Hyperplane section 𝕆 ℙ 0 2 of the complex Cayley plane as the homogeneous space F 4 / P 4

Karel Pazourek; Vít Tuček; Peter Franek

Commentationes Mathematicae Universitatis Carolinae (2011)

  • Volume: 52, Issue: 4, page 535-549
  • ISSN: 0010-2628

Abstract

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We prove that the exceptional complex Lie group F 4 has a transitive action on the hyperplane section of the complex Cayley plane 𝕆 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 Spin ( 9 , ℂ ) ≤ F 4 . Moreover, we identify the stabilizer of the F 4 -action as a parabolic subgroup P 4 (with Levi factor B 3 T 1 ) of the complex Lie group F 4 . In the real case we obtain an analogous realization of F 4 ( - 20 ) / ℙ .

How to cite

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Pazourek, Karel, Tuček, Vít, and Franek, Peter. "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 52.4 (2011): 535-549. <http://eudml.org/doc/246265>.

@article{Pazourek2011,
abstract = {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 $\{\mathrm \{Spin\}\}(9,\mathbb \{C\})\le \{\mathrm \{F\}_4\}$. 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 $\{\mathrm \{F\}_4\}^\{(-20)\}/\mathbb \{P\}$.},
author = {Pazourek, Karel, Tuček, Vít, Franek, Peter},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Cayley plane; octonionic contact structure; twistor fibration; parabolic geometry; Severi varieties; hyperplane section; exceptional geometry; Cayley plane; octonionic contact structure; twistor fibration; parabolic geometry; Severi variety; hyperplane section; exceptional geometry},
language = {eng},
number = {4},
pages = {535-549},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Hyperplane section $\{\mathbb \{O\}\mathbb \{P\}\}^2_0$ of the complex Cayley plane as the homogeneous space $\mathrm \{F_4/P_4\}$},
url = {http://eudml.org/doc/246265},
volume = {52},
year = {2011},
}

TY - JOUR
AU - Pazourek, Karel
AU - Tuček, Vít
AU - Franek, Peter
TI - Hyperplane section ${\mathbb {O}\mathbb {P}}^2_0$ of the complex Cayley plane as the homogeneous space $\mathrm {F_4/P_4}$
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2011
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 52
IS - 4
SP - 535
EP - 549
AB - 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 ${\mathrm {Spin}}(9,\mathbb {C})\le {\mathrm {F}_4}$. 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 ${\mathrm {F}_4}^{(-20)}/\mathbb {P}$.
LA - eng
KW - Cayley plane; octonionic contact structure; twistor fibration; parabolic geometry; Severi varieties; hyperplane section; exceptional geometry; Cayley plane; octonionic contact structure; twistor fibration; parabolic geometry; Severi variety; hyperplane section; exceptional geometry
UR - http://eudml.org/doc/246265
ER -

References

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