A characterization of coboundary Poisson Lie groups and Hopf algebras

Stanisław Zakrzewski

Banach Center Publications (1997)

  • Volume: 40, Issue: 1, page 273-278
  • ISSN: 0137-6934

Abstract

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We show that a Poisson Lie group (G,π) is coboundary if and only if the natural action of G×G on M=G is a Poisson action for an appropriate Poisson structure on M (the structure turns out to be the well known π + ). We analyze the same condition in the context of Hopf algebras. A quantum analogue of the π + structure on SU(N) is described in terms of generators and relations as an example.

How to cite

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Zakrzewski, Stanisław. "A characterization of coboundary Poisson Lie groups and Hopf algebras." Banach Center Publications 40.1 (1997): 273-278. <http://eudml.org/doc/252240>.

@article{Zakrzewski1997,
abstract = {We show that a Poisson Lie group (G,π) is coboundary if and only if the natural action of G×G on M=G is a Poisson action for an appropriate Poisson structure on M (the structure turns out to be the well known $π _\{+\}$). We analyze the same condition in the context of Hopf algebras. A quantum analogue of the $π _\{+\}$ structure on SU(N) is described in terms of generators and relations as an example.},
author = {Zakrzewski, Stanisław},
journal = {Banach Center Publications},
keywords = {Poisson Lie group; Poisson structure; Hopf algebras; quantum groups},
language = {eng},
number = {1},
pages = {273-278},
title = {A characterization of coboundary Poisson Lie groups and Hopf algebras},
url = {http://eudml.org/doc/252240},
volume = {40},
year = {1997},
}

TY - JOUR
AU - Zakrzewski, Stanisław
TI - A characterization of coboundary Poisson Lie groups and Hopf algebras
JO - Banach Center Publications
PY - 1997
VL - 40
IS - 1
SP - 273
EP - 278
AB - We show that a Poisson Lie group (G,π) is coboundary if and only if the natural action of G×G on M=G is a Poisson action for an appropriate Poisson structure on M (the structure turns out to be the well known $π _{+}$). We analyze the same condition in the context of Hopf algebras. A quantum analogue of the $π _{+}$ structure on SU(N) is described in terms of generators and relations as an example.
LA - eng
KW - Poisson Lie group; Poisson structure; Hopf algebras; quantum groups
UR - http://eudml.org/doc/252240
ER -

References

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  1. [1] V. G. Drinfeld, Hamiltonian structures on Lie groups, Lie bialgebras and the meaning of the classical Yang-Baxter equations, Soviet Math. Dokl. 27 (1983), 68-71. 
  2. [2] V. G. Drinfeld, Quantum groups, Proc. ICM, Berkeley, 1986, vol.1, 789-820. 
  3. [3] M. A. Semenov-Tian-Shansky, Dressing transformations and Poisson Lie group actions, Publ. Res. Inst. Math. Sci., Kyoto University 21 (1985), 1237-1260. Zbl0673.58019
  4. [4] J.-H. Lu and A. Weinstein, Poisson Lie Groups, Dressing Transformations and Bruhat Decompositions, J. Diff. Geom. 31 (1990), 501-526. Zbl0673.58018
  5. [5] J.-H. Lu, Multiplicative and affine Poisson structures on Lie groups, Ph.D. Thesis, University of California, Berkeley (1990). 
  6. [6] S. Zakrzewski, Poisson structures on the Lorentz group, Lett. Math. Phys. 32 (1994), 11-23. Zbl0827.17016
  7. [7] S. Zakrzewski, Poisson homogeneous spaces, in: 'Quantum Groups, Formalism and Applications', Proceedings of the XXX Winter School on Theoretical Physics 14-26 February 1994, Karpacz, J. Lukierski, Z. Popowicz, J. Sobczyk (eds.), Polish Scientific Publishers PWN, Warsaw 1995, pp. 629-639. 
  8. [8] S. L. Woronowicz, Tannaka-Krein duality for compact matrix pseudogroups. Twisted SU(N) groups, Invent. Math. 93 (1988), 35. Zbl0664.58044
  9. [9] J.-H. Lu, On the Drinfeld double and the Heisenberg double of a Hopf algebra, Duke Math. Journ. 74, No.3 (1994), 763-776. Zbl0815.16020

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