Looking for Groebner Basis Theory for (Almost) Skew 2-Nomial Algebras

TL;DR

This paper introduces (almost) skew 2-nomial algebras and investigates the existence of one-sided and two-sided Gr"obner basis theories for them, highlighting subclasses relevant to quantum binomial algebras and solutions to the Yang-Baxter equation.

Contribution

It establishes the existence of skew multiplicative bases and explores monomial orderings in (almost) skew 2-nomial algebras, expanding Gr"obner basis theory to new algebra classes.

Findings

Existence of skew multiplicative $K$-basis for skew 2-nomial algebras

Identification of subclasses with one-sided Gr"obner basis theory

Application to quantum binomial algebras related to Yang-Baxter solutions

Abstract

In this paper, we introduce (almost) skew 2-nomial algebras and look for a one-sided or two-sided Gr\"obner basis theory for such algebras at a modest level. That is, we establish the existence of a skew multiplicative $K$-basis for every skew 2-nomial algebra, and we explore the existence of a (left, right, or two-sided) monomial ordering for an (almost) skew 2-nomial algebra. As distinct from commonly recognized algebras holding a Gr\"obner basis theory (such as algebras of the solvable type [K-RW] and some of their homomorphic images), a subclass of skew 2-nomial algebras that have a left Gr\"obner basis theory but may not necessarily have a two-sided Gr\"obner basis theory, respectively a subclass of skew 2-nomial algebras that have a right Gr\"obner basis theory but may not necessarily have a two-sided Gr\"obner basis theory, are determined such that numerous quantum binomial algebras (which provide binomial solutions to the Yang-baxter equation [Laf], [G-I2]) are involved.