Groebner-Shirshov basis of the universal enveloping Rota-Baxter algebra of a Lie algebra

TL;DR

This paper establishes a PBW-type theorem for the universal enveloping Rota-Baxter algebra of a Lie algebra using Gr"obner-Shirshov bases, extending classical Lie algebra theory to include Rota-Baxter operators.

Contribution

It introduces a Gr"obner-Shirshov basis approach to the universal enveloping Rota-Baxter algebra, providing an operator analogue of the PBW theorem for Lie algebras.

Findings

Proved an operator PBW theorem for Rota-Baxter Lie algebras.

Constructed Gr"obner-Shirshov bases for these algebras.

Extended classical Lie algebra results to Rota-Baxter context.

Abstract

Consider the class RBLie of Lie algebras equipped with a Rota---Baxter operator. Then the forgetful functor RBLie --> Lie has a left adjoint one denoted by $U_{RB}(\cdot)$. We prove an "operator" analogue of the Poincare---Birkhoff---Witt theorem for $U_{RB}(L)$, where $L$ is an arbitrary Lie algebra, by means of Gr\"obner---Shirshov bases theory for Lie algebras with an additional operator.