Universal enveloping Rota-Baxter algebras of preassociative and postassociative algebras

TL;DR

This paper constructs universal enveloping Rota-Baxter algebras for preassociative and postassociative algebras, clarifying their algebraic relationships and answering a question posed by Li Guo.

Contribution

It establishes the PBW-pair relationship between certain Rota-Baxter algebras and pre/postassociative algebras, providing new insights into their algebraic structure.

Findings

The pair (Rota-Baxter algebras of nonzero weight, postassociative algebras) is a PBW-pair.

The pair (Rota-Baxter algebras of zero weight, preassociative algebras) is not a PBW-pair.

Universal enveloping algebras are explicitly constructed for these algebraic varieties.

Abstract

Universal enveloping Rota-Baxter algebras of preassociative and postassociative algebras are constructed. The question of Li Guo is answered: the pair of varieties (Rota-Baxter algebras of nonzero weight,postassociative algebras) is a PBW-pair and the pair (Rota-Baxter algebras of zero weight,preassociative algebras) is not.