Anti-quadratic nearly associative algebras

Abstract

An anti-quadratic nearly associative algebra (A, •, ω) is a nearly associative algebra equipped with a non-degenerate skew-symmetric bilinear form ω satisfying the anti-associativity condition ω(u • v, w) = −ω(u, v • w) for all u, v, w ∈ A. In this paper, we investigate the structural properties and extension theory of these algebras. We prove that any anti-quadratic nearly associative algebra is inherently nilpotent of index at most $6$. Furthermore, we introduce a novel T T^*-extension method and show that every anti-quadratic nearly associative algebra arises from this construction. We also introduce the notion of a double extension and prove that every such algebra can be constructed by iterated double extensions. Finally, as an application, we give a complete classification of anti-quadratic nearly associative algebras of dimension at most $4$.

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BibTeX RIS

BibTeX

@misc{bouarroudj2026antiquadratic,
  title = {{Anti-quadratic nearly associative algebras}},
  author = {Sofiane Bouarroudj and Hamza El Ouali},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.41465},
  eprint = {2610.41465},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.41465},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Anti-quadratic nearly associative algebras
AU  - Sofiane Bouarroudj
AU  - Hamza El Ouali
PY  - 2026
DA  - 2026/10/30
PB  - arXiv
DO  - 10.48550/arxiv.2610.41465
UR  - https://arxiv.org/abs/2610.41465
AB  - An anti-quadratic nearly associative algebra (A, •, ω) is a nearly associative algebra equipped with a non-degenerate skew-symmetric bilinear form ω satisfying the anti-associativity condition ω(u • v, w) = −ω(u, v • w) for all u, v, w ∈ A. In this paper, we investigate the structural properties and extension theory of these algebras. We prove that any anti-quadratic nearly associative algebra is inherently nilpotent of index at most $6$. Furthermore, we introduce a novel T T^*-extension method and show that every anti-quadratic nearly associative algebra arises from this construction. We also introduce the notion of a double extension and prove that every such algebra can be constructed by iterated double extensions. Finally, as an application, we give a complete classification of anti-quadratic nearly associative algebras of dimension at most $4$.
ER  -

Abstract

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