Gr\"obner-Shirshov bases for Lie algebras over a commutative algebra

TL;DR

This paper develops a Gr"obner-Shirshov bases theory for Lie algebras over commutative rings, enabling the classification of special and non-special Lie algebras and providing algorithms for embedding and property checking.

Contribution

It introduces a new Gr"obner-Shirshov bases framework for Lie algebras over commutative rings, with applications to embedding and classification.

Findings

Cohn's Lie algebras over certain characteristics are non-special

An algorithm to determine non-speciality of Cohn's Lie algebras

Any finitely or countably generated Lie algebra embeds into a two-generated Lie algebra

Abstract

In this paper we establish a Gr\"{o}bner-Shirshov bases theory for Lie algebras over commutative rings. As applications we give some new examples of special Lie algebras (those embeddable in associative algebras over the same ring) and non-special Lie algebras (following a suggestion of P.M. Cohn (1963) \cite{Conh}). In particular, Cohn's Lie algebras over the characteristic $p$ are non-special when $p=2,\ 3,\ 5$. We present an algorithm that one can check for any $p$, whether Cohn's Lie algebras is non-special. Also we prove that any finitely or countably generated Lie algebra is embeddable in a two-generated Lie algebra.