A Poincar\'{e}-Birkhoff-Witt theorem for the universal enveloping algebra of a Rota-Baxter Lie algebra

TL;DR

This paper proves a Poincaré-Birkhoff-Witt theorem for the universal enveloping algebra of Rota-Baxter Lie algebras, using operated algebras and Gr"obner-Shirshov bases, advancing the understanding of their algebraic structure.

Contribution

It establishes a PBW theorem for Rota-Baxter Lie algebras' universal enveloping algebra, solving a problem posed by Gubarev.

Findings

Positive proof of the PBW theorem for Rota-Baxter Lie algebras

Application of Gr"obner-Shirshov bases to operated algebras

Enhanced understanding of Rota-Baxter algebra structures

Abstract

Rota-Baxter associative algebras and Rota-Baxter Lie algebras are both important in mathematics and mathematical physics, with the former a basic structure in quantum field renormalization and the latter a operator form of the classical Yang-Baxter equation. An outstanding problem posed by Gubarev is to determine whether there is a Poincar\'e-Birkhoff-Witt theorem for the universal enveloping Rota-Baxter associative algebra of a Rota-Baxter Lie algebra. This paper resolves this problem positively, working with operated algebras and applying the method of Gr\"obner-Shirshov bases.