Anti-commutative anti-associative algebras. Acaa-algebras
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866912311238721536 |
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| author | Remm, Elisabeth |
| author_facet | Remm, Elisabeth |
| contents | Let $(A,μ)$ be a nonassociative algebra over a field of characteristic zero. The polarization process allows us to associate two other algebras, and this correspondence is one-one, one commutative, the other anti-commutative. Assume that $μ$ satisfies a quadratic identity $\sum_{σ\in Σ_3} a_σμ(μ(x_{σ(i)},x_{σ(j)}),x_{σ(k)}-a_σμ(x_{σ(i)},μ(x_{σ(j)},x_{σ(k)})=0.$ Under certain conditions, the polarization of such a multiplication determines an anticommutative multiplication also verifying a quadratic identity. Now only two identities are possible, the first is the Jacobi identity which makes this anticommutative multiplication a Lie algebra and the multiplication $μ$ is Lie admissible, the second, less classical is given by $[[x,y],z]=[[y,z],x]=[[z,x],y].$ Such a multiplication is here called Acaa for Anticommutative and Antiassociative. We establish some properties of this type of algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_04092 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Anti-commutative anti-associative algebras. Acaa-algebras Remm, Elisabeth Rings and Algebras 17Dxx 17Bxx Let $(A,μ)$ be a nonassociative algebra over a field of characteristic zero. The polarization process allows us to associate two other algebras, and this correspondence is one-one, one commutative, the other anti-commutative. Assume that $μ$ satisfies a quadratic identity $\sum_{σ\in Σ_3} a_σμ(μ(x_{σ(i)},x_{σ(j)}),x_{σ(k)}-a_σμ(x_{σ(i)},μ(x_{σ(j)},x_{σ(k)})=0.$ Under certain conditions, the polarization of such a multiplication determines an anticommutative multiplication also verifying a quadratic identity. Now only two identities are possible, the first is the Jacobi identity which makes this anticommutative multiplication a Lie algebra and the multiplication $μ$ is Lie admissible, the second, less classical is given by $[[x,y],z]=[[y,z],x]=[[z,x],y].$ Such a multiplication is here called Acaa for Anticommutative and Antiassociative. We establish some properties of this type of algebras. |
| title | Anti-commutative anti-associative algebras. Acaa-algebras |
| topic | Rings and Algebras 17Dxx 17Bxx |
| url | https://arxiv.org/abs/2504.04092 |