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 [Formula: see text], where [Formula: see text] is a simple Lie algebra and [Formula: see text] is a reductive Lie algebra, which is not isomorphic to [Formula: see text]. We also show that there is no post-associative algebra structure on a pair [Formula: see text] arising from a Rota–Baxter operator of [Formula: see text], where [Formula: see text] is a semisimple associative algebra and [Formula: see text] is not semisimple. The proofs use results on Rota–Baxter operators and decompositions of algebras.

中文翻译：

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代数和后关联代数结构的分解

我们介绍了后关联代数结构并研究了它们与后李代数结构、Rota-Baxter 算子以及关联代数和李代数分解的关系。我们展示了关于这种结构存在的几个结果。特别地，我们证明了在一对 [Formula: see text] 上不存在 post-Lie 代数结构，其中 [Formula: see text] 是一个简单的李代数，而 [Formula: see text] 是一个约简李代数，其中不与 [公式：见正文] 同构。我们还表明，在一对 [Formula: see text] 上没有后关联代数结构，它由 [Formula: see text] 的 Rota-Baxter 算子产生，其中 [Formula: see text] 是半单关联代数，并且 [公式：见正文]不是半简单的。证明使用Rota-Baxter 算子和代数分解的结果。