The algebraic and geometric classification of right alternative and semi-alternative algebras
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| Format: | Preprint |
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2025
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| author | Abdelwahab, Hani Kaygorodov, Ivan Lubkov, Roman |
| author_facet | Abdelwahab, Hani Kaygorodov, Ivan Lubkov, Roman |
| contents | The algebraic and geometric classifications of complex $3$-dimensional right alternative and semi-alternative algebras are given. As corollaries, we have the algebraic and geometric classification of complex $3$-dimensional $\mathfrak{perm}$, binary $\mathfrak{perm}$, associative, $(-1,1)$-, binary $(-1,1)$-, and assosymmetric algebras. In particular, we proved that the first example of non-associative right alternative algebras appears in dimension $3;$ the first example of non-associative assosymmetric algebras appears in dimension $3;$ the first example of non-assosymmetric semi-alternative algebras appears in dimension $4;$ the first example of binary $(-1,1)$-algebras, which is non-$(-1,1)$-, appears in dimension $4;$ the first example of right alternative algebras, which is not binary $(-1,1)$-, appears in dimension $4;$ the first example of binary $\mathfrak{perm}$ non-$\mathfrak{perm}$ algebras appears in dimension $4.$ As a byproduct, we give a more easy answer to problem 2.109 from the Dniester Notebook, previously resolved by Shestakov and Arenas. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_00720 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The algebraic and geometric classification of right alternative and semi-alternative algebras Abdelwahab, Hani Kaygorodov, Ivan Lubkov, Roman Rings and Algebras 17A30 (primary), 17D15, 14L30 (secondary) The algebraic and geometric classifications of complex $3$-dimensional right alternative and semi-alternative algebras are given. As corollaries, we have the algebraic and geometric classification of complex $3$-dimensional $\mathfrak{perm}$, binary $\mathfrak{perm}$, associative, $(-1,1)$-, binary $(-1,1)$-, and assosymmetric algebras. In particular, we proved that the first example of non-associative right alternative algebras appears in dimension $3;$ the first example of non-associative assosymmetric algebras appears in dimension $3;$ the first example of non-assosymmetric semi-alternative algebras appears in dimension $4;$ the first example of binary $(-1,1)$-algebras, which is non-$(-1,1)$-, appears in dimension $4;$ the first example of right alternative algebras, which is not binary $(-1,1)$-, appears in dimension $4;$ the first example of binary $\mathfrak{perm}$ non-$\mathfrak{perm}$ algebras appears in dimension $4.$ As a byproduct, we give a more easy answer to problem 2.109 from the Dniester Notebook, previously resolved by Shestakov and Arenas. |
| title | The algebraic and geometric classification of right alternative and semi-alternative algebras |
| topic | Rings and Algebras 17A30 (primary), 17D15, 14L30 (secondary) |
| url | https://arxiv.org/abs/2505.00720 |