Hyperplane section ${\mathbb{O}\mathbb{P}}^2_0$ of the complex Cayley plane as the homogeneous space $\mathrm{F_4/P_4}$

Commentationes mathematicae Universitatis Carolinae

Karel Pazourek¹ 

关键词: Cayley plane;    octonionic contact structure;    twistor fibration;    parabolic geometry;    Severi varieties;    hyperplane section;    exceptional geometry;   

DOI  :  

学科分类：物理化学和理论化学

来源: Univerzita Karlova v Praze * Matematicko-Fyzikalni Fakulta / Charles University in Prague, Faculty of Mathematics and Physics

【 摘 要 】

We prove that the exceptional complex Lie group ${\mathrm{F}_4}$ has a transitive action on the hyperplane section of the complex Cayley plane ${\mathbb{O}\mathbb{P}}^2$. Although the result itself is not new, our proof is elementary and constructive. We use an explicit realization of the vector and spin actions of ${\mathrm{Spin}}(9,\mathbb{C})\leq {\mathrm{F}_4}$. Moreover, we identify the stabilizer of the ${\mathrm{F}_4}$-action as a parabolic subgroup ${\mathrm{P}_4}$ (with Levi factor $\mathrm{B_3T_1}$) of the complex Lie group ${\mathrm{F}_4}$. In the real case we obtain an analogous realization of ${\mathrm{F}_4}^{(-20)}/\P$.

【 授权许可】

CC BY