# Multiplier Hopf algebras and duality

Banach Center Publications (1997)

- Volume: 40, Issue: 1, page 51-58
- ISSN: 0137-6934

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topvan Daele, A.. "Multiplier Hopf algebras and duality." Banach Center Publications 40.1 (1997): 51-58. <http://eudml.org/doc/252199>.

@article{vanDaele1997,

abstract = {We define a category containing the discrete quantum groups (and hence the discrete groups and the duals of compact groups) and the compact quantum groups (and hence the compact groups and the duals of discrete groups). The dual of an object can be defined within the same category and we have a biduality theorem. This theory extends the duality between compact quantum groups and discrete quantum groups (and hence the one between compact abelian groups and discrete abelian groups). The objects in our category are multiplier Hopf algebras, with invertible antipode, admitting invariant functionals (integrals), satisfying some extra condition (to take care of the non-abelianness of the underlying algebras). If we start with a multiplier Hopf *-algebra with positive invariant functionals, then also the dual is a multiplier Hopf *-algebra with positive invariant functionals. This makes it possible to formulate this duality also within the framework of C*-algebras.},

author = {van Daele, A.},

journal = {Banach Center Publications},

keywords = {biduality theorem; multiplier Hopf algebras; multiplier Hopf *-algebra},

language = {eng},

number = {1},

pages = {51-58},

title = {Multiplier Hopf algebras and duality},

url = {http://eudml.org/doc/252199},

volume = {40},

year = {1997},

}

TY - JOUR

AU - van Daele, A.

TI - Multiplier Hopf algebras and duality

JO - Banach Center Publications

PY - 1997

VL - 40

IS - 1

SP - 51

EP - 58

AB - We define a category containing the discrete quantum groups (and hence the discrete groups and the duals of compact groups) and the compact quantum groups (and hence the compact groups and the duals of discrete groups). The dual of an object can be defined within the same category and we have a biduality theorem. This theory extends the duality between compact quantum groups and discrete quantum groups (and hence the one between compact abelian groups and discrete abelian groups). The objects in our category are multiplier Hopf algebras, with invertible antipode, admitting invariant functionals (integrals), satisfying some extra condition (to take care of the non-abelianness of the underlying algebras). If we start with a multiplier Hopf *-algebra with positive invariant functionals, then also the dual is a multiplier Hopf *-algebra with positive invariant functionals. This makes it possible to formulate this duality also within the framework of C*-algebras.

LA - eng

KW - biduality theorem; multiplier Hopf algebras; multiplier Hopf *-algebra

UR - http://eudml.org/doc/252199

ER -

## References

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- [10] A. Van Daele, Multiplier Hopf Algebra, Trans. Amer. Math. Soc. 342 (1994) 917-932.
- [11] A. Van Daele, Discrete Quantum Groups, J. of Algebra 180 (1996) 431-444. Zbl0864.17012
- [12] A. Van Daele, An Algebraic Framework for Group Duality, preprint K.U. Leuven (1996). Zbl0933.16043
- [13] B. Drabant & A. Van Daele, Pairing and The Quantum Double of Multiplier Hopf Algebras, preprint K.U. Leuven (1996).
- [14] J. Kustermans & A. Van Daele, C*-algebraic Quantum Groups arising from Algebraic Quantum Groups, preprint K.U. Leuven (1996). Zbl1009.46038
- [15] S. L. Woronowicz, Compact Matrix Pseudo Groups, Comm. Math. Phys. 111 (1987) 613-665. Zbl0627.58034
- [16] S. L. Woronowicz, Compact Quantum Groups, preprint University of Warsaw (1992).

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