Abstract
The principal objective of the present paper is the study of the automorphisms and groups of automorphisms of composition algebras, that is, the algebras arising from quadratic forms which permit composition. These algebras are mainly quaternion algebras and Cayley algebras. The problem of determining the quadratic forms which permit composition (Huryitz’s problem) has been treated by many authors (2). In spite of this, there does not appear in any one place a complete solution of this problem in its most general form — which amounts to the determination of the algebras for an arbitrary field and not just to the determination of the possible dimensionalities. We give such a solution here for the case of characteristic not two. Aside from its intrinsic interest and applications to other fields (for example Jordan algebras, absolute valued algebras) we have still another reason for treating the Hurvitz problem again, namely: The analysis of the composition algebras is essential for our study of their automorphisms.
This research was supported in part by the Air Force Office of Scientific Research under contract AF 49 (638) 110.
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Bibliography
A. A. Albert, Quadratic froms permitting composition, Ann. of Math., vol. 43 (1942) pp. 161–177.
A. A. Albert and N. Jacobson, On reduced exceptional simple Jordan algebras, Ann. of Math., vol. 66 (1957), pp. 400–417.
E. Artin, Geometric Algebra, New York, 1957.
N. Bourbaki, Éléments de Matématique, Livre II, Algèbre, Paris, 1950.
C. Chevalley, Theory of Lie Groups, Princeton, 1946.
C. Chevalley, Théorie des Groupes de Lie, Paris, 1951.
C. Chevalley, The Algebraic Theory of Spinors, New York, 1954.
C. Chevalley, Sur certains groupes simples, Tohoku Math. J., second series, vol. 7 (1955), pp. 14–66.
L. E. Dickson, Theory of linear groups in en arbitrary field, Trans. Amer. Math. Soc., vol. 2 (1901), pp. 363–394.
J. Dieudonné, Sur les Groupes Classiques, Paris, 1948.
J. Dieudonné, La Géométrie des Groupes Classiques, Ergebn. series, Berlin, 1955.
H. Freudenthal, Okiaven, Ausnahmegruppen und Oktavengeometrie, Utrecht, 1951.
A. Hurvitz, Über die Composition der quadratischen Formen von beliebig vielen Variablen, Gott. Nachrichten, 1898, pp. 309–316.
N. Jacobson, Cayley numbers and normal simple Lie algebras of type G, Duke Math. Jour., vol. 5 (1939), pp. 775–783.
N. Jacobson, Abstract derivation and Lie algebras, Trans. Amer. Math. Soc., vol. 42 (1937), pp. 206–224.
I. Kaplansky, Infinite dimensional quadratic forms permitting composition, Proc. Amer. Math. Soc., vol. 4 (1953), pp. 956–960.
F. Kasch, Über den Automorphismenring einfacher Algebren, Arch. Math., vol. 6 (1954), pp. 59–64.
F. Levi and v. d. Waerden, Über eine bisondere Klasse von Gruppen, Hamb. Abhandl., vol. 9 (1933), pp. 154–158.
Y. V. Linnik, Quaternions and Cayley numbers; some applications of the arithmetic of quaternions, Uspehi Matem. Nauk. vol. 4 no. 5 (1949), pp. 49–98 (Russian).
M. Zorn, Alternativkorpern und quadratische Systeme, Hamb. Abhandl., vol. 9 (1953), pp. 393–402.
M. Zorn, The automorphisms of Cayley’s non-associative algebra, Proc. Nat. Acad. Sci., vol. 21 (1935), pp. 355–358.
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© 1989 Birkhäuser Boston
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Jacobson, N. (1989). Composition Algebras and Their Automorphisms. In: Nathan Jacobson Collected Mathematical Papers. Contemporary Mathematicians. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-3694-8_24
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DOI: https://doi.org/10.1007/978-1-4612-3694-8_24
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