Hodge type decomposition

Wojciech Koz/lowski¹

¹ Institute of Mathematics Polish Academy of Sciences /Lódź Branch and Faculty of Mathematics /Lódź University Banacha 22 90-238 /Lódź, Poland

Annales Polonici Mathematici, Tome 90 (2007) no. 2, pp. 99-104.

Résumé

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.

Détail

DOI : 10.4064/ap90-2-1

Keywords: space mit lambda polynomial p forms mathbb introduce special inner product mathbf space polynomial p forms which closed co closed prove purely algebraic mit lambda splits direct sum star mit lambda oplus delta star mit lambda p oplus mathbf where star resp delta star denotes adjoint operator resp delta respect inner product