A Cayley-Dickson process for a class of structurable algebras

Allison, B. N.; Faulkner, J. R.

doi:10.1090/S0002-9947-1984-0735416-2

A Cayley-Dickson process for a class of structurable algebras
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by B. N. Allison and J. R. Faulkner

Trans. Amer. Math. Soc. 283 (1984), 185-210

DOI: https://doi.org/10.1090/S0002-9947-1984-0735416-2

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Abstract:

In this paper, we study the class of all simple structurable algebras with the property that the space of skew-hermitian elements has dimension $1$. These algebras with involution have arisen in the study of Lie algebra constructions. The reduced algebras are isotopic to $2 \times 2$ matrix algebras. We study a Cayley-Dickson process for rationally constructing some algebras in the class including division algebras and nonreduced nondivision algebras. An important special case of the process endows the direct sum of two copies of a $28$-dimensional degree $4$ central simple Jordan algebra $\mathcal {B}$ with the structure of an algebra with involution. In preparatory work, we obtain a procedure for giving the space ${\mathcal {B}_0}$ of trace zero elements of any such Jordan algebra $\mathcal {B}$ the structure of a $27$-dimensional exceptional Jordan algebra.

References

B. N. Allison, A class of nonassociative algebras with involution containing the class of Jordan algebras, Math. Ann. 237 (1978), no. 2, 133–156. MR 507909, DOI 10.1007/BF01351677
B. N. Allison, Models of isotropic simple Lie algebras, Comm. Algebra 7 (1979), no. 17, 1835–1875. MR 547712, DOI 10.1080/00927877908822432
B. N. Allison and W. Hein, Isotopes of some nonassociative algebras with involution, J. Algebra 69 (1981), no. 1, 120–142. MR 613862, DOI 10.1016/0021-8693(81)90132-0
Hel Braun and Max Koecher, Jordan-Algebren, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Band 128, Springer-Verlag, Berlin-New York, 1966 (German). MR 0204470
Robert B. Brown, A new type of nonassociative algebras, Proc. Nat. Acad. Sci. U.S.A. 50 (1963), 947–949. MR 158913, DOI 10.1073/pnas.50.5.947
Robert B. Brown, Groups of type $E_{7}$, J. Reine Angew. Math. 236 (1969), 79–102. MR 248185, DOI 10.1515/crll.1969.236.79
J. C. Ferrar, Strictly regular elements in Freudenthal triple systems, Trans. Amer. Math. Soc. 174 (1972), 313–331 (1973). MR 374223, DOI 10.1090/S0002-9947-1972-0374223-1
Hans Freudenthal, Sur le groupe exceptionnel $E_7$, Nederl. Akad. Wetensch. Proc. Ser. A. 56=Indagationes Math. 15 (1953), 81–89 (French). MR 0054609
Nathan Jacobson, Lie algebras, Interscience Tracts in Pure and Applied Mathematics, No. 10, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0143793
Nathan Jacobson, Structure and representations of Jordan algebras, American Mathematical Society Colloquium Publications, Vol. XXXIX, American Mathematical Society, Providence, R.I., 1968. MR 0251099
I. L. Kantor, Models of the exceptional Lie algebras, Dokl. Akad. Nauk SSSR 208 (1973), 1276–1279 (Russian). MR 0349779
K. McCrimmon, Norms and noncommutative Jordan algebras, Pacific J. Math. 15 (1965), 925–956. MR 204477
Kevin McCrimmon, A proof of Schafer’s conjecture for infinite-dimensional forms admitting composition, J. Algebra 5 (1967), 72–83. MR 204478, DOI 10.1016/0021-8693(67)90026-9
Kurt Meyberg, Eine Theorie der Freudenthalschen Tripelsysteme. I, II, Nederl. Akad. Wetensch. Proc. Ser. A 71=Indag. Math. 30 (1968), 162–174, 175–190 (German). MR 0225838
Richard D. Schafer, An introduction to nonassociative algebras, Pure and Applied Mathematics, Vol. 22, Academic Press, New York-London, 1966. MR 0210757
George B. Seligman, Rational methods in Lie algebras, Lecture Notes in Pure and Applied Mathematics, Vol. 17, Marcel Dekker, Inc., New York-Basel, 1976. MR 0427394
T. A. Springer, On a class of Jordan algebras, Nederl. Akad. Wetensch. Proc. Ser. A 62 = Indag. Math. 21 (1959), 254–264. MR 0110739
T. A. Springer, Characterization of a class of cubic forms, Nederl. Akad. Wetensch. Proc. Ser. A 65 = Indag. Math. 24 (1962), 259–265. MR 0138661
J. Tits, Classification of algebraic semisimple groups, Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965) Amer. Math. Soc., Providence, R.I., 1966, pp. 33–62. MR 0224710