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 $SL(n, \mathbb{C})$.
Citation: |
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
V. Chari and A. Pressley, A Guide to Quantum Groups, Cambridge University Press, 1994.
![]() ![]() |
|
T. DeDonder, Theorie Invariantive du Calcul des Variations, Gauthier-Villars, Paris, 1935.
![]() |
|
V. De Smedt
, Existence of a Lie bialgebra structure on every Lie algebra, Lett. Math. Phys., 31 (1994)
, 225-231.
doi: 10.1007/BF00761714.![]() ![]() ![]() |
|
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.
![]() ![]() |
|
S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, GSM, Vol. 34, AMS, first edition 2001
doi: 10.1090/gsm/034.![]() ![]() ![]() |
|
J. Kijowski and W. M. Tulczyjew, A Symplectic Framework for Field Theories, Lec. Notes Phys., 107 Springer, 1979.
![]() ![]() |
|
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.![]() ![]() ![]() |
|
H. J. Lu, Multiplicative and Affine Poisson Structures on Lie Groups, PhD thesis, University of California, Berkeley, 1990.
![]() ![]() |
|
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.![]() ![]() ![]() |
|
C. L. Rogers, Higher Symplectic Geometry, PhD thesis, University of California, 2011.
![]() ![]() |
|
H. Weyle
, Geodesic fields in the calculus of variation for multiple integrals, Ann. Math., 36 (1935)
, 607-629.
doi: 10.2307/1968645.![]() ![]() ![]() |