ON THE STRUCTURE OF GRADED LIE TRIPLE SYSTEMS

Title & Authors
ON THE STRUCTURE OF GRADED LIE TRIPLE SYSTEMS
Martin, Antonio Jesus Calderon;

Abstract
We study the structure of an arbitrary graded Lie triple system $\small{\mathfrak{T}}$ with restrictions neither on the dimension nor the base field. We show that $\small{\mathfrak{T}}$ is of the form $\mathfrak{T} Keywords Lie triple system;grading;simple component;structure theory; Language English Cited by 1. On the structure of graded Leibniz triple systems, Linear Algebra and its Applications, 2016, 496, 496 References 1. Y. A. Bahturin and M. V. Zaicev, Group gradings on simple Lie algebras of type A, J. Lie Theory 16 (2006), no. 4, 719-742. 2. V. K. Balachandran, Real L*-algebras, Indian J. Pure Appl. Math. 3 (1972), no. 6, 1224-1246. 3. G. Benkart, A. Elduque, and G. Martinez, A(n, n)-graded Lie superalgebras, J. Reine Angew. Math. 573 (2004), 139-156. 4. M. Boussahel and N. Mebarki, Graded Lie algebra and the$U(3)_L{\times}U(1)_N$gauge model, Internat. J. Modern Phys. A 26 (2011), no. 5, 873-909. 5. C. Boyallian and V. Meinardi, Quasifinite representations of the Lie superalgebra of quantum pseudodifferential operators, J. Math. Phys. 49 (2008), no. 2, 023505, 13 pp. 6. A. J. Bruce, Tulczyjew triples and higher Poisson/Schouten structures on Lie algebroids, Rep. Math. Phys. 66 (2010), no. 2, 251-276. 7. A. J. Calderon Martin, On split Lie algebras with symmetric root systems, Proc. Indian Acad. Sci. Math. Sci. 118 (2008), no. 3, 351-356. 8. A. J. Calderon Martin On split Lie triple systems, Proc. Indian Acad. Sci. Math. Sci. 119 (2009), no. 2, 165-177. 9. A. J. Calderon Martin, On the structure of graded Lie algebras, J. Math. Phys. 50 (2009), no. 10, 103513, 8 pp. 10. A. J. Calderon, C. Draper, and C. Martin, Gradings on the real forms of the Albert algebra, of$g_2$, and of$f_4$, J. Math. Phys. 51 (2010), no. 5, 053516, 21 pp. 11. A. J. Calderon, C. Draper, and C. Martin, Gradings on Lie triple systems related to exceptional Lie algebras, J. Pure Appl. Algebra 217 (2013), no. 4, 672-688. 12. A. J. Calderon Martin and M. Forero Piulestan, On split Lie triple systems II, Proc. Indian Acad. Sci. Math. Sci. 120 (2010), no. 2, 185-198. 13. A. J. Calderon Martin and M. Forero Piulestan, Split 3-Lie algebras, J. Math. Phys. 52 (2011), no. 12, 123503, 16 pp. 14. A. J. Calderon and J. M. Sanchez, On the structure of graded Lie superalgebras, Modern Phys. Lett. A 27 (2012), no. 25, 1250142, 18 pp. 15. A. J. Calderon and J. M. Sanchez, Split Leibniz superalgebras, Linear Algebra Appl. 438 (2013), no. 12, 4709-4725. 16. A. J. Calderon and J. M. Sanchez, On the structure of graded Leibniz algebras, Algebra Colloquium. In press. 17. M. Chaves and D. Singleton, Phantom energy from graded algebras, Modern Phys. Lett. A 22 (2007), no. 1, 29-40. 18. R. Coquereaux, G. Esposito-Farese, and F. Scheck, Noncommutative geometry and graded algebras in electroweak interactions, Internat. J. Modern Phys. A 7 (1992), no. 26, 6555-6593. 19. C. Draper, C. Martin, and A. Elduque, Fine gradings on exceptional simple Lie superalgebras , Internat. J. Math. 22 (2011), no. 12, 1823-1855. 20. C. Draper and A. Viruel, Group gradings on o(8,$\mathbb{C}$), Rep. Math. Phys. 61 (2008), no. 2, 265-280. 21. A. Ebadian, N. Ghobadipour, and H. Baghban, Stability of bi-$\theta$-derivations on JB*-triples, Int. J. Geom. Methods Mod. Phys. 9 (2012), no. 7, 1250051, 12 pp. 22. A. Elduque and M. Kochetov, Gradings on the exceptional Lie algebras$F_4$and$G_2$revisited, Rev. Mat. Iberoam. 28 (2012), no. 3, 775-815. 23. H. Grosse and R. Wulkenhaar, 8D-spectral triple on 4D-Moyal space and the vacuum of noncommutative gauge theory, J. Geom. Phys. 62 (2012), no. 7, 1583-1599. 24. U. Gunther and S. Kuzhel, PT-symmetry, Cartan decompositions, Lie triple systems and Krein space-related Clifford algebras, J. Phys. A: Math. Theor. 43 (2010), 392002, 10 pp. 25. M. Havlicek, J. Patera, E. Pelatonova, and J. Tolar, On fine gradings and their symmetries , Czechoslovak J. Phys. 51 (2001), 383-391. 26. K. Iohara and Y. Koga, Note on spin modules associated to$\mathbb{Z}$-graded Lie superalgebras, J. Math. Phys. 50 (2009), no. 10, 103508, 9 pp. 27. P. Jordan, Uber Verallgemeinerungsm oglichkeiten des Formalismus der Quantenmechanik , Nachr. Ges. Wiss. Gottingen (1933), 209-214. 28. J. Kaad and R. Senior, A twisted spectral triple for quantum SU(2), J. Geom. Phys. 62 (2012), no. 4, 731-739. 29. A. K. Kwasniewski, On maximally graded algebras and Walsh functions, Rep. Math. Phys. 26 (1988), no. 1, 137-142. 30. J. Palmkvist, Three-algebras, triple systems and 3-graded Lie superalgebras, J. Phys. A 43 (2010), no. 1, 015205, 15 pp. 31. E. Poletaeva, Embedding of the Lie superalgebra D(2, 1;${\alpha}$) into the Lie superalgebra of pseudodifferential symbols on$S^{1{\mid}2}\$, J. Math. Phys. 48 (2007), no. 10, 103504, 17 pp.

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