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On the Identity and Group Problems for Complex Heisenberg Matrices

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Reachability Problems (RP 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14235))

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Abstract

We study the Identity Problem, the problem of determining if a finitely generated semigroup of matrices contains the identity matrix; see Problem 3 (Chapter 10.3) in “Unsolved Problems in Mathematical Systems and Control Theory” by Blondel and Megretski (2004). This fundamental problem is known to be undecidable for \(\mathbb {Z}^{4 \times 4}\) and decidable for \(\mathbb {Z}^{2 \times 2}\). The Identity Problem has been recently shown to be in polynomial time by Dong for the Heisenberg group over complex numbers in any fixed dimension with the use of Lie algebra and the Baker-Campbell-Hausdorff formula. We develop alternative proof techniques for the problem making a step forward towards more general problems such as the Membership Problem. We extend our techniques to show that the fundamental problem of determining if a given set of Heisenberg matrices generates a group, can also be decided in polynomial time.

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Notes

  1. 1.

    Note by a result of [1], the Membership Problem is decidable in polynomial time for commuting matrices. However, the authors prefer to have a self-contained proof.

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Author information

Authors and Affiliations

  1. Keele University, Keele, UK

    Paul C. Bell

  2. Liverpool John Moores University, Liverpool, UK

    Reino Niskanen & Pavel Semukhin

  3. University of Liverpool, Liverpool, UK

    Igor Potapov

Authors
  1. Paul C. Bell
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  2. Reino Niskanen
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  3. Igor Potapov
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  4. Pavel Semukhin
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Corresponding author

Correspondence to Reino Niskanen .

Editor information

Editors and Affiliations

  1. École Polytechnique, Paris, France

    Olivier Bournez

  2. Université Côte d’Azur, Nice, France

    Enrico Formenti

  3. University of Liverpool, Liverpool, UK

    Igor Potapov

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Bell, P.C., Niskanen, R., Potapov, I., Semukhin, P. (2023). On the Identity and Group Problems for Complex Heisenberg Matrices. In: Bournez, O., Formenti, E., Potapov, I. (eds) Reachability Problems. RP 2023. Lecture Notes in Computer Science, vol 14235. Springer, Cham. https://doi.org/10.1007/978-3-031-45286-4_4

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  • DOI: https://doi.org/10.1007/978-3-031-45286-4_4

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