Multiplier Hopf algebras and duality

A. van Daele

Banach Center Publications (1997)

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

Abstract

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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.

How to cite

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van 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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  1. [1] E. Abe, Hopf Algebras, Cambridge University Press (1977). 
  2. [2] V. G. Drinfel'd, Quantum groups, Proceedings ICM Berkeley (1986) 798-820. 
  3. [3] E. Effros & Z.-J. Ruan, Discrete Quantum Groups I. The Haar Measure, Int. J. Math. 5 (1994) 681-723. Zbl0824.17020
  4. [4] Y. Nakagami, T. Masuda & S. L. Woronowicz, (in preparation). 
  5. [5] P. Podleś & S. L. Woronowicz, Quantum Deformation of Lorentz Group, Comm. Math. Phys. 130 (1990) 381-431. Zbl0703.22018
  6. [6] E. M. Sweedler, Hopf Algebras, Benjamin (1969). 
  7. [7] A. Van Daele, Dual Pairs of Hopf *-algebras, Bull. London Math. Soc. 25 (1993) 209-230. 
  8. [8] A. Van Daele, The Haar Measure on Finite Quantum Groups, to appear in Proc. Amer. Math. Soc. Zbl0888.16023
  9. [9] A. Van Daele, The Haar Measure on Compact Quantum Groups, Proc. Amer. Math. Soc. 123 (1995) 3125-3128. Zbl0844.46032
  10. [10] A. Van Daele, Multiplier Hopf Algebra, Trans. Amer. Math. Soc. 342 (1994) 917-932. 
  11. [11] A. Van Daele, Discrete Quantum Groups, J. of Algebra 180 (1996) 431-444. Zbl0864.17012
  12. [12] A. Van Daele, An Algebraic Framework for Group Duality, preprint K.U. Leuven (1996). Zbl0933.16043
  13. [13] B. Drabant & A. Van Daele, Pairing and The Quantum Double of Multiplier Hopf Algebras, preprint K.U. Leuven (1996). 
  14. [14] J. Kustermans & A. Van Daele, C*-algebraic Quantum Groups arising from Algebraic Quantum Groups, preprint K.U. Leuven (1996). Zbl1009.46038
  15. [15] S. L. Woronowicz, Compact Matrix Pseudo Groups, Comm. Math. Phys. 111 (1987) 613-665. Zbl0627.58034
  16. [16] S. L. Woronowicz, Compact Quantum Groups, preprint University of Warsaw (1992). 

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