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Quadratic forms, compositions and triality over \(\mathbb F_1\). (English) Zbl 1343.17006

Authors’ abstract: “According to Tits, the quadric of dimension \(6\) over the “field” \(\mathbb{F}_1\) with one element is a set of \(8\) points structured by a permutation of order \(2\) without fixed points. Subsets that are disjoint from their image under the permutation are the subspaces of the quadric. As in the classical case of hyperbolic quadrics over an arbitrary field, maximal subspaces come in two different types. We define a geometric triality to be a permutation of order \(3\) of the set consisting of points and maximal subspaces, carrying points to maximal subspaces of one type and maximal subspaces of the other type to points while preserving the incidence relations. We show analogues over \(\mathbb{F}_1\) of the one-to-one correspondence between geometric trialities, trialitarian automorphisms of algebraic groups of type \(D_4\), and symmetric composition algebras of dimension \(8\). Here, the algebraic groups of type \(D_4\) are replaced by their Weyl group, which is the semi-direct product \(\mathfrak{S}_2^3\rtimes\mathfrak{S}_4\), and composition algebras by a certain type of binary operation on a quadric to which a \(0\) is adjoined. As in the classical case, we show that there are two types of trialities, one related to octonions and the other to Okubo algebras.”

MSC:

17A75 Composition algebras
12K99 Generalizations of fields
20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures
51E24 Buildings and the geometry of diagrams
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[1] August Weiss, Ernst, Oktaven, Engelscher Komplex, Trialitätsprinzip, Math. Z., 44, 580-611 (1938) · JFM 64.0658.02
[2] Bannai, Eiichi, Automorphisms of irreducible Weyl groups, J. Fac. Sci. Univ. Tokyo Sect. I, 16, 273-286 (1969) · Zbl 0196.05005
[3] Cartan, Élie, Le principe de dualité et la théorie des groupes simples et semi-simples, Bull. Sci. Math., 49, 361-374 (1925) · JFM 51.0322.02
[4] Cartan, Élie, Leçons sur la Théorie des Spineurs, The Theory of Spinors (1981), Hermann: Hermann Paris: Dover Publications Inc.: Hermann: Hermann Paris: Dover Publications Inc. New York, English transl.: · Zbl 0019.36301
[6] Chevalley, Claude C., The Algebraic Theory of Spinors (1954), Columbia University Press: Columbia University Press New York, Also in: Collected Works, Vol. 2, Springer-Verlag, Berlin, 1996 · Zbl 0057.25901
[7] Cohn, Henry, Projective geometry over \(F_1\) and the Gaussian binomial coefficients, Amer. Math. Monthly, 111, 6, 487-495 (2004) · Zbl 1060.05013
[9] Franzsen, William N.; Howlett, Robert B., Automorphisms of nearly finite Coxeter groups, Adv. Geom., 3, 3, 301-338 (2003) · Zbl 1044.20022
[11] Knus, Max-Albert; Merkurjev, Alexander; Rost, Markus; Tignol, Jean-Pierre, (The Book of Involutions. The Book of Involutions, American Mathematical Society Colloquium Publications (1998), American Mathematical Society: American Mathematical Society Providence, RI), With a preface in French by J. Tits · Zbl 0955.16001
[12] Knus, Max-Albert; Tignol, Jean-Pierre, Triality and étale algebras, (Quadratic Forms, Linear Algebraic Groups, and Cohomology. Quadratic Forms, Linear Algebraic Groups, and Cohomology, Dev. Math., vol. 18 (2010), Springer: Springer New York), 259-286 · Zbl 1219.13004
[13] Lorscheid, Oliver; Peña, Javier López, Mapping \(F_1\)-land: an overview of geometries over the field with one element, (Noncommutative Geometry, Arithmetic, and Related Topics (2011), Johns Hopkins Univ. Press: Johns Hopkins Univ. Press Baltimore, MD), 241-265 · Zbl 1271.14003
[14] Study, Ernst, Grundlagen und Ziele der analytischen Kinematik, Sitzungsber. Berl. Math. Ges., 12, 36-60 (1913) · JFM 44.0791.03
[15] Tits, Jacques, Sur les analogues algébriques des groupes semi-simples complexes, (Colloque d’Algèbre Supérieure, tenu à Bruxelles du 19 au 22 Décembre 1956. Colloque d’Algèbre Supérieure, tenu à Bruxelles du 19 au 22 Décembre 1956, Centre Belge de Recherches Mathématiques, Établissements Ceuterick Louvain (1957), Librairie Gauthier-Villars: Librairie Gauthier-Villars Paris), 261-289 · Zbl 0084.15902
[16] Tits, Jacques, Sur la trialité et certains groupes qui s’en déduisent, Publ. Math. Inst. Hautes Études Sci., 2, 14-60 (1959) · Zbl 0088.37204
[17] van der Blij, Frederik; Springer, Tonny A., Octaves and triality, Nieuw Arch. Wisk. (3), 8, 158-169 (1960) · Zbl 0127.11804
[18] Zorn, Max, Theorie der alternativen Ringe, Abh. Math. Seit. Hamb. Univ., 8, 123-147 (1930) · JFM 56.0140.01
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