# Poisson Lie groups and their relations to quantum groups

Banach Center Publications (1995)

- Volume: 34, Issue: 1, page 55-64
- ISSN: 0137-6934

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topGrabowski, Janusz. "Poisson Lie groups and their relations to quantum groups." Banach Center Publications 34.1 (1995): 55-64. <http://eudml.org/doc/251303>.

@article{Grabowski1995,

abstract = {The notion of Poisson Lie group (sometimes called Poisson Drinfel'd group) was first introduced by Drinfel'd [1] and studied by Semenov-Tian-Shansky [7] to understand the Hamiltonian structure of the group of dressing transformations of a completely integrable system. The Poisson Lie groups play an important role in the mathematical theories of quantization and in nonlinear integrable equations. The aim of our lecture is to point out the naturality of this notion and to present basic facts about Poisson Lie groups together with some relations to the recent work on quantum groups.},

author = {Grabowski, Janusz},

journal = {Banach Center Publications},

keywords = {quantum spaces; Lie bialgebras; Poisson bracket; group of dressing transformations; completely integrable system; Poisson Lie groups; nonlinear integrable equations; quantum groups},

language = {eng},

number = {1},

pages = {55-64},

title = {Poisson Lie groups and their relations to quantum groups},

url = {http://eudml.org/doc/251303},

volume = {34},

year = {1995},

}

TY - JOUR

AU - Grabowski, Janusz

TI - Poisson Lie groups and their relations to quantum groups

JO - Banach Center Publications

PY - 1995

VL - 34

IS - 1

SP - 55

EP - 64

AB - The notion of Poisson Lie group (sometimes called Poisson Drinfel'd group) was first introduced by Drinfel'd [1] and studied by Semenov-Tian-Shansky [7] to understand the Hamiltonian structure of the group of dressing transformations of a completely integrable system. The Poisson Lie groups play an important role in the mathematical theories of quantization and in nonlinear integrable equations. The aim of our lecture is to point out the naturality of this notion and to present basic facts about Poisson Lie groups together with some relations to the recent work on quantum groups.

LA - eng

KW - quantum spaces; Lie bialgebras; Poisson bracket; group of dressing transformations; completely integrable system; Poisson Lie groups; nonlinear integrable equations; quantum groups

UR - http://eudml.org/doc/251303

ER -

## References

top- [1]. V. G. Drinfel'd, Hamiltonian structures on Lie groups, Lie bialgebras and the geometric meaning of the classical Yang-Baxter equations, Soviet Math. Dokl. 27 (1983), 68-71.
- [2]. V. G. Drinfel'd, Quantum groups, Proc. ICM, Berkeley, Vol. 1, Amer. Math. Soc. 1986, 789-820.
- [3]. J. Grabowski, Quantum SU(2) group of Woronowicz and Poisson structures, in: Differential Geometry and its Application, Proc. Conf. Brno 1989, Eds. J. Janyška and D. Krupka, World Scientific 1990, 313-322.
- [4]. J. Grabowski, Abstract Jacobi and Poisson structures. Quantization and star-products, J. Geom. Phys. 9 (1992), 45-73. Zbl0761.16012
- [5] Y. Kosmann-Schwarzbach, Poisson-Drinfel'd groups, Publ. IRMA, Lille, Vol. 5, No. 12, 1987.
- [6] J.-H. Lu and A. Weinstein, Poisson Lie groups, dressing transformations, and Bruhat decompositions, J. Differential Geom. 31 (1990), 501-526. Zbl0673.58018
- [7] M. A. Semenov-Tian-Shansky, Dressing transformations and Poisson Lie group actions, Publ. Res. Inst. Math. Sci. 21 (1985), 1237-1260. Zbl0673.58019
- [8] J. Vey, Déformation du crochet de Poisson sur une variété symplectique, Comment. Math. Helv. 50 (1975), 421-454. Zbl0351.53029
- [9] S. L. Woronowicz, Twisted SU(2) group. An example of a non-commutative differential calculus, Publ. Res. Inst. Math. Sci. 23 (1987), 117-181. Zbl0676.46050

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