebra structures on certain nilpotent Lie algebras, showing they are associative and relate to Poisson-admissible algebras, thus advancing understanding of algebraic structures in Lie theory.

## Contribution

It provides explicit descriptions of CPA-structures on specific nilpotent Lie algebras, revealing their associative nature and connection to Poisson algebras.

## Key findings

- All CPA-structures on non-metabelian filiform nilpotent Lie algebras are associative.
- CPA-structures on Lie algebras of strictly upper-triangular matrices are associative.
- These structures induce Poisson-admissible algebras.

## Abstract

We give an explicit description of commutative post-Lie algebra structures on some classes of nilpotent Lie algebras. For non-metabelian filiform nilpotent Lie algebras and Lie algebras of strictly upper-triangular matrices we show that all CPA-structures are associative and induce an associated Poisson-admissible algebra.