Letters in Mathematical Physics

, Volume 107, Issue 3, pp 475–503 | Cite as

Lie algebra type noncommutative phase spaces are Hopf algebroids

  • Stjepan Meljanac
  • Zoran Škoda
  • Martina Stojić


For a noncommutative configuration space whose coordinate algebra is the universal enveloping algebra of a finite-dimensional Lie algebra, it is known how to introduce an extension playing the role of the corresponding noncommutative phase space, namely by adding the commuting deformed derivatives in a consistent and nontrivial way; therefore, obtaining certain deformed Heisenberg algebra. This algebra has been studied in physical contexts, mainly in the case of the kappa-Minkowski space-time. Here, we equip the entire phase space algebra with a coproduct, so that it becomes an instance of a completed variant of a Hopf algebroid over a noncommutative base, where the base is the enveloping algebra.


Universal enveloping algebra Noncommutative phase space Deformed derivative Hopf algebroid Completed tensor product 

Mathematics Subject Classification

16S30 16S32 16S35 16Txx 



We thank L. El Kaoutit, V. Roubtsov, T. Brzeziński, T. Maszczyk and G. Böhm for reading fragments of this work and advice. We thank A. Borowiec, J. Lukierski and A. Pachoł for discussions. S.M. has been supported by Croatian Science Foundation, project IP-2014-09-9582. A part of the work has been done at IRB, Zagreb, and a substantial progress has also been made during the visit of Z.Š. to IHÉS in November 2012. Z.Š. thanks G. Garkusha for the invitation to present a related Algebra and topology seminar at Swansea, July 15, 2014.


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Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  • Stjepan Meljanac
    • 1
  • Zoran Škoda
    • 2
  • Martina Stojić
    • 3
  1. 1.Theoretical Physics DivisionInstitute Rudjer BoškovićZagrebCroatia
  2. 2.Faculty of ScienceUniversity of Hradec KrálovéHradec KraloveCzech Republic
  3. 3.Department of MathematicsUniversity of ZagrebZagrebCroatia

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