Composition-Diamond Lemma for Tensor Product of Free Algebras

TL;DR

This paper extends the Composition-Diamond lemma to tensor products of free algebras, providing a foundation for constructing Groebner-Shirshov bases in these algebraic structures.

Contribution

It introduces a Composition-Diamond lemma for tensor products of free algebras and demonstrates how to lift Groebner-Shirshov bases in this context.

Findings

Established Composition-Diamond lemma for tensor products of free algebras.

Constructed Groebner-Shirshov basis in tensor product algebra.

Provided a method to lift bases from commutative to noncommutative tensor products.

Abstract

In this paper, we establish Composition-Diamond lemma for tensor product $k< X> \otimes k< Y>$ of two free algebras over a field. As an application, we construct a Groebner-Shirshov basis in $k< X> \otimes k< Y>$ by lifting a Groebner-Shirshov basis in $k[X] \otimes k< Y>$, where $k[X]$ is a commutative algebra.