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Commutative algebras are ideals of evolution algebras

  • Cristina Costoya
  • Amir Fernández Ouaridi
  • Antonio Viruel
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Abstract

We prove that every finite-dimensional commutative algebra can be embedded as an ideal of a finite-dimensional evolution algebra. We then determine the smallest possible dimension of an evolution algebra admitting such an embedding and study some structural properties preserved by the construction. Over fields of characteristic different from two, we relate this minimal dimension to simultaneous Waring decompositions of the quadratic map x↦ x2. In particular, for unital commutative algebras, the smallest possible dimension of an evolution algebra containing the given algebra coincides with the Waring rank of its squaring map. As an application, we determine this minimal dimension for the truncated polynomial algebras 𝔽[t]/(tn). Finally, we study embeddings for which additional structure is preserved, including automorphisms and multiplicative bases.

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Cite

BibTeXRIS
BibTeX
@misc{costoya2026commutative,
  title = {{Commutative algebras are ideals of evolution algebras}},
  author = {Cristina Costoya and Amir Fernández Ouaridi and Antonio Viruel},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.32784},
  eprint = {2609.32784},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.32784},
}
RIS
TY  - UNPB
TI  - Commutative algebras are ideals of evolution algebras
AU  - Cristina Costoya
AU  - Amir Fernández Ouaridi
AU  - Antonio Viruel
PY  - 2026
DA  - 2026/09/26
PB  - arXiv
DO  - 10.48550/arxiv.2609.32784
UR  - https://arxiv.org/abs/2609.32784
AB  - We prove that every finite-dimensional commutative algebra can be embedded as an ideal of a finite-dimensional evolution algebra. We then determine the smallest possible dimension of an evolution algebra admitting such an embedding and study some structural properties preserved by the construction. Over fields of characteristic different from two, we relate this minimal dimension to simultaneous Waring decompositions of the quadratic map x↦ x^2. In particular, for unital commutative algebras, the smallest possible dimension of an evolution algebra containing the given algebra coincides with the Waring rank of its squaring map. As an application, we determine this minimal dimension for the truncated polynomial algebras 𝔽[t]/(t^n). Finally, we study embeddings for which additional structure is preserved, including automorphisms and multiplicative bases.
ER  -