Polynomial Identities and Codimensions of Two- and Three-Dimensional Metabelian Non-Lie Leibniz Algebras
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908708658741248 |
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| author | Fertunani, Luis Fideles, Claudemir Muniz, Airton |
| author_facet | Fertunani, Luis Fideles, Claudemir Muniz, Airton |
| contents | Over an arbitrary field, we conduct a comprehensive study of the polynomial identities and codimensions of two- and three-dimensional metabelian non-Lie Leibniz algebras. In addition, we compute the images of multihomogeneous polynomials on two-dimensional Leibniz algebras and, as a consequence, prove that the image of any multilinear polynomial evaluated on such algebras is always a vector space. Our analysis includes the three nontrivial isomorphism classes in dimension two and the ten isomorphism classes in dimension three, all of which are metabelian. In particular, we determine finite bases for their corresponding $T$-ideals and provide explicit bases for the associated relatively free graded algebras. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_12282 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Polynomial Identities and Codimensions of Two- and Three-Dimensional Metabelian Non-Lie Leibniz Algebras Fertunani, Luis Fideles, Claudemir Muniz, Airton Rings and Algebras 17A32 (16R10, 17A30, 17B01) Over an arbitrary field, we conduct a comprehensive study of the polynomial identities and codimensions of two- and three-dimensional metabelian non-Lie Leibniz algebras. In addition, we compute the images of multihomogeneous polynomials on two-dimensional Leibniz algebras and, as a consequence, prove that the image of any multilinear polynomial evaluated on such algebras is always a vector space. Our analysis includes the three nontrivial isomorphism classes in dimension two and the ten isomorphism classes in dimension three, all of which are metabelian. In particular, we determine finite bases for their corresponding $T$-ideals and provide explicit bases for the associated relatively free graded algebras. |
| title | Polynomial Identities and Codimensions of Two- and Three-Dimensional Metabelian Non-Lie Leibniz Algebras |
| topic | Rings and Algebras 17A32 (16R10, 17A30, 17B01) |
| url | https://arxiv.org/abs/2512.12282 |