Holonomic $A\mathcal{V}$-modules for the affine space
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866912090019594240 |
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| author | Billig, Yuly Rocha, Henrique |
| author_facet | Billig, Yuly Rocha, Henrique |
| contents | We study the growth of representations of the Lie algebra of vector fields on the affine space that admit a compatible action of the polynomial algebra. We establish the Bernstein inequality for these representations, enabling us to focus on modules with minimal growth, known as holonomic modules. We show that simple holonomic modules are isomorphic to the tensor product of a holonomic module over the Weyl algebra and a finite-dimensional $\mathfrak{gl}_n$-module. We also prove that holonomic modules have a finite length and that the representation map associated with a holonomic module is a differential operator. Finally, we present examples illustrating our results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_21121 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Holonomic $A\mathcal{V}$-modules for the affine space Billig, Yuly Rocha, Henrique Representation Theory Rings and Algebras 17B10, 17B66 We study the growth of representations of the Lie algebra of vector fields on the affine space that admit a compatible action of the polynomial algebra. We establish the Bernstein inequality for these representations, enabling us to focus on modules with minimal growth, known as holonomic modules. We show that simple holonomic modules are isomorphic to the tensor product of a holonomic module over the Weyl algebra and a finite-dimensional $\mathfrak{gl}_n$-module. We also prove that holonomic modules have a finite length and that the representation map associated with a holonomic module is a differential operator. Finally, we present examples illustrating our results. |
| title | Holonomic $A\mathcal{V}$-modules for the affine space |
| topic | Representation Theory Rings and Algebras 17B10, 17B66 |
| url | https://arxiv.org/abs/2410.21121 |