On geometric structures associated with triple systems

Kamiya, Noriaki

doi:10.2478/v10309-012-0039-y

Abstract

In this paper, we show that geometric phenomena can be charac- terized using the concept of triple systems. In particular, we study a complex structure associated with triple systems

References

[1] W.Bertram, Complex and quatemionic structures on symmetric spaces -correspondence with Freudenthal-Kantor triple systems, Sophia Kokyuroku in Math., 45, Theory of Lie Groups and Manifolds, ed. R.Miyaoka and T.Tamaru, 57-76, 2002.
Search in Google Scholar
[2] A.Elduque, N.Kamiya and S.Okubo, Simple (-1,-1) balanced Freudenthal- Kantor triple systems , Glasgow Math. J., 45, 353-372, 2003.10.1017/S0017089503001290
Search in Google Scholar
[3] A.Elduque, N.Kamiya and S.Okubo, (-1,-1) balanced Freudenthal-Kantor triple systems and noncommutative Jordan algebras, J. Alg., 294, 19-40, 2005.
Search in Google Scholar
[4] H.Freudenthal and H.de Vries, Linear Lie Groups, Acad. Press, New York, 19G9.
Search in Google Scholar
[5] S.Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Acad. Press, New York, 1978.
Search in Google Scholar
[6] R.Iordanescu, Jordan structures in Analysis, Geometry and Physics, Ed- itura Acad. Romane, Bucureşti, 2009.
Search in Google Scholar
[7] N. Jacobson, Structures and Representations of Jordan Algebras , Amer. Math. Soc. Colloq. 39, Amer. Math. Soc., Providence, RI, 19G8.
Search in Google Scholar
[8] N.Kamiya, A structure theory of Freudenthal-Kantor triple systems, J. Alg., 110, 108-123, 1987.10.1016/0021-8693(87)90038-X
Search in Google Scholar
[9] N.Kamiya, A structure theory of Freudenthal-Kantor triple systems II, Comm. Math. Univ. Sancti., 38, 41-60, 1989.
Search in Google Scholar
[10] N.Kamiya, On radicals of triple systems, Groups, Rings, Lie and Hopf Algebras, (St. John’s 2001), 75-83, Math. Appl., 555, Kluwer Acad. Publ., Dordrecht, 2003.10.1007/978-1-4613-0235-3_5
Search in Google Scholar
[11] N.Kamiya, On a realization of the exceptional simple graded Lie algebras of second kind and Freudenthal-Kantor triple systems, Polish Academy of Sciences Math. 40, no.l, 55-65, 1998.
Search in Google Scholar
[12] N.Kamiya, Examples of Peirce decomposition of generalized Jordan triple systems of second order - Balanced cases-, Contemporary Mathematics, 391. A.M.S., 157-166, 2005.10.1090/conm/391/07326
Search in Google Scholar
[13] N.Kamiya, On a generalization of structurable algebras, Algebras, Groups and Geometries, 9, 35-47, 1992.
Search in Google Scholar
[14] I.L.Kantor and N.Kamiya, A Peirce decomposition for generalized Jordan triple systems of second order, Comm. Alg., 31, no.12, 5875-5913, 2003.10.1081/AGB-120024858
Search in Google Scholar
[15] N.Kamiya and D.Mondoc, Examples of Peirce decomposion of Kantor triple systems, Algebras, Groups and Geometries, 24, 325-348, 2007.
Search in Google Scholar
[16] N.Kamiya and D.Mondoc, A new class of nonassociative algebras with involution, Proc. Japan Acad. Ser.A, 84, no.5, 68-72, 2008.10.3792/pjaa.84.68
Search in Google Scholar
[17] N.Kamiya and S.Okubo, On δ-Lie Supertriple Systems Associated with (ε,δ)-Freudenthal-Kantor Supertriple Systems, Proc. Edinburgh Math. Soc., 43, 243-260, 2000.10.1017/S0013091500020903
Search in Google Scholar
[18] N.Kamiya and S.Okubo, Construction of Lie superalgcbras D(2, l;a), G(3) and F(4) from some triple systems, Proc. Edinburgh Math. Soc., 46, 87-98, 2003.10.1017/S0013091501000876
Search in Google Scholar
[19] N.Kamiya and S.Okubo, On generalized Freudenthal-Kantor triple systems and Yang-Baxter equations, Proc. XXIV International Coll. Group Theoretical Methods in Physics, Inst. Physics Conf. Ser., 173, 815-818, 2003.
Search in Google Scholar
[20] N.Kamiya and S.Okubo, On composition, quadratic and some triple systems, Lecture Notes in Pure and App. Math., 246, Taylor (CRC), 205-231, 2006.10.1201/9781420003451.ch16
Search in Google Scholar
[21] O.Loos, Symmetric Spaces, Benjamin, London, 1969.
Search in Google Scholar
[22] S.Okubo, Introduction to Octonion and other Non-associative Algebras in Physics, Cambridge Univ. Press, Cambridge, 1995.10.1017/CBO9780511524479
Search in Google Scholar
[23] S.Okubo and N.Kamiya, Jordan-Lie superalgcbras and Jordan-Lie triple systems, J. Alg., 198, 388-411, 1997.10.1006/jabr.1997.7144
Search in Google Scholar
[24] I.Satake, Algebraic Structures of Symmtric Domains, Princeton Univ. Press, Tokyo, 1980.10.1515/9781400856800
Search in Google Scholar
[25] M.Scheunert, The Theory of Lie Superalgebras, Lecture Notes in Math. 716, Springer, New York, 1979. 10.1007/BFb0070929
Search in Google Scholar

On geometric structures associated with triple systems

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