De Rham 2-Cohomology of Real Flag Manifolds

Symmetry Integrability and Geometry-Methods and Applications

article

Viviana del Barco¹  Luiz Antonio Barrera San Martin² 

[1] UNR-CONICET;IMECC-UNICAMP

关键词: flag manifold;    cellular homology;    Schubert cell;    de Rham cohomology;    characteristic classes;   

DOI  :  10.3842/SIGMA.2019.051

来源: National Academy of Science of Ukraine

【 摘 要 】

Let $\mathbb{F}_{\Theta }=G/P_{\Theta }$ be a flag manifold associated to a non-compact real simple Lie group $G$ and the parabolic subgroup $P_{\Theta }$. This is a closed subgroup of $G$ determined by a subset $\Theta $ of simple restricted roots of $\mathfrak{g}=\operatorname{Lie}(G)$. This paper computes the second de Rham cohomology group of $\mathbb{F}_\Theta$. We prove that it is zero in general, with some rare exceptions. When it is non-zero, we give a basis of $H^2(\mathbb{F}_\Theta,\mathbb{R})$ through the Weil construction of closed 2-forms as characteristic forms of principal fiber bundles. The starting point is the computation of the second homology group of $\mathbb{F}_{\Theta }$ with coefficients in a ring $R$.

【 授权许可】

Unknown