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A STRUCTURE THEORY OF (−1,−1)-FREUDENTHAL KANTOR TRIPLE SYSTEMS

Part of: General nonassociative rings Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  02 October 2009

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

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In this paper we discuss the simplicity criteria of (−1,−1)-Freudenthal Kantor triple systems and give examples of such triple systems, from which we can construct some Lie superalgebras. We also show that we can associate a Jordan triple system to any (ε,δ)-Freudenthal Kantor triple system. Further, we introduce the notion of δ-structurable algebras and connect them to (−1,δ)-Freudenthal Kantor triple systems and the corresponding Lie (super)algebra construction.

Keywords

Lie superalgebrastriple systems

MSC classification

Secondary: 17A40: Ternary compositions 17B60: Lie (super)algebras associated with other structures (associative, Jordan, etc.)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 81 , Issue 1 , February 2010 , pp. 132 - 155
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2009

Footnotes

The first author’s research was partially supported by Grant-in-Aid for Scientific Research (No. 19540042 (C),(2)), Japan Society for the Promotion of Science. The second author’s research was partially supported by Japan Society for the Promotion of Science (No. PE 07601). The third author’s research was supported by U.S. Department of Energy Grant No. DE-FG02-91ER40685.

References

[1]Allison, B. N., ‘A class of nonassociative algebras with involution containing the class of Jordan algebras’, Math. Ann. 237 (1978), 133–156.CrossRefGoogle Scholar
[2]Allison, B. N., ‘Models of isotropic simple Lie algebras’, Comm. Algebra 7 (1979), 1835–1875.CrossRefGoogle Scholar
[3]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
[4]Asano, H. and Yamaguti, K., ‘A construction of Lie algebras by generalized Jordan triple systems of second order’, Nederl. Akad. Wetensch. Indag. Math. 42 (1980), 249–253.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, Sophia Kokyuroku in Mathematics, 45 (eds. Miyaoka, R. and Tamaru, H.) (Sophia University, Tokyo, 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 (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 (2005), 19–40.CrossRefGoogle Scholar
[8]Faulkner, J. R., ‘Structurable triples, Lie triples, and symmetric spaces’, Forum Math. 6 (1994), 637–650.CrossRefGoogle Scholar
[9]Frappat, L., Sciarrino, A. and Sorba, P., Dictionary on Lie Algebras and Superalgebras (Academic Press, San Diego, CA, 2000).Google Scholar
[10]Jacobson, N., ‘Lie and Jordan triple systems’, Amer. J. Math. 71 (1949), 149–170.CrossRefGoogle Scholar
[11]Jacobson, N., Structure and Representations of Jordan Algebras, American Mathematical Society Colloquium Publications, XXXIX (American Mathematical Society, Providence, RI, 1968).CrossRefGoogle Scholar
[12]Kac, V. G., ‘Lie superalgebras’, Adv. Math. 26 (1977), 8–96.CrossRefGoogle Scholar
[13]Kamiya, N., ‘A structure theory of Freudenthal–Kantor triple systems’, J. Algebra 110 (1987), 108–123.CrossRefGoogle Scholar
[14]Kamiya, N., ‘A construction of anti-Lie triple systems from a class of triple systems’, Mem. Fac. Sci. Eng. Shimane Univ. 22 (1988), 51–62.Google Scholar
[15]Kamiya, N., ‘A structure theory of Freudenthal–Kantor triple systems. II’, Comment. Math. Univ. St. Pauli. 38 (1989), 41–60.Google Scholar
[16]Kamiya, N., ‘On (ϵ,δ)-Freudenthal–Kantor triple systems’, in: Nonassociative Algebras and Related Topics (Hiroshima, 1990) (World Scientific Publications, River Edge, NJ, 1991), pp. 65–75.Google Scholar
[17]Kamiya, N., The Construction of All Simple Lie Algebras over ℂ from Balanced Freudenthal–Kantor Triple Systems, Contributions to General Algebra (Vienna, 1990), 7 (Hölder–Pichler–Tempsky, Vienna, 1991), pp. 205–213.Google Scholar
[18]Kamiya, N., ‘On Freudenthal–Kantor triple systems and generalized structurable algebras’, in: Non-associative Algebra and its Applications (Oviedo, 1993), Mathematiques & Applications, 303 (Kluwer Academic Publications, Dordrecht, 1994), pp. 198–203.CrossRefGoogle Scholar
[19]Kamiya, N., ‘On the Peirce decompositions for Freudenthal–Kantor triple systems’, Comm. Algebra 25 (1997), 1833–1844.CrossRefGoogle Scholar
[20]Kamiya, N., ‘On a realization of the exceptional simple graded Lie algebras of the second kind and Freudenthal–Kantor triple systems’, Bull. Pol. Acad. Sci. Math. 46 (1998), 55–65.Google Scholar
[21]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, Contemporary Mathematics, 391 (American Mathematical Society, Providence, RI, 2005), pp. 157–165.CrossRefGoogle Scholar
[22]Kamiya, N. and Mondoc, D., ‘A new class of nonassociative algebras with involution’, Proc. Japan Acad. Ser. A 84 (2008), 68–72.CrossRefGoogle Scholar
[23]Kamiya, N. and Okubo, S., ‘On δ-Lie supertriple systems associated with (ϵ,δ)-Freudenthal–Kantor supertriple systems’, Proc. Edinb. Math. Soc. 43 (2000), 243–260.CrossRefGoogle Scholar
[24]Kamiya, N. and Okubo, S., ‘A construction of Jordan superalgebras from Jordan–Lie triple systems’, in: Nonassociative Algebra and its Applications, Lecture Notes in Pure and Applied Mathematics, 211 (eds. R. Costa, and L. A. Peresi, ) (Marcel Dekker Inc., New York, 2002),pp. 171–176.Google Scholar
[25]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 (2002), 339–348.CrossRefGoogle Scholar
[26]Kamiya, N. and Okubo, S., ‘Construction of Lie superalgebras D(2,1;α),G(3) and F(4) from some triple systems’, Proc. Edinb. Math. Soc. 46 (2003), 87–98.CrossRefGoogle Scholar
[27]Kamiya, N. and Okubo, S., ‘On generalized Freudenthal–Kantor triple systems and Yang–Baxter equations’, in: Proc. XXIV Inter. Colloq. Group Theoretical Methods in Physics, IPCS, 173 (2003) pp. 815–818.Google Scholar
[28]Kamiya, N. and Okubo, S., ‘A construction of simple Lie superalgebras of certain types from triple systems’, Bull. Aust. Math. Soc. 69 (2004), 113–123.CrossRefGoogle Scholar
[29]Kamiya, N. and Okubo, S., ‘Composition, quadratic, and some triple systems’, in: Non-associative Algebra and its Applications, Lecture Notes Pure and Applied Mathematics, 246 (eds. Sabinin, L., Sbitneva, L. and Shestakov, I.) (Chapman & Hall/CRC, Boca Raton, FL, 2006), pp. 205–231.Google Scholar
[30]Kaneyuki, S. and Asano, H., ‘Graded Lie algebras and generalized Jordan triple systems’, Nagoya Math. J. 112 (1988), 81–115.CrossRefGoogle Scholar
[31]Kantor, I. L., ‘Graded Lie algebras’, Trudy Sem. Vect. Tens. Anal. 15 (1970), 227–266.Google Scholar
[32]Kantor, I. L., ‘Some generalizations of Jordan algebras’, Trudy Sem. Vect. Tens. Anal. 16 (1972), 407–499.Google Scholar
[33]Kantor, I. L., ‘Models of exceptional Lie algebras’, Soviet Math. Dokl. 14 (1973), 254–258.Google Scholar
[34]Kantor, I. L., ‘A generalization of the Jordan approach to symmetric Riemannian spaces’, in: The Monster and Lie Algebras (Columbus, OH, 1996), Ohio State University Mathematical Research Institute Publications, 7 (de Gruyter, Berlin, 1998), pp. 221–234.CrossRefGoogle Scholar
[35]Kantor, I. L. and Kamiya, N., ‘A Peirce decomposition for generalized Jordan triple systems of second order’, Comm. Algebra 31 (2003), 5875–5913.CrossRefGoogle Scholar
[36]Koecher, M., ‘Embedding of Jordan algebras into Lie algebras I. II’, Amer. J. Math. 89 (1967), 787–816; 90 (1968), 476–510.CrossRefGoogle Scholar
[37]Lister, W. G., ‘A structure theory of Lie triple systems’, Trans. Amer. Math. Soc. 72 (1952), 217–242.CrossRefGoogle Scholar
[38]Meyberg, K., Lectures on Algebras and Triple Systems, Notes on a Course of Lectures Given During the Academic Year 1971–1972 (The University of Virginia, Charlottesville, VA, 1972).Google Scholar
[39]Mondoc, D., ‘Models of compact simple Kantor triple systems defined on a class of structurable algebras of skew-dimension one’, Comm. Algebra 34 (2006), 3801–3815.CrossRefGoogle Scholar
[40]Mondoc, D., ‘On compact realifications of exceptional simple Kantor triple systems’, J. Gen. Lie Theory Appl. 1 (2007), 29–40.CrossRefGoogle Scholar
[41]Mondoc, D., ‘Compact realifications of exceptional simple Kantor triple systems defined on tensor products of composition algebras’, J. Algebra 307 (2007), 917–929.CrossRefGoogle Scholar
[42]Mondoc, D., ‘Compact exceptional simple Kantor triple systems defined on tensor products of composition algebras’, Comm. Algebra 35 (2007), 3699–3712.CrossRefGoogle Scholar
[43]Neher, E., Jordan Triple Systems by the Grid Approach, Lecture Notes in Mathematics, 1280 (Springer, Berlin, 1987).CrossRefGoogle Scholar
[44]Okubo, S., Introduction to Octonion and Other Non-associative Algebras in Physics, Montroll Memorial Lecture Series in Mathematical Physics, 2 (Cambridge University Press, Cambridge, 1995).CrossRefGoogle Scholar
[45]Okubo, S., ‘Symmetric triality relations and structurable algebras’, Linear Algebra Appl. 396 (2005), 189–222.CrossRefGoogle Scholar
[46]Okubo, S. and Kamiya, N., ‘Jordan–Lie superalgebra and Jordan–Lie triple system’, J. Algebra 198 (1997), 388–411.CrossRefGoogle Scholar
[47]Okubo, S. and Kamiya, N., ‘Quasi-classical Lie superalgebras and Lie supertriple systems’, Comm. Algebra 30 (2002), 3825–3850.CrossRefGoogle Scholar
[48]Scheunert, M., The Theory of Lie Superalgebras. An Introduction, Lecture Notes in Mathematics, 716 (Springer, Berlin, 1979), p. x+271.CrossRefGoogle Scholar
[49]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
[50]Yamaguti, K. and Ono, A., ‘On representations of Freudenthal–Kantor triple systems U(ϵ,δ)’, Bull. Fac. School Ed. Hiroshima Univ. 7 (1984), 43–51.Google Scholar
[51]Zelmanov, E., ‘Primary Jordan triple systems’, Sibirsk. Mat. Zh. 24 (1983), 23–37.Google Scholar
[52]Zhevlakov, K. A., Slinko, A. M., Shestakov, I. P. and Shirshov, A. I., Rings that are Nearly Associative (Academic Press, New York, London, 1982).Google Scholar