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Fusion rules from the Norton inequality

  • Alonso Castillo-Ramirez
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Abstract

The Norton inequality is one of the fundamental axioms in the theory of Majorana and axial algebras, yet its precise structural consequences have remained only partially understood. In this paper, we show that the Norton inequality alone forces the $0$- and $1$-eigenspace fusion rules for arbitrary idempotents in a commutative real algebra A equipped with a Frobenius form. More precisely, if the Frobenius form is nondegenerate (as in Majorana algebras), we prove that the eigenspaces A0(e) and A1(e) of an arbitrary idempotent e ∈ A are subalgebras and annihilate one another: A0(e)A1(e)={0}, while in the degenerate case the corresponding inclusions hold modulo the radical of the Frobenius form. This answers a question of T. M. Mudziiri Shumba and S. Shpectorov concerning the closure of the $0$-eigenspace A0(e).

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Cite

BibTeXRIS
BibTeX
@misc{castilloramirez2026fusion,
  title = {{Fusion rules from the Norton inequality}},
  author = {Alonso Castillo-Ramirez},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2607.03237},
  eprint = {2607.03237},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.03237},
}
RIS
TY  - UNPB
TI  - Fusion rules from the Norton inequality
AU  - Alonso Castillo-Ramirez
PY  - 2026
DA  - 2026/10/15
PB  - arXiv
DO  - 10.48550/arxiv.2607.03237
UR  - https://arxiv.org/abs/2607.03237
AB  - The Norton inequality is one of the fundamental axioms in the theory of Majorana and axial algebras, yet its precise structural consequences have remained only partially understood. In this paper, we show that the Norton inequality alone forces the $0$- and $1$-eigenspace fusion rules for arbitrary idempotents in a commutative real algebra A equipped with a Frobenius form. More precisely, if the Frobenius form is nondegenerate (as in Majorana algebras), we prove that the eigenspaces A_0(e) and A_1(e) of an arbitrary idempotent e ∈ A are subalgebras and annihilate one another: A_0(e)A_1(e)={0}, while in the degenerate case the corresponding inclusions hold modulo the radical of the Frobenius form. This answers a question of T. M. Mudziiri Shumba and S. Shpectorov concerning the closure of the $0$-eigenspace A_0(e).
ER  -