Operadic Gröbner Bases: An Implementation

  • Vladimir Dotsenko
  • Mikael Vejdemo-Johansson
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6327)


In an upcoming paper [1], the first author and Anton Khoroshkin define the concept of a Gröbner basis for finitely presented operads, prove the diamond lemma for these Gröbner bases, and demonstrate that having a quadratic Gröbner basis is equivalent to the existence of a Poincaré-Birkhoff-Witt basis. As demonstrated by Eric Hoffbeck [2], an operad with a PBW basis is Koszul. Thus, out of this emerges an entirely computational framework for proving Koszulness, as well as the possibility to build tools for exploration of operads by means of explicit calculation.


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  1. 1.
    Dotsenko, V., Khoroshkin, A.: Gröbner bases for operads. Duke Math. Journal 153(2), 363–396 (2010)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Hoffbeck, E.: A Poincaré–Birkhoff–Witt criterion for Koszul operads. Manuscripta Mathematica 131, 87–110 (2010)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Dotsenko, V., Vejdemo-Johansson, M.: Implementing Gröbner bases for operads. Séminaires et Congrès (2009) (to appear)Google Scholar
  4. 4.
    Jones, S.P. (ed.): Haskell 98 language and libraries: the revised report. Cambridge Univ. Pr., Cambridge (2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Vladimir Dotsenko
    • 1
  • Mikael Vejdemo-Johansson
    • 2
  1. 1.Dublin Institute for Advanced Studies and School of MathematicsTrinity CollegeDublinIreland
  2. 2.Department of MathematicsStanford University 

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