Abstract
The notions of the rank or principal polynomial of an associative algebra and the corresponding notions of trace and norm are classical. These notions have been generalized recently by the author ([13,I]) to apply to strictly power associative algebras, and we have renamed these concepts the generic minimal polynomial, trace and norm, since this terminology appears to be more in keeping with present day usage in analogous situations. In our paper we investigated the groups of linear transformations which preserve the norms in special central simple Jordan algebras. This applies to central simple associative algebras as a special case. In a later paper ([13, III]) we studied the norm preserving groups of exceptional central simple Jordan algebras. The groups obtained in this way are generalizations of the complex Lie group E 6 and include certain geometrically defined subgroups of the collineation groups of Cayley planes.
This research has been supported by the U.S. Air Force under grant SAR-G-AFOSR-61-29.
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Bibliography
A. A. Albert: A theory of power associative commutative algebras, Trans. Amer. Math. Soc. 69 (1950), 503–527.
A. A. Albert: A construction of exceptional Jordan division algebras, Ann. of Math. 67 (1958), 1–28.
A. A. Albert and N. Jacobson: Reduced exceptional simple Jordan algebras, Ann. of Math. 66 (1957), 400–417.
G. Ancochea: On semi-automorphisms of division algebras, Ann. of Math. 48 (1947), 147–154.
M. Bôcher: Introduction to Higher Algebra, New York, Macmillan, 1929.
C. Chevalley: Théorie des Groupes de Lie, Tome II, Paris, Hermann, 1951.
P. M. Cohn: Special Jordan algebras, Canad. J. Math. 6 (1954), 253–264.
J. Dieudonné: Sur le polynôme principal d’une algèbre. Arch. Math, 8 (1957), 81–84,
H. Flanders: The norm function of an algebraic field extension, I, Pacific J. Math. 3 (1953), 103–112
II, ibid. 5 (1955), 519–528.
N. Jacobson: An application of E. H. Moores’ determinant of a hermitian matrix, Bull. Amer. Math. Soc. 45 (1939), 745–748.
N. Jacobson: Isomorphism of Jordan rings, Amer. J. Math. 70 (1948), 317–326.
N. Jacobson: Derivation algebras and multiplication algebras of semi-simple Jordan algebras, Ann. of Math. 50 (1949), 866–874.
N. Jacobson: Some groups of linear transformations defined by Jordan algebras, I, J. Reine Angew, Math. 201 (1959), 178–195
II, ibid. 204 (1960), 74–98
III, ibid. 207 (1961), 61–95.
N. Jacobson: MacDonald’s theorem on Jordan algebras, Arch. Math. 13 (1962), 241–250.
N. Jacobson: A coordinatization theorem for Jordan algebras, Proc. Nat. Acad. Sci. U.S.A. 48 (1962), 1154–1160.
N. Jacobson and F. D. Jacobson: Classification and representation of semi-simple Jordan algebras, Trans. Amer. Math. Soc. 65 (1949), 141–169.
M. Koecher: Jordan algebras and their applications, Multilithed notes, University of Minnesota, 1962.
I. G. MacDonald: Jordan algebras with three generators, Proc. London Math. Soc. III, 10 (1960), 395–408.
R. D. Schafer: On forms of degree n permitting composition, forthcoming in J. Math, and Mech.
A. I. Shirshov: On special J-rings, Mat. Sb. 80 (1960), 149–166 (in Russian).
J. Tits: forthcoming in Proc. Amer. Math. Soc.
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Dedicated to Professor K. Shoda on his sixtieth birthday
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© 1989 Birkhäuser Boston
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Jacobson, N. (1989). Generic Norm of an Algebra. In: Nathan Jacobson Collected Mathematical Papers. Contemporary Mathematicians. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-3694-8_33
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DOI: https://doi.org/10.1007/978-1-4612-3694-8_33
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