Two examples of affine homogeneous varieties
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The aim is to describe two homogeneous affine open sets in some projective spaces. The sets are well known, their groups of automorphisms contain simple exceptional groups of types \(E_6\) or \(E_7\), although the total groups of automorphisms are infinite-dimensional and both the sets are flexible. We prove existence of automorphisms not belonging to the connected components of unity, construct extended Weyl groups including these non-tame automorphisms. Our methods are based on classical combinatorics associated with 27 lines on non-singular cubic surfaces and with 56 exceptional curves on Del Pezzo surfaces of degree 2. Some traditional applications of Jordan algebras are used.
KeywordsFlexible set Exceptional groups \(E_6 , E_7\) Jordan algebra Weyl group
Mathematics Subject Classification14R10 14R20 14L30 17B22 20G41
First of all the author would like to express his gratitude to Ernest Vinberg, who attracted the attention of listeners to the subject during his lectures at the Moscow conference “Lie algebras, Algebraic Groups and Theory of Invariants” (January 30–February 4, 2017). His lecture from February, 4 was devoted to exceptional groups and the cubic form \(F_3\). The author also wishes to thank Ivan Cheltsov who encouraged to contribute to the collection dedicated to William Edge. The author is truly grateful to an anonymous referee for his/her efforts, remarks and comments.
- 7.Freudenthal, H.: Oktaven, Ausnahmegruppen und Oktavengeometrie. In: Springer, T.A., van Dalen, D. (eds.) Hans Freudenthal Selecta, pp. 214–269. European Mathematical Society (EMS), Zürich (2009)Google Scholar
- 8.Freudenthal, H.: Oktaven, Ausnahmegruppen und Oktavengeometrie (an extended Russian translation by B.A. Rosenfeld). Matematika 1(1), 117–153 (1957)Google Scholar
- 10.Freudenthal, H.: Zur ebenen Oktavengeometrie. In: Springer, T.A., van Dalen, D. (eds.) Hans Freudenthal Selecta, pp. 288–293. European Mathematical Society (EMS), Zürich (2009)Google Scholar
- 11.Henderson, A.: The Twenty-Seven Lines Upon the Cubic Surface. Ph.D. Thesis, University of Chicago (1915)Google Scholar
- 12.Jacobson, N.: Structure and Representations of Jordan Algebras. American Mathematical Society Colloquium Publications, vol. 39. American Mathematical Society, Providence (1968)Google Scholar
- 14.Manin, Yu.I.: Cubic Forms. Nauka, Moscow (1972); English translation by M. Hazewinlel, North-Holland Mathematical Library, vol. 4. North-Holland, Amsterdam (1986)Google Scholar
- 17.Salmon, G.: A Treatise on the Higher Plane Curves, 3rd edn. Hodges, Foster, Dublin (1879)Google Scholar