Operator identities on Lie algebras, rewriting systems and Gr\"obner-Shirshov bases

TL;DR

This paper extends Rota's program on algebraic operator identities to Lie algebras, using rewriting systems and Groebner-Shirshov bases to classify differential and Rota-Baxter operators.

Contribution

It formulates a Lie algebra analog of Rota's program, demonstrating its compatibility with associative algebra cases and applying it to classify key operators.

Findings

Lie algebra operator identities are characterized via rewriting systems.

A classification of differential type operators on Lie algebras is provided.

Rota-Baxter operators on Lie algebras are systematically classified.

Abstract

Motivated by the pivotal role played by linear operators, many years ago Rota proposed to determine algebraic operator identities satisfied by linear operators on associative algebras, later called Rota's program on algebraic operators. Recent progresses on this program have been achieved in the contexts of operated algebra, rewriting systems and Groebner-Shirshov bases. These developments also suggest that Rota's insight can be applied to determine operator identities on Lie algebras, and thus to put the various linear operators on Lie algebras in a uniform perspective. This paper carries out this approach, utilizing operated polynomial Lie algebras spanned by non-associative Lyndon-Shirshov bracketed words. The Lie algebra analog of Rota's program was formulated in terms convergent rewriting systems and equivalently in terms of Groebner-Shirshov bases. This Lie algebra analog is shown to be compatible with Rota's program for associative algebras. As applications, a classification of differential type operators and Rota-Baxter operators are presented.