Differential identities of matrix algebras
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866911797926166528 |
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| author | Brox, Jose Rizzo, Carla |
| author_facet | Brox, Jose Rizzo, Carla |
| contents | We study the differential identities of the algebra $M_k(F)$ of $k\times k$ matrices over a field $F$ of characteristic zero when its full Lie algebra of derivations, $L=\mbox{Der}(M_k(F))$, acts on it. We determine a set of 2 generators of the ideal of differential identities of $M_k(F)$ for $k\geq 2$. Moreover, we obtain the exact values of the corresponding differential codimensions and differential cocharacters. Finally we prove that, unlike the ordinary case, the variety of differential algebras with $L$-action generated by $M_k(F)$ has almost polynomial growth for all $k\geq 2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_09337 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Differential identities of matrix algebras Brox, Jose Rizzo, Carla Rings and Algebras Primary 16R10, 16R50, 17B10, Secondary 16W25, 16P90, 16S30, 17B35, 17B20, 16G30, 15B30 We study the differential identities of the algebra $M_k(F)$ of $k\times k$ matrices over a field $F$ of characteristic zero when its full Lie algebra of derivations, $L=\mbox{Der}(M_k(F))$, acts on it. We determine a set of 2 generators of the ideal of differential identities of $M_k(F)$ for $k\geq 2$. Moreover, we obtain the exact values of the corresponding differential codimensions and differential cocharacters. Finally we prove that, unlike the ordinary case, the variety of differential algebras with $L$-action generated by $M_k(F)$ has almost polynomial growth for all $k\geq 2$. |
| title | Differential identities of matrix algebras |
| topic | Rings and Algebras Primary 16R10, 16R50, 17B10, Secondary 16W25, 16P90, 16S30, 17B35, 17B20, 16G30, 15B30 |
| url | https://arxiv.org/abs/2403.09337 |