Nonassociative algebras of anti-biderivation-type
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912443070939136 |
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| author | Benayadi, Saïd Boulmane, Said Kaygorodov, Ivan |
| author_facet | Benayadi, Saïd Boulmane, Said Kaygorodov, Ivan |
| contents | The main purpose of this paper is to study the class of Jacobi-Jordan-admissible algebras, such that its product is an anti-biderivation of the related Jacobi-Jordan algebra. We called it as $\mathcal A{\rm BD}$-algebras. First, we provide characterizations of algebras in this class. Furthermore, we show that this class of nonassociative algebras includes Jacobi-Jordan algebras, symmetric anti-Leibniz algebras, and anti-${\rm LR}$-algebras. In particular, we proved that anti-${\rm LR}$-algebras under the commutator product give $\mathfrak{s}_4$-algebras, which were recently introduced by Filippov and Dzhumadildaev. In addition, we then study $\mathcal A$flexible ${\mathcal A}{\rm BD}$-algebras. Then, we introduce the post-Jacobi-Jordan structures on Jacobi-Jordan algebras and establish results that each Jacobi-Jordan algebra admits a non-trivial post-Jacobi-Jordan structure. At the end of the paper, we give the algebraic classification of complex $3$-dimensional $\mathcal A{\rm BD}$-algebras. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_17228 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Nonassociative algebras of anti-biderivation-type Benayadi, Saïd Boulmane, Said Kaygorodov, Ivan Rings and Algebras The main purpose of this paper is to study the class of Jacobi-Jordan-admissible algebras, such that its product is an anti-biderivation of the related Jacobi-Jordan algebra. We called it as $\mathcal A{\rm BD}$-algebras. First, we provide characterizations of algebras in this class. Furthermore, we show that this class of nonassociative algebras includes Jacobi-Jordan algebras, symmetric anti-Leibniz algebras, and anti-${\rm LR}$-algebras. In particular, we proved that anti-${\rm LR}$-algebras under the commutator product give $\mathfrak{s}_4$-algebras, which were recently introduced by Filippov and Dzhumadildaev. In addition, we then study $\mathcal A$flexible ${\mathcal A}{\rm BD}$-algebras. Then, we introduce the post-Jacobi-Jordan structures on Jacobi-Jordan algebras and establish results that each Jacobi-Jordan algebra admits a non-trivial post-Jacobi-Jordan structure. At the end of the paper, we give the algebraic classification of complex $3$-dimensional $\mathcal A{\rm BD}$-algebras. |
| title | Nonassociative algebras of anti-biderivation-type |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2506.17228 |