On two-generated non-commutative algebras subject to the affine relation

Abstract

We consider algebras over a field double-struck K, generated by two variables x and y subject to the single relation yx = qxy + αx + βy + γ for and q ∈ double-struck K* and α, β, γ ∈ double-struck 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 ym·xn in terms of standard monomials xi yj 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. © 2011 Springer-Verlag.