Viktor Levandovskyy, Christoph Koutschan, Oleksandr Motsak,
"On Two-generated Non-commutative Algebras Subject to the Affine Relation"
, in Vladimir Gerdt, Wolfram Koepf, Ernst W. Mayr, Evgenii Vorozhtsov: Proceedings of CASC 2011, Serie Lecture Notes in Computer Science, Vol. 6885, Springer, Seite(n) 309--320, 2011, ISBN: 978-3-642-23567-2
On Two-generated Non-commutative Algebras Subject to the Affine Relation
Sprache des Titels:
Proceedings of CASC 2011
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.