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A note on $2$-plectic vector spaces | ||
| Journal of Mahani Mathematical Research | ||
| دوره 13، شماره 1 - شماره پیاپی 26، بهمن 2023، صفحه 443-455 اصل مقاله (490.15 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22103/jmmr.2023.20889.1389 | ||
| نویسنده | ||
| Mohammad Shafiee* | ||
| Department of Mathematics, Vali-e-Asr University of Rafsanjan, Rafsanjan, Iran | ||
| چکیده | ||
| Among the $2$-plectic structures on vector spaces, the canonical ones and the $2$-plectic structures induced by the Killing form on semisimple Lie algebras are more interesting. In this note, we show that the group of linear preservers of the canonical $2$-plectic structure is noncompact and disconnected and its dimension is computed. Also, we show that the group of automorphisms of a compact semisimple Lie algebra preserving its $2$-plectic structure, is compact. Furthermore, it is shown that the $2$-plectic structure on a semisimple Lie algebra $\mathfrak{g}$ is canonical if and only if it has an abelian Lie subalgebra whose dimension satisfies in a special condition. As a consequence, we conclude that the $2$-plectic structures induced by the Killing form on some important classical Lie algebras are not canonical. | ||
| کلیدواژه‌ها | ||
| $2$-plectic structure؛ Canonical $2$-plectic structure؛ Semisimple Lie algebra | ||
| مراجع | ||
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[1] Baez, J. C., Ho nung, A. E., & Rogers, C. L. (2010). Categori ed Symplectic Geometry and the Classical String, Comm. Math.Phys. 293 , 701-725.
[2] Cantrijn, F., Ibort, A., & DeLeon, M. (1999). On the Geometry of Multisymplectic Manifolds, J. Austral. Math. Soc.(Series A), 66, 303-330.
[3] DeDonder, T. (1935). Theorie Invariantive du Calcul des Variations, Gauthier-Villars, Paris.
[4] Fulton, W., & Harris, J. (1991). Representation Theory, A First Course, Springer NewYork.
[5] Hussin, V., Winternitz, P., & Zassenhaus, H., (1988) Maximal Abelian Subalgebras of Complex Orthogonal lie Algebras, Linear Algebra Appl. 141, 183-220.
[6] Martin, G. (1988). A Darboux Theorem for Multisymplectic Manifolds, Lett. Math. Phys. 27, 571-585.
[7] McDu , D., & Salamon, D. (1995) Introduction to Symplectic Topology, Oxford Mathematical Monographs, Oxford University Press.
[8] Onishchik, A. (2004). Lectures on Real Semisimple Lie Algebras and Their Representation, ESI Lectures in Mathematics and Physics. European Mathematical Society (EMS), Z¨urich.
[9] Rogers, C. L. (2011) Higher Symplectic Geometry, University of California, Reverside D. Phil. thesis, available as arXiv:1106.4068v1.
[10] Saemann, C., & Szabo, R. (2011). Groupoid Quantization of Loop Spaces, PoS CORFU2011 046. e-Print:1203.5921. DOI:10.22323/1.155.0046.
[11] Saemann,C., & Szabo, R. (2011). Quantization of 2-plectic manifolds, Volume 155 - Proceedings of the Corfu Summer Institute 2011 (CORFU2011) - Workshop on Non-commutative Field Theory and Gravity, https://doi.org/10.22323/1.155.0046.
[12] Weyle, H. (1935). Geodesic Fields in the Calculus of Variation for Multiple Integrals, Ann. Math. 36, 607-629. | ||
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