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

We consider algebras over a field K, generated by two variables x and y subject to the single relation yx = qxy + αx + βy + γ for q ∈ K ∗ and α, β, γ ∈ 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 e. g. establishing formulas of binomial type and in an implementation of 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