American Institute of Mathematical Sciences

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

A note on $2$-plectic homogeneous manifolds

Mohammad Shafiee 1,

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.
Keywords: homogeneous space., $2$-plectic structure.
Mathematics Subject Classification: 53D05, 53C30, 11F2.
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.  doi: 10.1007/s00220-009-0951-9.  Google Scholar

[2]

F. Cantrijn, A. Ibort and M. DeLeon, Hamiltonian structures on multisymplectic manifolds, Rend. Sem. Mat. Univ. Pol. Torino, 54 (1996), 225.   Google Scholar

[3]

F. Cantrijn, A. Ibort and M. DeLeon, On the geometry of multisymplectic manifolds,, J. Austral. Math. Soc.(Series A), 66 (1999), 303.  doi: 10.1017/S1446788700036636.  Google Scholar

[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.  doi: 10.1016/0926-2245(91)90013-Y.  Google Scholar

[5]

M. Gotay, J. Isenberg, J. Marsden and R. Montgomery, Momentum maps and classical relativistic fields, Part I: Covariant field theory,, , ().   Google Scholar

[6]

J. Kijowski, A finite-dimensional canonical formalism in the classical field theory,, Commun. Math. Phys., 30 (1973), 99.  doi: 10.1007/BF01645975.  Google Scholar

[7]

T. B. Madsen and A. Swann, Multi-Moment maps,, Adv. Math., 229 (2012), 2287.  doi: 10.1016/j.aim.2012.01.002.  Google Scholar

[8]

G. Martin, A Darboux theorem for multisymplectic manifolds,, Lett. Math. Phys., 16 (1988), 133.  doi: 10.1007/BF00402020.  Google Scholar

[9]

C. L. Rogers, Higher Symplectic Geometry,, Ph.D thesis, ().   Google Scholar

[10]

M. Shafiee, On compact semisimple Lie groups as $2$-plectic manifolds,, J. Geom., 105 (2014), 615.  doi: 10.1007/s00022-014-0223-5.  Google Scholar

[11]

Ph. B. Zwart and W. M. Boothby, On compact homogeneous symplectic manifolds,, Ann. Inst. Fourier, 30 (1980), 129.   Google Scholar

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.  doi: 10.1007/s00220-009-0951-9.  Google Scholar

[2]

F. Cantrijn, A. Ibort and M. DeLeon, Hamiltonian structures on multisymplectic manifolds, Rend. Sem. Mat. Univ. Pol. Torino, 54 (1996), 225.   Google Scholar

[3]

F. Cantrijn, A. Ibort and M. DeLeon, On the geometry of multisymplectic manifolds,, J. Austral. Math. Soc.(Series A), 66 (1999), 303.  doi: 10.1017/S1446788700036636.  Google Scholar

[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.  doi: 10.1016/0926-2245(91)90013-Y.  Google Scholar

[5]

M. Gotay, J. Isenberg, J. Marsden and R. Montgomery, Momentum maps and classical relativistic fields, Part I: Covariant field theory,, , ().   Google Scholar

[6]

J. Kijowski, A finite-dimensional canonical formalism in the classical field theory,, Commun. Math. Phys., 30 (1973), 99.  doi: 10.1007/BF01645975.  Google Scholar

[7]

T. B. Madsen and A. Swann, Multi-Moment maps,, Adv. Math., 229 (2012), 2287.  doi: 10.1016/j.aim.2012.01.002.  Google Scholar

[8]

G. Martin, A Darboux theorem for multisymplectic manifolds,, Lett. Math. Phys., 16 (1988), 133.  doi: 10.1007/BF00402020.  Google Scholar

[9]

C. L. Rogers, Higher Symplectic Geometry,, Ph.D thesis, ().   Google Scholar

[10]

M. Shafiee, On compact semisimple Lie groups as $2$-plectic manifolds,, J. Geom., 105 (2014), 615.  doi: 10.1007/s00022-014-0223-5.  Google Scholar

[11]

Ph. B. Zwart and W. M. Boothby, On compact homogeneous symplectic manifolds,, Ann. Inst. Fourier, 30 (1980), 129.   Google Scholar

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