Hodge type decomposition

Volume 90 / 2007

Wojciech Koz/lowski Annales Polonici Mathematici 90 (2007), 99-104 MSC: 33C55, 35J99, 53C43. DOI: 10.4064/ap90-2-1


In the space ${\mit\Lambda}^p$ of polynomial $p$-forms in $\mathbb{R}^n$ we introduce some special inner product. Let $\mathbf{H}^p$ be the space of polynomial $p$-forms which are both closed and co-closed. We prove in a purely algebraic way that ${\mit\Lambda}^p$ splits as the direct sum $d^\star({\mit\Lambda}^{p+1})\oplus \delta^\star({\mit\Lambda}^{p-1}) \oplus \mathbf{H}^p$, where $d^\star$ (resp. $\delta^\star$) denotes the adjoint operator to $d$ (resp. $\delta$) with respect to that inner product.


  • Wojciech Koz/lowskiInstitute of Mathematics
    Polish Academy of Sciences
    /Lódź Branch
    Faculty of Mathematics
    /Lódź University
    Banacha 22
    90-238 /Lódź, Poland

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