| Title:
|
Hyperplane section ${\mathbb{O}\mathbb{P}}^2_0$ of the complex Cayley plane as the homogeneous space $\mathrm{F_4/P_4}$ (English) |
| Author:
|
Pazourek, Karel |
| Author:
|
Tuček, Vít |
| Author:
|
Franek, Peter |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
52 |
| Issue:
|
4 |
| Year:
|
2011 |
| Pages:
|
535-549 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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 ${\mathrm{F}_4}^{(-20)}/\P$. (English) |
| Keyword:
|
Cayley plane |
| Keyword:
|
octonionic contact structure |
| Keyword:
|
twistor fibration |
| Keyword:
|
parabolic geometry |
| Keyword:
|
Severi varieties |
| Keyword:
|
hyperplane section |
| Keyword:
|
exceptional geometry |
| MSC:
|
14M17 |
| MSC:
|
32M12 |
| idZBL:
|
Zbl 1249.32019 |
| idMR:
|
MR2863997 |
| . |
| Date available:
|
2011-12-16T14:14:37Z |
| Last updated:
|
2015-02-11 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/141745 |
| . |
| Reference:
|
[1] Armstrong S., Biquard O.: Einstein metrics with anisotropic boundary behaviour.arXiv:0901.1051v1 [math.DG]. MR 2646355 |
| Reference:
|
[2] Atiyah M., Berndt J.: Projective Planes, Severi Varieties and Spheres.Surveys in Differential Geometry, International Press of Boston Inc., 2003. Zbl 1057.53040, MR 2039984 |
| Reference:
|
[3] Baez J.C.: The octonions.Bull. Amer. Math. Soc. 39 (2001), no. 2, 145–205. Zbl 1026.17001, MR 1886087, 10.1090/S0273-0979-01-00934-X |
| Reference:
|
[4] Biquard O.: Asymptotically Symmetric Einstein Metrics.SMF/AMS Texts and Monographs, American Mathematical Society, Providence, 2006. Zbl 1112.53001, MR 2260400 |
| Reference:
|
[5] Bourbaki N.: Lie Groups and Lie Algebras. Chapters 4–6.Elements of Mathematics, Springer, Berlin, 2002. Zbl 1145.17001, MR 1890629 |
| Reference:
|
[6] Čap A., Slovák J.: Parabolic Geometries: Background and General Theory.Mathematical Surveys and Monographs, AMS Bookstore, 2009. MR 2532439 |
| Reference:
|
[7] Dray T., Manogue C.A.: Octonionic Cayley spinors and $\mathrm{E}_6$.Comment. Math. Univ. Carolin. 51 (2010), 193–207. MR 2682473 |
| Reference:
|
[8] Friedrich T.: Weak Spin$(9)$-structures on $16$-dimensional Riemannian manifolds.Asian J. Math. 5 (2001), 129–160; arXiv:math/9912112v1 [math.DG]. Zbl 1021.53028, MR 1868168 |
| Reference:
|
[9] Goodman R., Wallach N.R.: Representations and Invariants of the Classical Groups.Cambridge University Press, Cambridge, 1998. Zbl 1173.22001, MR 1606831 |
| Reference:
|
[10] Harvey F.R.: Spinors and Calibration.Academic Press, San Diego, 1990. MR 1045637 |
| Reference:
|
[11] Held R., Stavrov I., VanKoten B.: (Semi-)Riemannian geometry of (para-)octonionic projective planes.Differential Geom. Appl. 27 (2009), 464–481. Zbl 1175.53064, MR 2547826, 10.1016/j.difgeo.2009.01.007 |
| Reference:
|
[12] Humphreys J.E.: Linear Algebraic Groups.Graduate Texts in Mathematics, 21, Springer, 1975. Zbl 0471.20029, MR 0396773 |
| Reference:
|
[13] Jacobson N.: Structure and Representations of Jordan Algebras.AMS Bookstore, 2008. Zbl 0218.17010 |
| Reference:
|
[14] Jordan P., von Neumann J., Wigner E.: On an algebraic generalization of the quantum mechanical formalism.Ann. of Math. (2) 35 (1934), no. 1, 29–64. Zbl 0008.42103, MR 1503141, 10.2307/1968117 |
| Reference:
|
[15] Krýsl S.: Classification of $1^{st}$ order symplectic spinor operators over contact projective geometries.Differential Geom. Appl. 26 (2008), 553–565. MR 2458281, 10.1016/j.difgeo.2007.11.037 |
| Reference:
|
[16] Krýsl S.: Classification of $\mathfrak{p}$-homomorphisms between higher symplectic spinors.Rend. Circ. Mat. Palermo (2), Suppl. no. 79 (2006), 117–127. MR 2287131 |
| Reference:
|
[17] Krýsl S.: BGG diagrams for contact graded odd dimensional orthogonal geometries.Acta Univ. Carolin. Math. Phys. 45 (2004), no. 1, 67–77. MR 2109695 |
| Reference:
|
[18] Landsberg J.M., Manivel L.: On the projective geometry of rational homogenous varieties.Comment. Math. Helv. 78 (2003), no. 1, 65–100. MR 1966752 |
| Reference:
|
[19] Landsberg J.M., Manivel L.: The projective geometry of Freudenthal's magic square.J. Algebra 239 (2001), 477–512. Zbl 1064.14053, MR 1832903, 10.1006/jabr.2000.8697 |
| Reference:
|
[20] van Leeuwen M.A.A., Cohen A.M., Lisser B.: LiE, A Package for Lie Group Computations.Computer Algebra Nederland, Amsterdam, 1992, available at http://www-math.univ-poitiers.fr/$\sim $maavl/LiE/. |
| Reference:
|
[21] Moufang R.: Alternativkörper und der Satz vom vollständigen Vierseit.Abhandl. Math. Univ. Hamburg 9 (1933), 207–222. Zbl 0007.07205, 10.1007/BF02940648 |
| Reference:
|
[22] Springer T.A., Veldkamp F.D.: Octonions, Jordan Algebras and Exceptional Groups.Springer Monographs in Mathematics, Springer, Berlin, 2000. Zbl 1087.17001, MR 1763974 |
| Reference:
|
[23] Yokota I.: Exceptional Lie groups.arXiv.org:0902.0431 [math.DG], 2009. Zbl 1145.22002 |
| . |
