@**inproceedings**{RISC4374,author = {Viktor Levandovskyy and Christoph Koutschan and Oleksandr Motsak},

title = {{On Two-generated Non-commutative Algebras Subject to the Affine Relation}},

booktitle = {{Proceedings of CASC 2011}},

language = {english},

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 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.},

series = {Lecture Notes in Computer Science},

volume = {6885},

pages = {309--320},

publisher = {Springer},

isbn_issn = {ISBN 978-3-642-23567-2},

year = {2011},

editor = {Vladimir Gerdt and Wolfram Koepf and Ernst W. Mayr and Evgenii Vorozhtsov},

refereed = {yes},

length = {12}

}