Rings of differential operators on singular generalized multi-cusp algebras
The aim of the paper is to study the ring of differential operators \(\mathcal{D}(A(m))\) on the generalized multi-cusp algebra \(A(m)\) where \(m\in \mathbb{N}^n\) (of Krull dimension \(n\)). The algebra \(A(m)\) is singular apart from the single case when \(m=(1, \ldots , 1)\). In this case, the a...
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Lugansk National Taras Shevchenko University
2025
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admjournalluguniveduua-article-23502025-01-19T19:44:59Z Rings of differential operators on singular generalized multi-cusp algebras Bavula, Volodymyr V. Hakami, Khalil the ring of differential operators, the generalized multicusp algebra, the generalized Weyl algebra, the global dimension, the Krull dimension, the Gelfand-Kirillov dimension, simple module, the projective dimension, orbit, the projective resolution The aim of the paper is to study the ring of differential operators \(\mathcal{D}(A(m))\) on the generalized multi-cusp algebra \(A(m)\) where \(m\in \mathbb{N}^n\) (of Krull dimension \(n\)). The algebra \(A(m)\) is singular apart from the single case when \(m=(1, \ldots , 1)\). In this case, the algebra \(A(m)\) is a polynomial algebra in \(n\) variables. So, the \(n\)'th Weyl algebra \(A_n=\mathcal{D}(A(1, \ldots , 1))\) is a member of the family of algebras \(\mathcal{D}(A(m))\). We prove that the algebra \(\mathcal{D}(A(m))\) is a central, simple, \(\mathbb{Z}^n\)-graded, finitely generated Noetherian domain of Gelfand-Kirillov dimension \(2n\). Explicit finite sets of generators and defining relations is given for the algebra \(\mathcal{D}(A(m))\). We prove that the Krull dimension and the global dimension of the algebra \(\mathcal{D}(A(m))\) is \(n\). An analogue of the Inequality of Bernstein is proven. In the case when \(n = 1\), simple \(\mathcal{D}(A(m))\)-modules are classified. Lugansk National Taras Shevchenko University 2025-01-19 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2350 10.12958/adm2350 Algebra and Discrete Mathematics; Vol 38, No 2 (2024): A special issue 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2350/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/2350/1262 Copyright (c) 2025 Algebra and Discrete Mathematics |
| institution |
Algebra and Discrete Mathematics |
| baseUrl_str |
|
| datestamp_date |
2025-01-19T19:44:59Z |
| collection |
OJS |
| language |
English |
| topic |
the ring of differential operators the generalized multicusp algebra the generalized Weyl algebra the global dimension the Krull dimension the Gelfand-Kirillov dimension simple module the projective dimension orbit the projective resolution |
| spellingShingle |
the ring of differential operators the generalized multicusp algebra the generalized Weyl algebra the global dimension the Krull dimension the Gelfand-Kirillov dimension simple module the projective dimension orbit the projective resolution Bavula, Volodymyr V. Hakami, Khalil Rings of differential operators on singular generalized multi-cusp algebras |
| topic_facet |
the ring of differential operators the generalized multicusp algebra the generalized Weyl algebra the global dimension the Krull dimension the Gelfand-Kirillov dimension simple module the projective dimension orbit the projective resolution |
| format |
Article |
| author |
Bavula, Volodymyr V. Hakami, Khalil |
| author_facet |
Bavula, Volodymyr V. Hakami, Khalil |
| author_sort |
Bavula, Volodymyr V. |
| title |
Rings of differential operators on singular generalized multi-cusp algebras |
| title_short |
Rings of differential operators on singular generalized multi-cusp algebras |
| title_full |
Rings of differential operators on singular generalized multi-cusp algebras |
| title_fullStr |
Rings of differential operators on singular generalized multi-cusp algebras |
| title_full_unstemmed |
Rings of differential operators on singular generalized multi-cusp algebras |
| title_sort |
rings of differential operators on singular generalized multi-cusp algebras |
| description |
The aim of the paper is to study the ring of differential operators \(\mathcal{D}(A(m))\) on the generalized multi-cusp algebra \(A(m)\) where \(m\in \mathbb{N}^n\) (of Krull dimension \(n\)). The algebra \(A(m)\) is singular apart from the single case when \(m=(1, \ldots , 1)\). In this case, the algebra \(A(m)\) is a polynomial algebra in \(n\) variables. So, the \(n\)'th Weyl algebra \(A_n=\mathcal{D}(A(1, \ldots , 1))\) is a member of the family of algebras \(\mathcal{D}(A(m))\). We prove that the algebra \(\mathcal{D}(A(m))\) is a central, simple, \(\mathbb{Z}^n\)-graded, finitely generated Noetherian domain of Gelfand-Kirillov dimension \(2n\). Explicit finite sets of generators and defining relations is given for the algebra \(\mathcal{D}(A(m))\). We prove that the Krull dimension and the global dimension of the algebra \(\mathcal{D}(A(m))\) is \(n\). An analogue of the Inequality of Bernstein is proven. In the case when \(n = 1\), simple \(\mathcal{D}(A(m))\)-modules are classified. |
| publisher |
Lugansk National Taras Shevchenko University |
| publishDate |
2025 |
| url |
https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2350 |
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2025-12-02T15:50:10Z |
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2025-12-02T15:50:10Z |
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