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 · x in terms of standard monomials xy 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. In this paper we study non-commutative algebras in two generators obeying single affine relation. Many operator algebras, coming from different areas of natural sciences, are built from algebras in two generators, see Sect. 2.1 for examples. One of generators, say x, often corresponds to the operator of the multiplication with the function x. Another operator, say y, corresponds to a linear operator, acting on functions in the variable x. In the main Theorem we identify precisely five types of non-isomorphic algebras, which we call model algebras, among them. Despite the fact that many such algebras have been studied in the literature (see e. g. [3, 7, 2, 1], many aspects and properties are too scattered in the existing literature. Another point of this note is to search systematically for closed form of multiplication formulas on monomials. Such closed forms are needed, among other, in computer algebra, where many sophisticated algorithms heavily rely on basic multiplication among monomials. It is not enough to have such formulas just for model algebras, since isomorphisms do not preserve monomials but turn them into polynomials. It turned out, that there are still several cases, where we were not able to derive ? This paper is in the Proceedings of CASC 2011, by Vladimir Gerdt, Wolfram Koepf, Ernst W. Mayr, and Evgenii Vorozhtsov (eds.), Lecture Notes in Computer Science, vol. 6885. The final publication is available at www.springerlink.com. closed formulas in terms of standard monomials. With our approach, however, one is still able to derive formulas of certain type for them.