American Institute of Mathematical Sciences

Advanced
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

[1]

Katarzyna Grabowska, Marcin Zając. The Tulczyjew triple in mechanics on a Lie group. Journal of Geometric Mechanics, 2016, 8 (4) : 413-435. doi: 10.3934/jgm.2016014
[2]

Yusi Fan, Chenrui Yao, Liangyun Chen. Structure of sympathetic Lie superalgebras. Electronic Research Archive, 2021, 29 (5) : 2945-2957. doi: 10.3934/era.2021020
[3]

Franz W. Kamber and Peter W. Michor. Completing Lie algebra actions to Lie group actions. Electronic Research Announcements, 2004, 10: 1-10.

[4]

Ben Muatjetjeja, Dimpho Millicent Mothibi, Chaudry Masood Khalique. Lie group classification a generalized coupled (2+1)-dimensional hyperbolic system. Discrete & Continuous Dynamical Systems - S, 2020, 13 (10) : 2803-2812. doi: 10.3934/dcdss.2020219
[5]

Lakehal Belarbi. Ricci solitons of the $ \mathbb{H}^{2} \times \mathbb{R} $ Lie group. Electronic Research Archive, 2020, 28 (1) : 157-163. doi: 10.3934/era.2020010
[6]

Elena Celledoni, Brynjulf Owren. Preserving first integrals with symmetric Lie group methods. Discrete & Continuous Dynamical Systems, 2014, 34 (3) : 977-990. doi: 10.3934/dcds.2014.34.977
[7]

Emma Hoarau, Claire david@lmm.jussieu.fr David, Pierre Sagaut, Thiên-Hiêp Lê. Lie group study of finite difference schemes. Conference Publications, 2007, 2007 (Special) : 495-505. doi: 10.3934/proc.2007.2007.495
[8]

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
[9]

Michele Zadra, Elizabeth L. Mansfield. Using Lie group integrators to solve two and higher dimensional variational problems with symmetry. Journal of Computational Dynamics, 2019, 6 (2) : 485-511. doi: 10.3934/jcd.2019025
[10]

Andrew N. W. Hone, Matteo Petrera. Three-dimensional discrete systems of Hirota-Kimura type and deformed Lie-Poisson algebras. Journal of Geometric Mechanics, 2009, 1 (1) : 55-85. doi: 10.3934/jgm.2009.1.55
[11]

Robert I. McLachlan, Ander Murua. The Lie algebra of classical mechanics. Journal of Computational Dynamics, 2019, 6 (2) : 345-360. doi: 10.3934/jcd.2019017
[12]

Meera G. Mainkar, Cynthia E. Will. Examples of Anosov Lie algebras. Discrete & Continuous Dynamical Systems, 2007, 18 (1) : 39-52. doi: 10.3934/dcds.2007.18.39
[13]

Ville Salo, Ilkka Törmä. Recoding Lie algebraic subshifts. Discrete & Continuous Dynamical Systems, 2021, 41 (2) : 1005-1021. doi: 10.3934/dcds.2020307
[14]

Richard H. Cushman, Jędrzej Śniatycki. On Lie algebra actions. Discrete & Continuous Dynamical Systems - S, 2020, 13 (4) : 1115-1129. doi: 10.3934/dcdss.2020066
[15]

Javier Pérez Álvarez. Invariant structures on Lie groups. Journal of Geometric Mechanics, 2020, 12 (2) : 141-148. doi: 10.3934/jgm.2020007
[16]

André Caldas, Mauro Patrão. Entropy of endomorphisms of Lie groups. Discrete & Continuous Dynamical Systems, 2013, 33 (4) : 1351-1363. doi: 10.3934/dcds.2013.33.1351
[17]

Gerard Thompson. Invariant metrics on Lie groups. Journal of Geometric Mechanics, 2015, 7 (4) : 517-526. doi: 10.3934/jgm.2015.7.517
[18]

Hongliang Chang, Yin Chen, Runxuan Zhang. A generalization on derivations of Lie algebras. Electronic Research Archive, 2021, 29 (3) : 2457-2473. doi: 10.3934/era.2020124
[19]

Marco Zambon. Holonomy transformations for Lie subalgebroids. Journal of Geometric Mechanics, 2021, 13 (3) : 517-532. doi: 10.3934/jgm.2021016
[20]

K. C. H. Mackenzie. Drinfel'd doubles and Ehresmann doubles for Lie algebroids and Lie bialgebroids. Electronic Research Announcements, 1998, 4: 74-87.

2020 Impact Factor: 0.857

Tools

Metrics

  • PDF downloads (74)
  • HTML views (55)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy