On $3$-generated axial algebras of Jordan type $\frac{1}{2}$
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866917796354457600 |
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| author | Bildanov, Ravil Gorshkov, Ilya |
| author_facet | Bildanov, Ravil Gorshkov, Ilya |
| contents | Axial algebras of Jordan type $η$ are a special type of commutative non-associative algebras. They are generated by idempotents whose adjoint operators have the minimal polynomial dividing $(x-1)x(x-η)$, where $η$ is a fixed value that is not equal to $0$ or $1$. These algebras have restrictive multiplication rules that generalize the Peirce decomposition for idempotents in Jordan algebras.
A universal $3$-generated algebra of Jordan type $\frac{1}{2}$ as an algebra with $4$ parameters was constructed by I. Gorshkov and A. Staroletov. Depending on the value of the parameter, the universal algebra may contain a non-trivial form radical. In this paper, we describe all semisimple $3$-generated algebras of Jordan type $\frac{1}{2}$ over a quadratically closed field. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_10680 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On $3$-generated axial algebras of Jordan type $\frac{1}{2}$ Bildanov, Ravil Gorshkov, Ilya Rings and Algebras Group Theory Axial algebras of Jordan type $η$ are a special type of commutative non-associative algebras. They are generated by idempotents whose adjoint operators have the minimal polynomial dividing $(x-1)x(x-η)$, where $η$ is a fixed value that is not equal to $0$ or $1$. These algebras have restrictive multiplication rules that generalize the Peirce decomposition for idempotents in Jordan algebras. A universal $3$-generated algebra of Jordan type $\frac{1}{2}$ as an algebra with $4$ parameters was constructed by I. Gorshkov and A. Staroletov. Depending on the value of the parameter, the universal algebra may contain a non-trivial form radical. In this paper, we describe all semisimple $3$-generated algebras of Jordan type $\frac{1}{2}$ over a quadratically closed field. |
| title | On $3$-generated axial algebras of Jordan type $\frac{1}{2}$ |
| topic | Rings and Algebras Group Theory |
| url | https://arxiv.org/abs/2309.10680 |