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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| Accesso online: | https://arxiv.org/abs/2503.05337 |
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| _version_ | 1866909583922954240 |
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| author | Alvarez, María Alejandra Lopatin, Artem |
| author_facet | Alvarez, María Alejandra Lopatin, Artem |
| contents | We classify all two-dimensional simple algebras (which may be non-associative) over an algebraically closed field. For each two-dimensional algebra $\mathcal{A}$, we describe a minimal (with respect to inclusion) generating set for the algebra of invariants of the $m$-tuples of $\mathcal{A}$ in the case of characteristic zero. In particular, we establish that for any two-dimensional simple algebra $\mathcal{A}$ with a non-trivial automorphism group, the Artin--Procesi--Iltyakov Equality holds for $\mathcal{A}^m$; that is, the algebra of polynomial invariants of $m$-tuples of $\mathcal{A}$ is generated by operator traces. As a consequence, we describe two-dimensional algebras that admit a symmetric or skew-symmetric invariant nondegenerate bilinear form. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_05337 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Polynomial invariants for low dimensional algebras Alvarez, María Alejandra Lopatin, Artem Rings and Algebras 13A50, 15A72, 1630, 17A30, 17A36, 20F29 We classify all two-dimensional simple algebras (which may be non-associative) over an algebraically closed field. For each two-dimensional algebra $\mathcal{A}$, we describe a minimal (with respect to inclusion) generating set for the algebra of invariants of the $m$-tuples of $\mathcal{A}$ in the case of characteristic zero. In particular, we establish that for any two-dimensional simple algebra $\mathcal{A}$ with a non-trivial automorphism group, the Artin--Procesi--Iltyakov Equality holds for $\mathcal{A}^m$; that is, the algebra of polynomial invariants of $m$-tuples of $\mathcal{A}$ is generated by operator traces. As a consequence, we describe two-dimensional algebras that admit a symmetric or skew-symmetric invariant nondegenerate bilinear form. |
| title | Polynomial invariants for low dimensional algebras |
| topic | Rings and Algebras 13A50, 15A72, 1630, 17A30, 17A36, 20F29 |
| url | https://arxiv.org/abs/2503.05337 |