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

  • Viktor Levandovskyy
  • Christoph Koutschan
  • Oleksandr Motsak
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6885)


We consider algebras over a field \({\mathbb K}\), generated by two variables x and y subject to the single relation yx = q xy + αx + βy + γ for \(q\in{\mathbb K}^*\) and \(\alpha, \beta, \gamma \in {\mathbb K}\). 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.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Viktor Levandovskyy
    • 1
  • Christoph Koutschan
    • 2
  • Oleksandr Motsak
    • 3
  1. 1.Lehrstuhl D für MathematikRWTH AachenGermany
  2. 2.RISCJohannes Kepler UniversityLinzAustria
  3. 3.TU KaiserslauternGermany

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