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Multiplicative Loops of Quasifields Having Complex Numbers as Kernel

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Abstract

We determine the multiplicative loops of locally compact connected 4-dimensional quasifields Q having the field of complex numbers as their kernel. In particular, we turn our attention to multiplicative loops which have either a normal subloop of dimension one or which contain a subgroup isomorphic to \(Spin_3({\mathbb {R}})\). Although the 4-dimensional semifields Q are known, their multiplicative loops have interesting Lie groups generated by left or right translations. We determine explicitly the quasifields Q which coordinatize locally compact translation planes of dimension 8 admitting an at least 16-dimensional Lie group as automorphism group.

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Authors and Affiliations

  1. Dipartimento di Matematica e Informatica, via Archirafi 34, 90123, Palermo, Italy

    Giovanni Falcone

  2. Institute of Mathematics, University of Debrecen, P.O. Box 400, 4002, Debrecen, Hungary

    Ágota Figula

  3. Department Mathematik, Universität Erlangen-Nürnberg, Cauerstrasse 11, 91058, Erlangen, Germany

    Karl Strambach

Authors
  1. Giovanni Falcone
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  2. Ágota Figula
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  3. Karl Strambach
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Correspondence to Ágota Figula.

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In accordance with Karl’s will, we cordially dedicate this paper to the 75th birthday of our common friend Heinrich Wefelscheid.

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Falcone, G., Figula, Á. & Strambach, K. Multiplicative Loops of Quasifields Having Complex Numbers as Kernel. Results Math 72, 2129–2156 (2017). https://doi.org/10.1007/s00025-017-0699-z

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