On Two-generated Non-commutative Algebras Subject to the Affine Relation

TL;DR

This paper classifies all two-generated non-commutative algebras over a field with a specific affine relation, providing explicit multiplication formulas and analyzing their algebraic properties.

Contribution

It identifies five isomorphism classes of such algebras, introduces model algebras, and derives explicit multiplication formulas for computational and theoretical applications.

Findings

Five isomorphism classes of algebras identified

Explicit formulas for y^m*x^n derived

Analysis of centers and ring properties conducted

Abstract

We consider algebras over a field K, generated by two variables x and y subject to the single relation yx = qxy + ax + by + c for q in K^* and a, b, c in K. We prove, that among such algebras there are precisely five isomorphism classes. The representatives of these classes, which are ubiquitous operator algebras, are called model algebras. We derive explicit multiplication formulas for y^m*x^n in terms of standard monomials x^i*y^j for many algebras of the considered type. Such formulas are used in establishing formulas of binomial type and in implementing non-commutative multiplication in a computer algebra system. By using the formulas we also study centers and ring-theoretic properties of the non-commutative model algebras.