Universal enveloping commutative Rota-Baxter algebras of precommutative and postcommutative algebras

TL;DR

This paper constructs universal enveloping commutative Rota-Baxter algebras for pre- and postcommutative algebras and analyzes their algebraic pairings, revealing PBW-pair properties.

Contribution

It introduces explicit constructions of universal enveloping algebras for these structures and establishes PBW-pair relationships for specific algebraic varieties.

Findings

Proves the pair (commutative Rota-Baxter algebras of nonzero weight, postcommutative algebras) is a PBW-pair.

Shows the pair (commutative Rota-Baxter algebras of zero weight, precommutative algebras) is not a PBW-pair.

Constructs universal enveloping algebras for pre- and postcommutative algebras.

Abstract

Universal enveloping commutative Rota-Baxter algebras of pre- and postcommutative algebras are constructed. We prove that the pair of varieties (commutative Rota-Baxter algebras of nonzero weight,postcommutative algebras) is a PBW-pair and the pair (commutative Rota-Baxter algebras of zero weight,precommutative algebras) is not.