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March  2017, 9(1): 83-90. doi: 10.3934/jgm.2017003

The 2-plectic structures induced by the Lie bialgebras

Mohammad Shafiee

Department of Mathematics, Vali-e-Asr University of Rafsanjan, Rafsanjan, Iran

Received  June 2016 Revised  January 2017 Published  March 2017

In this paper we show that if the Lie algebra $\mathfrak{g}$ admits a Lie bialgebra structure and $\mathcal{D}$ is a Lie group with Lie algebra $\mathfrak{d}$, the double of $\mathfrak{g}$, then $\mathcal{D}$ or its quotient by a suitable Lie subgroup admits a $2$-plectic structure. In particular it is shown that the imaginary part of the Killing form on $\mathfrak{sl}(n, \mathbb{C})$ (as a real Lie algebra) induces a $2$-plectic structure on $SL(n, \mathbb{C})$.

Keywords: 2-plectic structure, Lie bialgebra, poisson Lie group, manin triple.
Mathematics Subject Classification: 53DXX, 17B37, 17B62.
Citation: Mohammad Shafiee. The 2-plectic structures induced by the Lie bialgebras. Journal of Geometric Mechanics, 2017, 9 (1) : 83-90. doi: 10.3934/jgm.2017003
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.  Google Scholar

[2]

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.  Google Scholar

[3] V. Chari and A. Pressley, A Guide to Quantum Groups, Cambridge University Press, 1994.   Google Scholar
[4] T. DeDonder, Theorie Invariantive du Calcul des Variations, Gauthier-Villars, Paris, 1935.   Google Scholar
[5]

V. De Smedt, Existence of a Lie bialgebra structure on every Lie algebra, Lett. Math. Phys., 31 (1994), 225-231.  doi: 10.1007/BF00761714.  Google Scholar

[6]

V. G. Drinfeld, Hamiltonian structures on Lie groups, Lie bialgebras and the geometric meaning of the classical Yang-Baxter equations, Dokl. Akad. Nauk SSSR, 268 (1983), 285-287.   Google Scholar

[7]

S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, GSM, Vol. 34, AMS, first edition 2001 doi: 10.1090/gsm/034.  Google Scholar

[8]

J. Kijowski and W. M. Tulczyjew, A Symplectic Framework for Field Theories, Lec. Notes Phys., 107 Springer, 1979.  Google Scholar

[9]

Y. Kosmann-Schwarzbach, Lie Bialgebras, Poisson Lie Groups and Dressing Transformations, in In Integrability of Nonlinear Systems (Pondicherry, 1996), volume 495 of Lecture Notes in Phys., pages 104-170. Springer, Berlin, 1997. doi: 10.1007/BFb0113695.  Google Scholar

[10]

H. J. Lu, Multiplicative and Affine Poisson Structures on Lie Groups, PhD thesis, University of California, Berkeley, 1990.  Google Scholar

[11]

M. A. Semenov-Tian-Shansky, Dressing transformations and Poisson Lie group actions, Publ. RIMS, Kyoto University, 21 (1985), 1237-1260.  doi: 10.2977/prims/1195178514.  Google Scholar

[12]

C. L. Rogers, Higher Symplectic Geometry, PhD thesis, University of California, 2011.  Google Scholar

[13]

H. Weyle, Geodesic fields in the calculus of variation for multiple integrals, Ann. Math., 36 (1935), 607-629.  doi: 10.2307/1968645.  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), 701-725.  doi: 10.1007/s00220-009-0951-9.  Google Scholar

[2]

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.  Google Scholar

[3] V. Chari and A. Pressley, A Guide to Quantum Groups, Cambridge University Press, 1994.   Google Scholar
[4] T. DeDonder, Theorie Invariantive du Calcul des Variations, Gauthier-Villars, Paris, 1935.   Google Scholar
[5]

V. De Smedt, Existence of a Lie bialgebra structure on every Lie algebra, Lett. Math. Phys., 31 (1994), 225-231.  doi: 10.1007/BF00761714.  Google Scholar

[6]

V. G. Drinfeld, Hamiltonian structures on Lie groups, Lie bialgebras and the geometric meaning of the classical Yang-Baxter equations, Dokl. Akad. Nauk SSSR, 268 (1983), 285-287.   Google Scholar

[7]

S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, GSM, Vol. 34, AMS, first edition 2001 doi: 10.1090/gsm/034.  Google Scholar

[8]

J. Kijowski and W. M. Tulczyjew, A Symplectic Framework for Field Theories, Lec. Notes Phys., 107 Springer, 1979.  Google Scholar

[9]

Y. Kosmann-Schwarzbach, Lie Bialgebras, Poisson Lie Groups and Dressing Transformations, in In Integrability of Nonlinear Systems (Pondicherry, 1996), volume 495 of Lecture Notes in Phys., pages 104-170. Springer, Berlin, 1997. doi: 10.1007/BFb0113695.  Google Scholar

[10]

H. J. Lu, Multiplicative and Affine Poisson Structures on Lie Groups, PhD thesis, University of California, Berkeley, 1990.  Google Scholar

[11]

M. A. Semenov-Tian-Shansky, Dressing transformations and Poisson Lie group actions, Publ. RIMS, Kyoto University, 21 (1985), 1237-1260.  doi: 10.2977/prims/1195178514.  Google Scholar

[12]

C. L. Rogers, Higher Symplectic Geometry, PhD thesis, University of California, 2011.  Google Scholar

[13]

H. Weyle, Geodesic fields in the calculus of variation for multiple integrals, Ann. Math., 36 (1935), 607-629.  doi: 10.2307/1968645.  Google Scholar

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