Some new operated Lie polynomial identities and Gr\"obner-Shirshov bases

TL;DR

This paper proves that all operated Lie polynomial identities corresponding to known operated associative identities are Gr"obner-Shirshov, advancing the understanding of algebraic operators in Lie algebras.

Contribution

It establishes that each operated Lie polynomial identity related to associative identities is Gr"obner-Shirshov, confirming a conjecture and enriching Rota's Program.

Findings

All operated Lie polynomial identities are Gr"obner-Shirshov.

Supports the extension of associative identities to Lie algebras.

Enhances the algebraic operator framework within Lie algebra theory.

Abstract

Bremner and Elgendy developed a classification of operated polynomial identities for linear operators on associative algebras, encompassing both classical and newly discovered cases. Within the framework of Rota's Program, each of these new operated associative polynomial identities was shown to be Gr\"obner-Shirshov. This naturally led to a question posed by Guo and collaborators: is each corresponding operated Lie polynomial identity also Gr\"obner-Shirshov? In this paper, we provide an affirmative answer by proving that each such Lie analogue indeed is Gr\"obner-Shirshov, thereby enriching the development of Rota's Program on algebraic operators within the Lie algebraic setting.