Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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COMBINATORIAL SUPERSYMMETRY: SUPERGROUPS, SUPERQUASIGROUPS, AND THEIR MULTIPLICATION GROUPS

  • Bokhee Im (Department of Mathematics Chonnam National University) ;
  • Jonathan D. H. Smith (Department of Mathematics Iowa State University)
  • Received : 2023.04.01
  • Accepted : 2023.10.25
  • Published : 2024.01.01
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Abstract

The Clifford algebra of a direct sum of real quadratic spaces appears as the superalgebra tensor product of the Clifford algebras of the summands. The purpose of the current paper is to present a purely settheoretical version of the superalgebra tensor product which will be applicable equally to groups or to their non-associative analogues - quasigroups and loops. Our work is part of a project to make supersymmetry an effective tool for the study of combinatorial structures. Starting from group and quasigroup structures on four-element supersets, our superproduct unifies the construction of the eight-element quaternion and dihedral groups, further leading to a loop structure which hybridizes the two groups. All three of these loops share the same character table.

Keywords

Acknowledgement

The authors are grateful to Petr Vojtechovsky for identifying the quatedral loop within GAP (compare Remark 4.20), and also to Connor Depies and an anonymous referee for other helpful comments on the paper.

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