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Classification of $3$-dimensional unital commutative algebras

  • Maqpal Eraliyeva
  • Ivan Kaygorodov
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Abstract

We present the complete classification of non-isomorphic complex $3$-dimensional unital commutative algebras. Every such algebra decomposes canonically as ℂe1⊕ V, where e1 is the unit and V is the kernel of the trace form; the induced traceless $2$-dimensional commutative algebra on V is one of four algebras, and the classification reduces to the classification of symmetric bilinear forms on V up to the automorphisms of this kernel. The answer consists of six algebras, three one-parameter families, one two-parameter family and one three-parameter family.

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Cite

BibTeXRIS
BibTeX
@misc{eraliyeva2026classification,
  title = {{Classification of $3$-dimensional unital commutative algebras}},
  author = {Maqpal Eraliyeva and Ivan Kaygorodov},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.18689},
  eprint = {2610.18689},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.18689},
}
RIS
TY  - UNPB
TI  - Classification of $3$-dimensional unital commutative algebras
AU  - Maqpal Eraliyeva
AU  - Ivan Kaygorodov
PY  - 2026
DA  - 2026/10/13
PB  - arXiv
DO  - 10.48550/arxiv.2610.18689
UR  - https://arxiv.org/abs/2610.18689
AB  - We present the complete classification of non-isomorphic complex $3$-dimensional unital commutative algebras. Every such algebra decomposes canonically as ℂe_1⊕ V, where e_1 is the unit and V is the kernel of the trace form; the induced traceless $2$-dimensional commutative algebra on V is one of four algebras, and the classification reduces to the classification of symmetric bilinear forms on V up to the automorphisms of this kernel. The answer consists of six algebras, three one-parameter families, one two-parameter family and one three-parameter family.
ER  -