The $2\times 2$ upper triangular matrix algebra and its generalized polynomial identities
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866929631189270528 |
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| author | Martino, F. Rizzo, C. |
| author_facet | Martino, F. Rizzo, C. |
| contents | Let $UT_2$ be the algebra of $2\times 2$ upper triangular matrices over a field $F$ of characteristic zero. Here we study the generalized polynomial identities of $UT_2$, i.e., identical relations holding for $UT_2$ regarded as $UT_2$-algebra. We determine a set of two generators of the $T_{UT_2}$-ideal of generalized polynomial identities of $UT_2$ and compute the exact values of the corresponding sequence of generalized codimensions. Moreover, we give a complete description of the space of multilinear generalized identities in $n$ variables in the language of Young diagrams through the representation theory of the symmetric group $S_n$. Finally, we prove that, unlike in the ordinary case, the generalized variety of $UT_2$-algebras generated by $UT_2$ has no almost polynomial growth; nevertheless, we exhibit two distinct generalized varieties of almost polynomial growth. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_02838 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The $2\times 2$ upper triangular matrix algebra and its generalized polynomial identities Martino, F. Rizzo, C. Rings and Algebras 16R10, 16R50 (Primary) 16P90, 20C30 (Secondary) Let $UT_2$ be the algebra of $2\times 2$ upper triangular matrices over a field $F$ of characteristic zero. Here we study the generalized polynomial identities of $UT_2$, i.e., identical relations holding for $UT_2$ regarded as $UT_2$-algebra. We determine a set of two generators of the $T_{UT_2}$-ideal of generalized polynomial identities of $UT_2$ and compute the exact values of the corresponding sequence of generalized codimensions. Moreover, we give a complete description of the space of multilinear generalized identities in $n$ variables in the language of Young diagrams through the representation theory of the symmetric group $S_n$. Finally, we prove that, unlike in the ordinary case, the generalized variety of $UT_2$-algebras generated by $UT_2$ has no almost polynomial growth; nevertheless, we exhibit two distinct generalized varieties of almost polynomial growth. |
| title | The $2\times 2$ upper triangular matrix algebra and its generalized polynomial identities |
| topic | Rings and Algebras 16R10, 16R50 (Primary) 16P90, 20C30 (Secondary) |
| url | https://arxiv.org/abs/2312.02838 |