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A CHARACTERIZATION OF (−1, −1)-FREUDENTHAL–KANTOR TRIPLE SYSTEMS

Published online by Cambridge University Press:  01 August 2011

NORIAKI KAMIYA,
DANIEL MONDOC and
SUSUMU OKUBO
NORIAKI KAMIYA
Affiliation:
Center for Mathematical Sciences, University of Aizu, 965-8580 Aizuwakamatsu, Japan e-mail: kamiya@u-aizu.ac.jp
DANIEL MONDOC
Affiliation:
Centre for Mathematical Sciences, Lund University, 22 100 Lund, Sweden e-mail: Daniel.Mondoc@math.lu.se
SUSUMU OKUBO
Affiliation:
Department of Physics, University of Rochester, Rochester, NY 14627, USA e-mail: okubo@pas.rochester.edu
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Abstract

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In this paper, we discuss a connection between (−1, −1)-Freudenthal–Kantor triple systems, anti-structurable algebras, quasi anti-flexible algebras and give examples of such structures. The paper provides the correspondence and characterization of a bilinear product corresponding a triple product.

Keywords

17A4017B60

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 53 , Issue 3 , September 2011, pp. 727 - 738
Copyright
Copyright © Glasgow Mathematical Journal Trust 2011

References

REFERENCES

1.Allison, B. N., A class of nonassociative algebras with involution containing the class of Jordan algebras, Math. Ann. 237 (2) (1978), 133–156.CrossRefGoogle Scholar
2.Allison, B. N., Models of isotropic simple Lie algebras, Commun. Algebra 7 (17) (1979), 1835–1875.CrossRefGoogle Scholar
3.Asano, H. and Yamaguti, K., A construction of Lie algebras by generalized Jordan triple systems of second order, Nederl. Akad. Wetensch. Indag. Math. 42 (3)(1980), 249–253.CrossRefGoogle Scholar
4.Asano, H., Classification of non-compact real simple generalized Jordan triple systems of the second kind, Hiroshima Math. J. 21 (1991), 463–489.CrossRefGoogle Scholar
5.Bertram, W., Complex and quaternionic structures on symmetric spaces – correspondence with Freudenthal–Kantor triple systems, in Theory of Lie groups and manifolds (Miyaoka, R. and Tamaru, H., Editors), Sophia Kokyuroku in Math., Vol. 45 (Sophia University, 2002), pp. 61–80.Google Scholar
6.Elduque, A., Kamiya, N. and Okubo, S., Simple (−1, −1) balanced Freudenthal–Kantor triple systems, Glasg. Math. J. 11 (2)(2003), 353–372.CrossRefGoogle Scholar
7.Elduque, A., Kamiya, N. and Okubo, S., (−1, −1) balanced Freudenthal–Kantor triple systems and noncommutative Jordan algebras, J. Algebra 294 (1) (2005), 19–40.CrossRefGoogle Scholar
8.Faulkner, J. R., Structurable triples, Lie triples, and symmetric spaces, Forum Math. 6 (1994), 637–50.CrossRefGoogle Scholar
9.Frappat, L., Sciarrino, A. and Sorba, P., Dictionary on Lie algebras and superalgebras (Academic Press, San Diego, CA, 2000), xxii+410 pp.Google Scholar
10.Freudenthal, H., Beziehungen der E 7 und E 8 zur Oktavenebene, I. Indag. Math. (1954), no. 16, 218–230.CrossRefGoogle Scholar
11.Jacobson, N., Lie and Jordan triple systems, Amer. J. Math. 71 (1949), 149–170.CrossRefGoogle Scholar
12.Jacobson, N., Structure and representations of Jordan algebras, Amer. Math. Soc. Colloq. Publ., Vol. 39 (Amer. Math. Soc., Providence, RI, 1968), x + 453 pp.Google Scholar
13.Kac, V. G., Lie superalgebras, Adv. Math. 26 (1) (1977), 8–96.CrossRefGoogle Scholar
14.Kamiya, N., A structure theory of Freudenthal–Kantor triple systems, J. Algebra 110 (1) (1987), 108–123.CrossRefGoogle Scholar
15.Kamiya, N., A construction of anti-Lie triple systems from a class of triple systems, Mem. Fac. Sci. Shimane Univ. 22 (1988), 51–62.Google Scholar
16.Kamiya, N., A structure theory of Freudenthal–Kantor triple systems. II. Comment, Math. Univ. St. Paul. 38 (1) (1989), 41–60.Google Scholar
17.Kamiya, N., On (ϵ, δ)-Freudenthal–Kantor triple systems, in Nonassociative algebras and related topics (Hiroshima, 1990), 65–75 (World Scientific, River Edge, NJ, 1991).Google Scholar
18.Kamiya, N., The construction of all simple Lie algebras over ℂ from balanced Freudenthal–Kantor triple systems, Contributions to general algebra, 7 (Vienna, 1990), 205–213, Hölder-Pichler-Tempsky, Vienna, 1991.Google Scholar
19.Kamiya, N., On Freudenthal–Kantor triple systems and generalized structurable algebras, Non-associative algebra and its applications (Oviedo, 1993), 198–203, Math. Appl., 303 (Kluwer, Dordrecht, 1994).CrossRefGoogle Scholar
20.Kamiya, N., On the Peirce decompositions for Freudenthal–Kantor triple systems, Commun. Algebra 25 (6) (1997), 1833–1844.CrossRefGoogle Scholar
21.Kamiya, N., On a realization of the exceptional simple graded Lie algebras of the second kind and Freudenthal–Kantor triple systems, Bull. Polish Acad. Sci. Math. 46 (1) (1998), 55–65.Google Scholar
22.Kamiya, N. and Okubo, S., On δ-Lie supertriple systems associated with (ϵ, δ)-Freudenthal–Kantor supertriple systems, Proc. Edinburgh Math. Soc. 43 (2) (2000), 243–260.CrossRefGoogle Scholar
23.Kamiya, N. and Okubo, S., A construction of Jordan superalgebras from Jordan-Lie triple systems, in Non-Associative Algebra and Its Applications (Lecture Notes in Pure and Applied Mathematics), Vol. 211 (Costa, R., Grishkov, A., Guzzo, H. and Peresi, L., Editors) (Marcel Dekker, New York, 2002), pp. 171–176.Google Scholar
24.Kamiya, N. and Okubo, S., A construction of simple Jordan superalgebra of F type from a Jordan–Lie triple system, Ann. Mat. Pura Appl. (4) 181 (3) (2002), 339–348.CrossRefGoogle Scholar
25.Kamiya, N. and Okubo, S., Construction of Lie superalgebras D(2, 1; α), G(3) and F(4) from some triple systems, Proc. Edinburgh Math. Soc. 46 (1) (2003), 87–98.CrossRefGoogle Scholar
26.Kamiya, N. and Okubo, S., On generalized Freudenthal–Kantor triple systems and Yang–Baxter equations, Proc. 24th Inter. Colloq. Group Theoretical Methods in Physics (Gazeau, J-P., Kerner, R., Antoine, J-P., Metens, S. and Thibon, J-Y., Editors) (IPCS, Paris, 2003) Vol. 173 (2003), 815–818.Google Scholar
27.Kamiya, N. and Okubo, S., A construction of simple Lie superalgebras of certain types from triple systems, Bull. Austral. Math. Soc. 69 (1) (2004), 113–123.CrossRefGoogle Scholar
28.Kamiya, N., Examples of Peirce decomposition of generalized Jordan triple system of second order-Balanced cases, in Noncommutative geometry and representation theory in mathematical physics, Contemp. Math., Vol. 391 (Fuchs, J., Mickelsson, J., Rosenblioum, G., Stolin, A. and Westerberg, A., Editors) (AMS, Providence, RI, 2005), 157–165.CrossRefGoogle Scholar
29.Kamiya, N. and Okubo, S., Composition, quadratic, and some triple systems, in Non-associative algebra and its applications, Lecture Notes Pure Appl. Math., Vol. 246 (Sabinin, L., Sbitneva, L. and Shestakov, I., Editors) (Chapman & Hall, Boca Raton, FL, 2006), 205–231.Google Scholar
30.Kamiya, N. and Mondoc, D., A new class of nonassociative algebras with involution, Proc. Japan Acad. Ser. A 84 (5) (2008), 68–72.Google Scholar
31.Kamiya, N., Mondoc, D. and Okubo, S., A structure theory of (−1, −1)-Freudenthal– Kantor triple systems, Bull. Australian Math. Soc. 81 (2010), 132–155.CrossRefGoogle Scholar
32.Kamiya, N. and Mondoc, D., On anti-structurable algebras and extended Dynkin diagrams, J. Gen. Lie Theory Appl. 3 (3) (2009), 185–192.CrossRefGoogle Scholar
33.Kaneyuki, S. and Asano, H., Graded Lie algebras and generalized Jordan triple systems, Nagoya Math. J. 112 (1988), 81–115.CrossRefGoogle Scholar
34.Kantor, I. L., Graded Lie algebras, Trudy Sem. Vect. Tens. Anal. 15 (1970), 227–266.Google Scholar
35.Kantor, I. L., Some generalizations of Jordan algebras, Trudy Sem. Vect. Tens. Anal. 16 (1972), 407–499.Google Scholar
36.Kantor, I. L., Models of exceptional Lie algebras, Soviet Math. Dokl. 14 (1) (1973), 254–258.Google Scholar
37.Kantor, I. L., A generalization of the Jordan approach to symmetric Riemannian spaces, in The Monster and Lie algebras (Columbus, OH, 1996), Ohio State Univ. Math. Res. Inst. Publ., 7 (Ferrar, J. and Harada, K., Editors) (de Gruyter, Berlin, 1998), 221–234.CrossRefGoogle Scholar
38.Kantor, I. L. and Kamiya, N., A Peirce decomposition for generalized Jordan triple systems of second order, Commun. Algebra 31 (12) (2003), 5875–5913.CrossRefGoogle Scholar
39.Koecher, M., Embedding of Jordan algebras into Lie algebras I, Amer. J. Math 89 (1967), 787–816.CrossRefGoogle Scholar
40.Koecher, M., Embedding of Jordan algebras into Lie algebras II, Amer. J. Math 90 (1968), 476–510.CrossRefGoogle Scholar
41.Lister, W. G., A structure theory of Lie triple systems, Trans. Amer. Math. Soc. 72 (1952), 217–242.CrossRefGoogle Scholar
42.Meyberg, K., Lectures on algebras and triple systems, Lecture Notes (The University of Virginia, Charlottesville, VA, 1972), v + 226 pp.Google Scholar
43.Mondoc, D., Models of compact simple Kantor triple systems defined on a class of structurable algebras of skew-dimension one, Commun. Algebra 34 (10) (2006), 3801–3815.CrossRefGoogle Scholar
44.Mondoc, D., On compact realifications of exceptional simple Kantor triple systems, J. Gen. Lie Theory Appl. 1 (1) (2007), 29–40.CrossRefGoogle Scholar
45.Mondoc, D., Compact realifications of exceptional simple Kantor triple systems defined on tensor products of composition algebras, J. Algebra 307 (2) (2007), 917–929.CrossRefGoogle Scholar
46.Mondoc, D., Compact exceptional simple Kantor triple systems defined on tensor products of composition algebras, Commun. Algebra 35 (11) (2007), 3699–3712.CrossRefGoogle Scholar
47.Neher, E., Jordan triple systems by the grid approach, Lecture Notes in Mathematics, Vol. 1280 (Springer, Berlin, 1987), xii + 193 pp.CrossRefGoogle Scholar
48.Okubo, S., Introduction to octonion and other non-associative algebras in physics, Montroll Memorial Lecture Series in Mathematical Physics, 2 (Cambridge University Press, Cambridge, MA, 1995), xii + 136 pp.CrossRefGoogle Scholar
49.Okubo, S. and Kamiya, N., Jordan–Lie superalgebra and Jordan–Lie triple system, J. Algebra 198 (2) (1997), 388–411.CrossRefGoogle Scholar
50.Okubo, S. and Kamiya, N., Quasi-classical Lie superalgebras and Lie supertriple systems, Commun. Algebra 30 (8) (2002), 3825–3850.CrossRefGoogle Scholar
51.Okubo, S., Symmetric triality relations and structurable algebras, Linear Algebra Appl. 396 (2005), 189–222.CrossRefGoogle Scholar
52.Scheunert, M., The theory of Lie superalgebras. An introduction. Lecture Notes in Mathematics, Vol. 716 (Springer, Berlin, 1979), x + 271 pp.CrossRefGoogle Scholar
53.Tits, J., Une classe d'algèbres de Lie en relation avec les algèbres de Jordan, Nederl. Acad. Wetensch. Proc. Ser. A 65 (1962), 530–535.Google Scholar
54.Yamaguti, K. and Ono, A., On representations of Freudenthal–Kantor triple systems U(ϵ, δ), Bull. Fac. School Ed. Hiroshima Univ. 7 (2) (1984), 43–51.Google Scholar
55.Zhevlakov, K. A., Slinko, A. M., Shestakov, I. P. and Shirshov, A. I., Rings that are nearly associative (Academic Press, New York, 1982), xi + 371 pp.Google Scholar
56.Zelmanov, E., Primary Jordan triple systems, Sibirsk. Mat. Zh. 24 (4) (1983), 23–37.Google Scholar