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arXiv · 2509.04698

Super homology groups of differential forms and vector fields on Euclidean line

Kentaro Mikami·Tadayoshi Mizutani
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Abstract

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 $n=1$.

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Kentaro Mikami, Tadayoshi Mizutani. 2025-09-04. Super homology groups of differential forms and vector fields on Euclidean line. https://arxiv.org/abs/2509.04698

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