# A characterization of coboundary Poisson Lie groups and Hopf algebras

Banach Center Publications (1997)

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

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topZakrzewski, 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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- [5] J.-H. Lu, Multiplicative and affine Poisson structures on Lie groups, Ph.D. Thesis, University of California, Berkeley (1990).
- [6] S. Zakrzewski, Poisson structures on the Lorentz group, Lett. Math. Phys. 32 (1994), 11-23. Zbl0827.17016
- [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] S. L. Woronowicz, Tannaka-Krein duality for compact matrix pseudogroups. Twisted SU(N) groups, Invent. Math. 93 (1988), 35. Zbl0664.58044
- [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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