Rings of differential operators on singular generalized multi-cusp algebras
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| Format: | Preprint |
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2023
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| author | Bavula, Volodymyr Hakami, K. |
| author_facet | Bavula, Volodymyr Hakami, K. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2312_17303 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Rings of differential operators on singular generalized multi-cusp algebras Bavula, Volodymyr Hakami, K. Rings and Algebras 16S32, 16E10, 16E05, 16D60, 16P40, 16S35, 16S15, 16P50, 16P90, 16D30 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. |
| title | Rings of differential operators on singular generalized multi-cusp algebras |
| topic | Rings and Algebras 16S32, 16E10, 16E05, 16D60, 16P40, 16S35, 16S15, 16P50, 16P90, 16D30 |
| url | https://arxiv.org/abs/2312.17303 |