Infinite dimensional quadratic forms admitting composition

Elduque, Alberto; Pérez, José

doi:10.1090/S0002-9939-97-03799-4

Infinite dimensional quadratic forms admitting composition
HTML articles powered by AMS MathViewer

by Alberto Elduque and José María Pérez PDF

Proc. Amer. Math. Soc. 125 (1997), 2207-2216 Request permission

Abstract:

Composition algebras with a unit element constitute a well-known class of algebras. In this paper, those composition algebras with a one–sided unit element are characterized and examples are given of arbitrary infinite dimension.

References

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
Morgan Ward and R. P. Dilworth, The lattice theory of ova, Ann. of Math. (2) 40 (1939), 600–608. MR 11, DOI 10.2307/1968944
José Antonio Cuenca Mira, On one-sided division infinite-dimensional normed real algebras, Publ. Mat. 36 (1992), no. 2A, 485–488 (1993). MR 1209820, DOI 10.5565/PUBLMAT_{3}62A92_{1}4
Alberto Elduque and Hyo Chul Myung, On flexible composition algebras, Comm. Algebra 21 (1993), no. 7, 2481–2505. MR 1218509, DOI 10.1080/00927879308824688
Mohamed Lamei el-Mallah and Artibano Micali, Sur les dimensions des algèbres absolument valuées, J. Algebra 68 (1981), no. 2, 237–246 (French, with English summary). MR 608533, DOI 10.1016/0021-8693(81)90262-3
J.M. Faillétaz, Compositions d’Espaces Bilinéaires par des Espaces Quadratiques, Doctoral dissertation, Université de Lausanne, 1992.
A. Hurwitz, Uber die Komposition der quadratischen Formen von beliebig vielen Variabeln, Nachr. Ges. Wiss. Göttingen (1898), 309–316.
A. Hurwitz, Uber die Komposition der quadratischen Formen, Math. Ann. 88 (1923), 1–25.
Nathan Jacobson, Structure theory of Jordan algebras, University of Arkansas Lecture Notes in Mathematics, vol. 5, University of Arkansas, Fayetteville, Ark., 1981. MR 634508
J. Junker, Das Hurwitz Problem für quadratische Formen über Körper der Charakteristik 2, Diplom thesis, Saarbrücken, 1980.
Sam Perlis, Maximal orders in rational cyclic algebras of composite degree, Trans. Amer. Math. Soc. 46 (1939), 82–96. MR 15, DOI 10.1090/S0002-9947-1939-0000015-X
T. Y. Lam, The algebraic theory of quadratic forms, Mathematics Lecture Note Series, W. A. Benjamin, Inc., Reading, Mass., 1973. MR 0396410
Julian Ławrynowicz and Jakub Rembieliński, Erratum. Reprint of the article: “On the composition of nondegenerate quadratic forms with an arbitrary index” [Ann. Fac. Sci. Toulouse Math. (5) 10 (1989), no. 1, 141–168], Ann. Fac. Sci. Toulouse Math. (5) 11 (1990), no. 1, 141–168 (English, with English and French summaries). MR 1191477
Hyo Chul Myung, Non-unital composition algebras, Lecture Notes Series, vol. 22, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1994. MR 1286256
J. Radon, Lineare scharen orthogonale Matrizen, Abh. Math. Sem. Univ. Hamburg 1 (1922), 1–14.
Ángel Rodríguez-Palacios, One-sided division absolute valued algebras, Publ. Mat. 36 (1992), no. 2B, 925–954 (1993). MR 1210027, DOI 10.5565/PUBLMAT_{3}62B92_{1}2
Daniel B. Shapiro, Spaces of similarities. I. The Hurwitz problem, J. Algebra 46 (1977), no. 1, 148–170. MR 453636, DOI 10.1016/0021-8693(77)90397-0
Daniel B. Shapiro, Spaces of similarities. I. The Hurwitz problem, J. Algebra 46 (1977), no. 1, 148–170. MR 453636, DOI 10.1016/0021-8693(77)90397-0
Daniel B. Shapiro, Spaces of similarities. IV. $(s,t)$-families, Pacific J. Math. 69 (1977), no. 1, 223–244. MR 476634
—, Composition of Quadratic Forms, (manuscript).
Kazimierz Urbanik and Fred B. Wright, Absolute-valued algebras, Proc. Amer. Math. Soc. 11 (1960), 861–866. MR 120264, DOI 10.1090/S0002-9939-1960-0120264-6
Daniel B. Shapiro, Spaces of similarities. I. The Hurwitz problem, J. Algebra 46 (1977), no. 1, 148–170. MR 453636, DOI 10.1016/0021-8693(77)90397-0
P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
K. A. Zhevlakov, A. M. Slin′ko, I. P. Shestakov, and A. I. Shirshov, Rings that are nearly associative, Pure and Applied Mathematics, vol. 104, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1982. Translated from the Russian by Harry F. Smith. MR 668355
R. Züger, Kompositionen von quadratischen Formen mit quadratischen, bilinearen oder hermiteschen Formen, Doctoral dissertation, Universität Bern, 1995.