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Covariant Hamiltonian field theories on manifolds with boundary: Yang-Mills theories
The 2-plectic structures induced by the Lie bialgebras
Department of Mathematics, Vali-e-Asr University of Rafsanjan, Rafsanjan, Iran |
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 $SL(n, \mathbb{C})$.
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,
On the geometry of multisymplectic manifolds, J. Austral. Math. Soc.(Series A), 66 (1999), 303-330.
doi: 10.1017/S1446788700036636. |
[3] |
V. Chari and A. Pressley, A Guide to Quantum Groups, Cambridge University Press, 1994.
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[4] |
T. DeDonder, Theorie Invariantive du Calcul des Variations, Gauthier-Villars, Paris, 1935.
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[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. |
[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.
|
[7] |
S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, GSM, Vol. 34, AMS, first edition 2001
doi: 10.1090/gsm/034. |
[8] |
J. Kijowski and W. M. Tulczyjew, A Symplectic Framework for Field Theories, Lec. Notes Phys., 107 Springer, 1979. |
[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. |
[10] |
H. J. Lu, Multiplicative and Affine Poisson Structures on Lie Groups, PhD thesis, University of California, Berkeley, 1990. |
[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. |
[12] |
C. L. Rogers, Higher Symplectic Geometry, PhD thesis, University of California, 2011. |
[13] |
H. Weyle,
Geodesic fields in the calculus of variation for multiple integrals, Ann. Math., 36 (1935), 607-629.
doi: 10.2307/1968645. |
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,
On the geometry of multisymplectic manifolds, J. Austral. Math. Soc.(Series A), 66 (1999), 303-330.
doi: 10.1017/S1446788700036636. |
[3] |
V. Chari and A. Pressley, A Guide to Quantum Groups, Cambridge University Press, 1994.
![]() ![]() |
[4] |
T. DeDonder, Theorie Invariantive du Calcul des Variations, Gauthier-Villars, Paris, 1935.
![]() |
[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. |
[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.
|
[7] |
S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, GSM, Vol. 34, AMS, first edition 2001
doi: 10.1090/gsm/034. |
[8] |
J. Kijowski and W. M. Tulczyjew, A Symplectic Framework for Field Theories, Lec. Notes Phys., 107 Springer, 1979. |
[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. |
[10] |
H. J. Lu, Multiplicative and Affine Poisson Structures on Lie Groups, PhD thesis, University of California, Berkeley, 1990. |
[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. |
[12] |
C. L. Rogers, Higher Symplectic Geometry, PhD thesis, University of California, 2011. |
[13] |
H. Weyle,
Geodesic fields in the calculus of variation for multiple integrals, Ann. Math., 36 (1935), 607-629.
doi: 10.2307/1968645. |
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