Super homology groups of differential forms and vector fields on Euclidean line
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915480198971392 |
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| author | Mikami, Kentaro Mizutani, Tadayoshi |
| author_facet | Mikami, Kentaro Mizutani, Tadayoshi |
| contents | The arXiv:2105.09738 claims several stuffs. In particular, we recall the following two. (1) Vector fields and differential forms become a Lie superalgebra structure for each manifold. (2) For an n-dimensional Euclidean space, vector fields and differential forms with polynomial coefficients become a double weighted Lie uperalgebra. By using Euler vector field, the Betti numbers are 0 except the last one if the primary weight and the secondary weight are different. Now, a simple question arises: What happens when the primary weight and the secondary weight are equal? This note shall give a complete answer to the question for the case $n=1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_04698 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Super homology groups of differential forms and vector fields on Euclidean line Mikami, Kentaro Mizutani, Tadayoshi Differential Geometry 17B10, 17B15 The arXiv:2105.09738 claims several stuffs. In particular, we recall the following two. (1) Vector fields and differential forms become a Lie superalgebra structure for each manifold. (2) For an n-dimensional Euclidean space, vector fields and differential forms with polynomial coefficients become a double weighted Lie uperalgebra. By using Euler vector field, the Betti numbers are 0 except the last one if the primary weight and the secondary weight are different. Now, a simple question arises: What happens when the primary weight and the secondary weight are equal? This note shall give a complete answer to the question for the case $n=1$. |
| title | Super homology groups of differential forms and vector fields on Euclidean line |
| topic | Differential Geometry 17B10, 17B15 |
| url | https://arxiv.org/abs/2509.04698 |