Geodesic
Poisson–Lie sigma models on Drinfel’d double
Archivum mathematicum, Tome 48 (2012) no. 5, pp. 423-447 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

Poisson sigma models represent an interesting use of Poisson manifolds for the construction of a classical field theory. Their definition in the language of fibre bundles is shown and the corresponding field equations are derived using a coordinate independent variational principle. The elegant form of equations of motion for so called Poisson-Lie groups is derived. Construction of the Poisson-Lie group corresponding to a given Lie bialgebra is widely known only for coboundary Lie bialgebras. Using the adjoint representation of Lie group and Drinfel’d double we show that Poisson-Lie group can be constructed for general Lie bialgebra.
Poisson sigma models represent an interesting use of Poisson manifolds for the construction of a classical field theory. Their definition in the language of fibre bundles is shown and the corresponding field equations are derived using a coordinate independent variational principle. The elegant form of equations of motion for so called Poisson-Lie groups is derived. Construction of the Poisson-Lie group corresponding to a given Lie bialgebra is widely known only for coboundary Lie bialgebras. Using the adjoint representation of Lie group and Drinfel’d double we show that Poisson-Lie group can be constructed for general Lie bialgebra.
DOI : 10.5817/AM2012-5-423
Classification : 53D17, 70G45
Keywords: Poisson sigma models; Poisson manifolds; Poisson-Lie groups; bundle maps 
@article{10_5817_AM2012_5_423,
     author = {Vysok\'y, Jan and Hlavat\'y, Ladislav},
     title = {Poisson{\textendash}Lie sigma models on {Drinfel{\textquoteright}d} double},
     journal = {Archivum mathematicum},
     pages = {423--447},
     year = {2012},
     volume = {48},
     number = {5},
     doi = {10.5817/AM2012-5-423},
     mrnumber = {3007623},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.5817/AM2012-5-423/}
}
TY  - JOUR
AU  - Vysoký, Jan
AU  - Hlavatý, Ladislav
TI  - Poisson–Lie sigma models on Drinfel’d double
JO  - Archivum mathematicum
PY  - 2012
SP  - 423
EP  - 447
VL  - 48
IS  - 5
UR  - http://geodesic.mathdoc.fr/articles/10.5817/AM2012-5-423/
DO  - 10.5817/AM2012-5-423
LA  - en
ID  - 10_5817_AM2012_5_423
ER  - 
%0 Journal Article
%A Vysoký, Jan
%A Hlavatý, Ladislav
%T Poisson–Lie sigma models on Drinfel’d double
%J Archivum mathematicum
%D 2012
%P 423-447
%V 48
%N 5
%U http://geodesic.mathdoc.fr/articles/10.5817/AM2012-5-423/
%R 10.5817/AM2012-5-423
%G en
%F 10_5817_AM2012_5_423
Vysoký, Jan; Hlavatý, Ladislav. Poisson–Lie sigma models on Drinfel’d double. Archivum mathematicum, Tome 48 (2012) no. 5, pp. 423-447. doi: 10.5817/AM2012-5-423

[1] Bojowald, M., Kotov, A., Strobl, T.: Lie algebroid morphisms, Poisson Sigma Models, and off-shell closed gauge symmetries. J. Geom. Phys. 54 (2005), 400–426. | DOI | MR | Zbl

[2] Calvo, I.: Poisson sigma models on surfaces with boundary: Classical and quantum aspects. Ph.D. thesis, University of Zaragoza, 2006.

[3] Calvo, I., Falceto, F., García–Álvarez, D.: Topological Poisson sigma models on Poisson–Lie groups. JHEP, 0310 (033), 2003. | MR

[4] Dufour, J–P., Zung, N. T.: Poisson Structures and Their Normal Forms. Progr. Math., vol. 242, Birkhäuser Verlag, 2005. | MR | Zbl

[5] Klimčík, C.: Yang–Baxter $\sigma $–models and $d{S}/{A}d{S}$ T–Duality. JHEP, 0212 (051), 2002.

[6] Klimčík, C., Ševera, P.: T–duality and the moment map. hep-th/9610198. | Zbl

[7] Klimčík, C., Ševera, P.: Poisson–Lie T–duality and loops of Drinfeld doubles. Phys. Lett. B 375 (1996), 65–71.

[8] Lu, J., Weinstein, A.: Poisson Lie groups, dressing transformations, and Bruhat decompositions. J. Differential Geom. 31 (1990), 501–526. | MR | Zbl

[9] Nakahara, M.: Geometry, Topology and Physics. Taylor & Francis, 2003. | MR | Zbl

[10] Schaller, P., Strobl, T.: Poisson–Sigma–Models: A generalization of 2–D gravity Yang–Mills–systems. hep-th/9411163.

[11] Schaller, P., Strobl, T.: Poisson structure induced (topological) field theories. Mod. Phys. Lett. A9 (1994), 3129–3136. | DOI | MR | Zbl

[12] Steenrod, N.: The Topology of Fibre Bundles. Princeton University Press, 1951. | MR | Zbl

[13] Vaisman, I.: Lectures on the Geometry of Poisson Manifolds. Progr. Math., vol. 118, Birkhäuser Verlag, 2005. | MR

Cité par Sources :