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

  • Viktor Levandovskyy
  • Christoph Koutschan
  • Oleksandr Motsak
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 = qxy + 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 ym ·xn in terms of standard monomials xiyj 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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