# American Institute of Mathematical Sciences

September  2015, 7(3): 389-394. doi: 10.3934/jgm.2015.7.389

## A note on $2$-plectic homogeneous manifolds

 1 Department of Mathematics, Vali-e-Asr University of Rafsanjan, Rafsanjan, P.O.Box 518, Iran

Received  February 2014 Revised  June 2015 Published  July 2015

In this note we study the existence of $2$-plectic structures on homogenous spaces. In particular we show that $S^{5}=\frac{SU(3)}{SU(2)}$, $\frac{SU(3)}{S^{1}}$, $\frac{SU(3)}{T^{2}}$ and $\frac{SO(4)}{S^{1}}$ admit a $2$-plectic structure. Furthermore, If $G$ is a Lie group with Lie algebra $\mathfrak{g}$ and $R$ is a closed Lie subgroup of $G$ corresponding to the nilradical of $\mathfrak{g}$, then $\frac{G}{R}$ is a $2$-plectic manifold.
Citation: Mohammad Shafiee. A note on $2$-plectic homogeneous manifolds. Journal of Geometric Mechanics, 2015, 7 (3) : 389-394. doi: 10.3934/jgm.2015.7.389
##### References:
 [1] J. C. Baez, A. E. Hoffnung and C. L. Rogers, Categorified symplectic geometry and the classical string, Comm. Math.Phys., 293 (2010), 701-725. doi: 10.1007/s00220-009-0951-9. [2] F. Cantrijn, A. Ibort and M. DeLeon, Hamiltonian structures on multisymplectic manifolds Rend. Sem. Mat. Univ. Pol. Torino, 54 (1996), 225-236 [3] F. Cantrijn, A. Ibort and M. DeLeon, On the geometry of multisymplectic manifolds, J. Austral. Math. Soc.(Series A), 66 (1999), 303-330. doi: 10.1017/S1446788700036636. [4] J. F. Carinena, M. Crampin and L. A. Ibort, On the multisymplectic formalism for first order field theories, Diff. Geom. Appl., 1 (1991), 345-374. doi: 10.1016/0926-2245(91)90013-Y. [5] M. Gotay, J. Isenberg, J. Marsden and R. Montgomery, Momentum maps and classical relativistic fields, Part I: Covariant field theory, arXiv:Physics/9801019. [6] J. Kijowski, A finite-dimensional canonical formalism in the classical field theory, Commun. Math. Phys., 30 (1973), 99-128. doi: 10.1007/BF01645975. [7] T. B. Madsen and A. Swann, Multi-Moment maps, Adv. Math., 229 (2012), 2287-2309, arXiv:1012.2048v2. doi: 10.1016/j.aim.2012.01.002. [8] G. Martin, A Darboux theorem for multisymplectic manifolds, Lett. Math. Phys., 16 (1988), 133-138. doi: 10.1007/BF00402020. [9] C. L. Rogers, Higher Symplectic Geometry, Ph.D thesis, University of California, Reverside D. Phil. available as arXiv:1106.4068v1. [10] M. Shafiee, On compact semisimple Lie groups as $2$-plectic manifolds, J. Geom., 105 (2014), 615-623. doi: 10.1007/s00022-014-0223-5. [11] Ph. B. Zwart and W. M. Boothby, On compact homogeneous symplectic manifolds, Ann. Inst. Fourier, 30 (1980), 129-157.

show all references

##### References:
 [1] J. C. Baez, A. E. Hoffnung and C. L. Rogers, Categorified symplectic geometry and the classical string, Comm. Math.Phys., 293 (2010), 701-725. doi: 10.1007/s00220-009-0951-9. [2] F. Cantrijn, A. Ibort and M. DeLeon, Hamiltonian structures on multisymplectic manifolds Rend. Sem. Mat. Univ. Pol. Torino, 54 (1996), 225-236 [3] F. Cantrijn, A. Ibort and M. DeLeon, On the geometry of multisymplectic manifolds, J. Austral. Math. Soc.(Series A), 66 (1999), 303-330. doi: 10.1017/S1446788700036636. [4] J. F. Carinena, M. Crampin and L. A. Ibort, On the multisymplectic formalism for first order field theories, Diff. Geom. Appl., 1 (1991), 345-374. doi: 10.1016/0926-2245(91)90013-Y. [5] M. Gotay, J. Isenberg, J. Marsden and R. Montgomery, Momentum maps and classical relativistic fields, Part I: Covariant field theory, arXiv:Physics/9801019. [6] J. Kijowski, A finite-dimensional canonical formalism in the classical field theory, Commun. Math. Phys., 30 (1973), 99-128. doi: 10.1007/BF01645975. [7] T. B. Madsen and A. Swann, Multi-Moment maps, Adv. Math., 229 (2012), 2287-2309, arXiv:1012.2048v2. doi: 10.1016/j.aim.2012.01.002. [8] G. Martin, A Darboux theorem for multisymplectic manifolds, Lett. Math. Phys., 16 (1988), 133-138. doi: 10.1007/BF00402020. [9] C. L. Rogers, Higher Symplectic Geometry, Ph.D thesis, University of California, Reverside D. Phil. available as arXiv:1106.4068v1. [10] M. Shafiee, On compact semisimple Lie groups as $2$-plectic manifolds, J. Geom., 105 (2014), 615-623. doi: 10.1007/s00022-014-0223-5. [11] Ph. B. Zwart and W. M. Boothby, On compact homogeneous symplectic manifolds, Ann. Inst. Fourier, 30 (1980), 129-157.
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