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PreprintCC BY 4.0arXiv

Fourier Analysis and Idempotents in Quandle Algebras

  • Mohamed Elhamdadi
  • Luc Ta
  • Bryce Virgin
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Abstract

In 2023, the first author, Nunez, Singh, and Swain [10] formulated an analogue of Kaplansky's idempotent conjecture for integral quandle rings. In this paper, we develop a new Fourier-analytic approach to quandle rings and use it to establish the conjecture for two major classes of quandles: Takasaki quandles, including dihedral quandles, and medial commutative quandles. A central contribution of the paper is the introduction of Fourier analysis on Alexander quandles, which provides a new framework for studying quandle rings and, in particular, their idempotents. Using this Fourier-analytic framework, we also prove that a previously known sufficient condition for the existence of counterexamples to the conjecture is in fact necessary, thereby resolving a problem of Jablonowski [12].

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Cite

BibTeXRIS
BibTeX
@misc{elhamdadi2026fourier,
  title = {{Fourier Analysis and Idempotents in Quandle Algebras}},
  author = {Mohamed Elhamdadi and Luc Ta and Bryce Virgin},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.03799},
  eprint = {2609.03799},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.03799},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Fourier Analysis and Idempotents in Quandle Algebras
AU  - Mohamed Elhamdadi
AU  - Luc Ta
AU  - Bryce Virgin
PY  - 2026
DA  - 2026/09/03
PB  - arXiv
DO  - 10.48550/arxiv.2609.03799
UR  - https://arxiv.org/abs/2609.03799
AB  - In 2023, the first author, Nunez, Singh, and Swain [10] formulated an analogue of Kaplansky's idempotent conjecture for integral quandle rings. In this paper, we develop a new Fourier-analytic approach to quandle rings and use it to establish the conjecture for two major classes of quandles: Takasaki quandles, including dihedral quandles, and medial commutative quandles. A central contribution of the paper is the introduction of Fourier analysis on Alexander quandles, which provides a new framework for studying quandle rings and, in particular, their idempotents. Using this Fourier-analytic framework, we also prove that a previously known sufficient condition for the existence of counterexamples to the conjecture is in fact necessary, thereby resolving a problem of Jablonowski [12].
ER  -