Hodge type decomposition
Tom 90 / 2007
Annales Polonici Mathematici 90 (2007), 99-104
MSC: 33C55, 35J99, 53C43.
DOI: 10.4064/ap90-2-1
Streszczenie
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.
