Just-Infinite Loops and Loop Algebras

Abstract

Let F be a field and let L be a loop. We call L just-infinite if it is infinite and every nontrivial normal subloop has finite index, and we call the possibly nonassociative loop algebra F[L] just-infinite if it is infinite-dimensional and every nonzero two-sided ideal has finite codimension. We first prove that just-infiniteness of F[L] always implies just-infiniteness of L. Next, using the Chein construction, we show for every infinite group G that M(G,2) is just-infinite if and only if G is just-infinite, and that $F[M(G,2)]$ is just-infinite if and only if F[G] is just-infinite. We extend the algebraic equivalence to the generalized Moufang doubles M(G,*,g₀) whenever G is infinite and nonabelian. Finally, we construct a single locally finite, residually finite, nonassociative Moufang loop L for which F[L] is residually finite-dimensional, locally finite-dimensional, and just-infinite over every field, and we explain why infinite nonassociative RA loops cannot be just-infinite.

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BibTeX RIS

BibTeX

@misc{paiva2026justinfinite,
  title = {{Just-Infinite Loops and Loop Algebras}},
  author = {Thales Fernando Vilamaior Paiva},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.21016},
  eprint = {2609.21016},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.21016},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Just-Infinite Loops and Loop Algebras
AU  - Thales Fernando Vilamaior Paiva
PY  - 2026
DA  - 2026/09/29
PB  - arXiv
DO  - 10.48550/arxiv.2609.21016
UR  - https://arxiv.org/abs/2609.21016
AB  - Let F be a field and let L be a loop. We call L just-infinite if it is infinite and every nontrivial normal subloop has finite index, and we call the possibly nonassociative loop algebra F[L] just-infinite if it is infinite-dimensional and every nonzero two-sided ideal has finite codimension. We first prove that just-infiniteness of F[L] always implies just-infiniteness of L. Next, using the Chein construction, we show for every infinite group G that M(G,2) is just-infinite if and only if G is just-infinite, and that $F[M(G,2)]$ is just-infinite if and only if F[G] is just-infinite. We extend the algebraic equivalence to the generalized Moufang doubles M(G,*,g_0) whenever G is infinite and nonabelian. Finally, we construct a single locally finite, residually finite, nonassociative Moufang loop L for which F[L] is residually finite-dimensional, locally finite-dimensional, and just-infinite over every field, and we explain why infinite nonassociative RA loops cannot be just-infinite.
ER  -

Abstract

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