# Decompositions of algebras and post-associative algebra structures

## TL;DR

This paper introduces post-associative algebra structures, explores their connections to other algebraic structures, and proves non-existence results for certain configurations involving Lie and associative algebras.

## Contribution

It defines post-associative algebra structures and establishes new non-existence results related to post-Lie and Rota--Baxter algebra structures.

## Key findings

- No post-Lie algebra structure exists between a simple and a reductive Lie algebra unless they are isomorphic.
- No post-associative algebra structure arises from a Rota--Baxter operator when the associative algebra is semisimple and the other algebra is not.
- Results connect algebraic decompositions with the existence of post-structures.

## Abstract

We introduce post-associative algebra structures and study their relationship to post-Lie algebra structures, Rota--Baxter operators and decompositions of associative algebras and Lie algebras. We show several results on the existence of such structures. In particular we prove that there exists no post-Lie algebra structure on a pair $(\mathfrak{g},\mathfrak{n})$, where $\mathfrak{n}$ is a simple Lie algebra and $\mathfrak{g}$ is a reductive Lie algebra, which is not isomorphic to $\mathfrak{n}$. We also show that there is no post-associative algebra structure on a pair $(A,B)$ arising from a Rota--Baxter operator of $B$, where $A$ is a semisimple associative algebra and $B$ is not semisimple. The proofs use results on Rota--Baxter operators and decompositions of algebras.