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Department of Mathematics, University of Aizu, Aizuwakamatsu, Japan

- Corresponding Author:
- Kamiya N

Department of Mathematics

University of Aizu, Aizuwakamatsu

965-8580, Japan

**Tel:**0242 37 2500

**E-mail:**[email protected]

**Received date:** July 14, 2015 **Accepted date:**July 16, 2015 **Published date:** July 29, 2015

**Citation: **Kamiya N (2015) A Survey of My Recent Research. J Generalized Lie Theory Appl 9:e102. doi:10.4172/1736-4337.1000e102

**Copyright:** ©2015 Kamiya N. This is an open-access article distributed under the
terms of the Creative Commons Attribution License, which permits unrestricted
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In this note, we will describe a short survey with respect to the recent works of Noriaki Kamiya (author) mainly.

The author study is in non associative algebras related with mathematical physics. In particular, we (author) are interesting for the subjects as following; triple systems, Lie algebras or super algebras constructed from triple systems, triality algebras (containing structurable algebras), triality groups (containing automorphism groups of Lie algebras).

It seems that the concept of a triple system (or called a ternary algebra) in non associative algebras started from the metasymplectic geometry due to Freudenthal. After a generalization of the concept has been studied by Tits, Koecher, Kantor, Yamaguti, Allison et al. [1-6]. Also it is well known the object of investigation of Jordan and Lie algebras with application to symmetric spaces or domains [7] and to physics [8,9].

Non associative algebras are rich in of mathematics, not only for pure algebra differential geometry, but also for representation theory and algebraic geometry. Specially, the Lie algebras and Jordan algebras plays an important role in many mathematical and physical objects. As a construction of Lie algebras as well as Jordan algebras, we are interested in characterizing the Lie algebras from view point of triple systems [1,10-12]. These imply that we are considering to structure of the subspace L_{1} of the five graded Lie (super) algebra satisfying

associated with an (ε, δ) Freudenthal-Kantor triple system which contains a class of Jordan triple systems related 3 graded Lie algebra For these considerations without utilizing properties of root systems or Cartan matrices, we would like to refer to the articles of the present author and earlier references quoted therein [2,3,13-17]. In particular, for an characterizing of Lie algebras, an applying to geometry and physics, we have introduced couple topics about a symmetry of Lie algebras and a definition of hermitian triple systems in "Examples of Freudenthal-Kantor triple systems, "published by JGLTA (2014) recently [18]. More precisely speaking, in the paper, first, the symmetry group of Lie algebras and super algebras constructed from (ε, δ) Freudenthal-Kantor triple systems has been studied. Especially, for a special (ε,ε) Freudenthal-Kantor triple, it is SL (2) group. Secondly, we give a definition of hermitian* generalized Jordan triple systems and the examples of their tripotents defined by elements c of triple systems satisfying ccc=c. This concept is a generalization of Hermitian Jordan triple systems related symmetric bounded domains.

In final, the author has several coworks with Prof. Okubo, Kantor, Elduque, Mondoc, Shibukawa, and Sato etc, (Europe, U.S.A., Japan) with respect to non associative algebras and mathematical physics.

Thus from these reasons, our fields will be glow up as the object in future.

- Kamiya N (1987) A structure theory of Freudenthal-Kantor triple systems. J Alg 110: 108-123.
- Kamiya N, Okubo S (2000) On d-Lie supertriple systems associated with (e,d) Freudenthal-Kantor triple systems. Proc Edinb Math Soc 43: 243-260.
- Kamiya N, Mondoc D, Okubo S (2010) A structure theory of (-1,
*-*1) Freudenthal-Kantor triple systems. Bull Aust Math Soc 81: 132-155. - Allison BN (1978) A Class of non associative algebras with involution containing the class of Jordan algebras. Math Ann 237: 133-156
- Allison BN, Benkart G, Gao Y (2002) Lie algebras graded by the root systems
*BCr; r*__>__2, Mem Amer Math Soc. - Allison BN, Faulkner JR (1993) Non-associative coecient algebras for Steinberg unitary Lie algebras. J.Algebra16: 1-19.
- Loos O (1977) Bounded symmetric domains and Jordan pairs, Mathematical Lectures, Univ. of California, Irvine, USA.
- Lohmus J, Paar E, Sorgsepp L (1994) Non associative algebras in physics. Hadronic Press, Palm Harbor, USA.
- Okubo S (1995) Introduction to octonion and other non associative algebras in physics. Cambridge Univ. Press, USA.
- Elduque A, Kamiya N, Okubo S (2013) Left unital Kantor triple systems andstructurable algebra. Linear and multilinear algebras.
- K Meyberg (1968) A theory of Freudenthal's Triple systems I, II. Nederl. Acad. Wetensch. Ser. A-71 Indag Math 30: 162-190.
- Yamaguti K, Ono S (1984) On representations of Freudenthal-Kantor triple system
*U*(e,d). Bull Fac School, Edn Hiroshima Univ. Ser II 7: 43-51 - Kamiya N (1991) A construction of simple lie algebras over
*C*from balanced Freudenthal-Kantor triple systems. Contributions to general algebras 7: 205-213. - Kantor IL, Kamiya N (2003) A Peirce decomposition for generalized Jordan triple systems of second order. Comm in Alg 31: 5875-5913.
- Kamiya N, Mondoc D (2008) A new class of non associative algebras with involution. Proc Japan Acad Ser A Math Sci 84: 68-72.
- Kamiya N, Okubo S (2014) Triality of structurable and pre-structurable algebras. Journal of Algebra 416: 58-83.
- Okubo S (2005) Symmetric triality relations and structurable algebra. Linear Algebra and its Application 396: 189-222.
- Kamiya N, Shibukawa Y (2011) Dynamical Yang-Baxter maps associated with homogeneous presystems. J Gen Lie Theory Appl.

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