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A Two-Dimensional Counterexample to Radical Equality in Primitive Axial Algebras

  • Bo Peng
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Abstract

Let R(A,X) denote the largest ideal of a primitive axial algebra (A,X) that contains no axis from the specified generating set X, and let J(A) be the intersection of the maximal ideals of A. Mamontov, Shpectorov, and Zhelyabin asked whether R(A,X)=J(A) always holds. We give a negative answer. Over every field of characteristic different from $2$, the two-dimensional commutative algebra with basis a,b and multiplication a2=a, ab=2b, and b2=b is a primitive axial algebra for an explicit fusion law and the generating set X=a,b. Its complete ideal lattice is 0<𝔽b<A, whence R(A,X)=0 and J(A)=𝔽b; in particular, the primitive axis b lies in J(A). Over ℂ, this axial presentation is equivalent to the previously classified D(−1),e2,a6 presentation with fusion law FD3. Thus the algebra and axial structure are known; the new point is the computation of its Jacobson radical and the resulting counterexample to the radical-equality question.

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Cite

BibTeXRIS
BibTeX
@misc{peng2026twodimensional,
  title = {{A Two-Dimensional Counterexample to Radical Equality in Primitive Axial Algebras}},
  author = {Bo Peng},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.28653},
  eprint = {2608.28653},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.28653},
}
RIS
TY  - UNPB
TI  - A Two-Dimensional Counterexample to Radical Equality in Primitive Axial Algebras
AU  - Bo Peng
PY  - 2026
DA  - 2026/08/20
PB  - arXiv
DO  - 10.48550/arxiv.2608.28653
UR  - https://arxiv.org/abs/2608.28653
AB  - Let R(A,X) denote the largest ideal of a primitive axial algebra (A,X) that contains no axis from the specified generating set X, and let J(A) be the intersection of the maximal ideals of A. Mamontov, Shpectorov, and Zhelyabin asked whether R(A,X)=J(A) always holds. We give a negative answer. Over every field of characteristic different from $2$, the two-dimensional commutative algebra with basis a,b and multiplication a^2=a, ab=2b, and b^2=b is a primitive axial algebra for an explicit fusion law and the generating set X=a,b. Its complete ideal lattice is 0<𝔽b<A, whence R(A,X)=0 and J(A)=𝔽b; in particular, the primitive axis b lies in J(A). Over ℂ, this axial presentation is equivalent to the previously classified D(−1),e_2,a_6 presentation with fusion law F_D3. Thus the algebra and axial structure are known; the new point is the computation of its Jacobson radical and the resulting counterexample to the radical-equality question.
ER  -