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Multiplicative Loops of Quasifields Having Complex Numbers as Kernel

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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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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Keywords

  • Multiplicative loops of locally compact quasifields
  • semifields
  • sections in Lie groups
  • translation planes
  • automorphism groups

Mathematics Subject Classification

  • 20N05
  • 22A30
  • 12K10
  • 12K99
  • 51A40
  • 57M60