Varieties of graded $W$-algebras and asymptotic behavior of codimension growth
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909930852712448 |
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| author | Busalacchi, Giovanni Martino, Fabrizio Rizzo, Carla |
| author_facet | Busalacchi, Giovanni Martino, Fabrizio Rizzo, Carla |
| contents | Let $W$ be a $G$-graded algebra over a field of characteristic zero, where $G$ is a finite group. We develope a theory of generalized $G$-graded polynomial identities satisfied by any finite-dimensional $W$-algebra $A$, by mean of the graded multiplier algebra of $A.$ In particular, we first prove that the graded generalized exponent exists and equals the ordinary one. Then, we explicitly compute the $G$-graded generalized identities of $UT_2,$ the $2 \times 2$ upper triangular matrix algebra equipped with its canonical $\mathbb{Z}_2$-grading, under all the possible graded $W$-actions. Finally, we exhibit examples of varieties of graded $W$-algebras with almost polynomial growth of the codimensions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_22602 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Varieties of graded $W$-algebras and asymptotic behavior of codimension growth Busalacchi, Giovanni Martino, Fabrizio Rizzo, Carla Rings and Algebras Primary 16R10, 16R50, 16W50, Secondary 16P90, 16D20 Let $W$ be a $G$-graded algebra over a field of characteristic zero, where $G$ is a finite group. We develope a theory of generalized $G$-graded polynomial identities satisfied by any finite-dimensional $W$-algebra $A$, by mean of the graded multiplier algebra of $A.$ In particular, we first prove that the graded generalized exponent exists and equals the ordinary one. Then, we explicitly compute the $G$-graded generalized identities of $UT_2,$ the $2 \times 2$ upper triangular matrix algebra equipped with its canonical $\mathbb{Z}_2$-grading, under all the possible graded $W$-actions. Finally, we exhibit examples of varieties of graded $W$-algebras with almost polynomial growth of the codimensions. |
| title | Varieties of graded $W$-algebras and asymptotic behavior of codimension growth |
| topic | Rings and Algebras Primary 16R10, 16R50, 16W50, Secondary 16P90, 16D20 |
| url | https://arxiv.org/abs/2511.22602 |