Composition-Diamond Lemma for Modules

TL;DR

This paper extends the Composition-Diamond lemma to modules, establishing relationships among Groebner-Shirshov bases in various algebraic structures and applying it to modules over Lie algebras and related algebraic systems.

Contribution

It introduces a generalized Composition-Diamond lemma for modules, unifying existing lemmas and applying to diverse algebraic modules and structures.

Findings

Chibrikov's Composition-Diamond lemma for modules established

Kang-Lee's lemma derived from the new lemma

Applications to modules over Lie and conformal algebras

Abstract

In this paper we give some relationships among the Groebner-Shirshov bases in free associative algebras, free left modules and "double-free" left modules (free modules over a free algebra). We give the Chibrikov's Composition-Diamond lemma for modules and show that Kang-Lee's Composition-Diamond lemma follows from this lemma. As applications, we also deal with highest weight module over the Lie algebra $sl_2$, Verma module over a Kac-Moody algebra, Verma module over Lie algebra of coefficients of a free conformal algebra and the universal enveloping module for a Sabinin algebra.