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Albert algebras over rings and related torsors

Part of: Jordan algebras Linear algebraic groups and related topics Algebraic groups

Published online by Cambridge University Press:  30 March 2020

Seidon Alsaody
Seidon Alsaody*
Affiliation:
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, ABT6G 2G1, Canada
*
e-mail: [email protected]
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Abstract

We study exceptional Jordan algebras and related exceptional group schemes over commutative rings from a geometric point of view, using appropriate torsors to parametrize and explain classical and new constructions, and proving that over rings, they give rise to nonisomorphic structures.

We begin by showing that isotopes of Albert algebras are obtained as twists by a certain $\mathrm F_4$ -torsor with total space a group of type $\mathrm E_6$ and, using this, that Albert algebras over rings in general admit nonisomorphic isotopes even in the split case, as opposed to the situation over fields. We then consider certain $\mathrm D_4$ -torsors constructed from reduced Albert algebras, and show how these give rise to a class of generalised reduced Albert algebras constructed from compositions of quadratic forms. Showing that this torsor is nontrivial, we conclude that the Albert algebra does not uniquely determine the underlying composition, even in the split case. In a similar vein, we show that a given reduced Albert algebra can admit two coordinate algebras which are nonisomorphic and have nonisometric quadratic forms, contrary, in a strong sense, to the case over fields, established by Albert and Jacobson.

Keywords

Albert algebraJordan algebrahomogeneous spacetorsorisotopecompositionoctonion

MSC classification

Primary: 17C40: Exceptional Jordan structures
Secondary: 20G41: Exceptional groups 14L30: Group actions on varieties or schemes (quotients) 20G10: Cohomology theory
Type
Article
Information
Canadian Journal of Mathematics , Volume 73 , Issue 3 , June 2021 , pp. 875 - 898
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Copyright
© Canadian Mathematical Society 2020

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Footnotes

An extensive part of this work was done while the author was a postdoctoral fellow at Institut Camille Jordan (Lyon, France), supported by the grant KAW 2015.0367 from the Knut and Alice Wallenberg Foundation. Part of this work was done while the author was partially supported by PIMS at the University of Alberta.

References

Alberca Bjerregaard, P., Loos, O., and Martín González, C., Derivations and automorphisms of Jordan algebras in characteristic two . J. Algebra 285(2005), 146–181.CrossRefGoogle Scholar
Albert, A. A. and Jacobson, N., On reduced exceptional simple Jordan algebras . Ann. Math. 66(1957), 400–417.CrossRefGoogle Scholar
Alsaody, S. and Gille, P., Isotopes of octonion algebras, ${\boldsymbol{G}}_2$ -torsors and triality . Adv. Math. 343(2019), 864–909.CrossRefGoogle Scholar
Bott, R. and Samelson, H., Applications of the theory of Morse to symmetric spaces . Amer. J. Math. 80(1958), 964–1029.CrossRefGoogle Scholar
Colliot-Thélène, J.-L. and Sansuc, J.-J., Fibrés quadratiques et composantes connexes réelles . Math. Ann. 244(1979), 105–134.CrossRefGoogle Scholar
Demazure, M. and Gabriel, P., Groupes algébriques. Masson, Paris, 1970.Google Scholar
M. Demazure and A. Grothendieck (eds.), Séminaire de Géométrie algébrique de l’I. H. E. S., 1963–1964, schémas en groupes. Lecture Notes in Math, 151, Springer, New York, 1970, pp. 151–153.Google Scholar
Gille, P., Octonion algebras over rings are not determined by their norms . Canad. Math. Bull. 57(2014), 303–309.CrossRefGoogle Scholar
Gille, P., Sur la classification des schémas en groupes semisimples, “Autour des schémas en groupes, III” . Panoramas et Synthèses 47(2015), 39–110.Google Scholar
Grothendieck, A. (avec la collaboration de J. Dieudonné), Eléments de Géométrie Algébrique IV. Publications mathématiques de l’I.H.É.S. nos. 20, 24, 28 and 32, 1964–1967.CrossRefGoogle Scholar
Jacobson, N., Some groups of transformations defined by Jordan algebras II. J. Reine Angew. Math. 204(1960), 74–98.CrossRefGoogle Scholar
Jacobson, N., Basic algebra I. Freeman, San Francisco, CA, 1974.Google Scholar
Knus, M.-A., Quadratic and Hermitian forms over rings. Grundlehren der Matematischen Wissenschaften, 294, Springer-Verlag, New York, NY, 1991.CrossRefGoogle Scholar
Knus, M.-A., Merkurjev, A., Rost, M., and Tignol, J.-P., The book of involutions. AMS Colloquium Publications, 44, Amer. Math. Soc., Providence, RI, 1998.Google Scholar
Loos, O., On algebraic groups defined by Jordan pairs . Nagoya Math. J. 74(1979), 23–66.CrossRefGoogle Scholar
McCrimmon, K., Axioms for inversion in Jordan algebras . J. Algebra 47(1977), 201–222.CrossRefGoogle Scholar
McCrimmon, K. and Zelmanov, E., The structure of strongly prime quadratic Jordan algebras . Adv. Math. 69(1988), 133–222.CrossRefGoogle Scholar
Mimura, M., The homotopy groups of lie groups of low rank. J. Math. Kyoto Univ. 6(1967), 131–176.Google Scholar
Petersson, H. P., A survey on Albert algebras . Transform. Groups 24(2019), 219–278.CrossRefGoogle Scholar
Petersson, H. P. and Racine, M., Octonions and Albert algebras over commutative rings. Monograph in progress, 2018.Google Scholar
Serre, J.-P., Quelques calculs de groupes d’homotopie . C. R. Acad. Sci. Paris 236(1953), 2475–2477.Google Scholar
Springer, T. A., Jordan algebras and algebraic groups. Ergebnisse der Mathematik un ihre Grenzgebiete, Band 75, Springer-Verlag, New York, NY, 1973.Google Scholar
Springer, T. A. and Veldkamp, F. D., Octonion algebras, Jordan algebras and exceptional groups. Springer Monographs in Mathematics, Springer-Verlag, New York, NY, 2000.CrossRefGoogle Scholar